Inhomogeneous Body and Surface Plane Waves
in a Generalized Viscoelastic Half—Space

By
Roger Duane Borcherdt
A (University of Colorado) 1963

.5.
M.A. (University of Wisconsin) 1965
M.S. (University of California) 1970

DISSERTATION

Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY

in
Engineering

in the
GRADUATE DIVISION
of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Approved:

    

Committee in Charge

DEGREE CONFERRED DEC.18,197

3081
1971

INHOMOGENLOUS BODY AND SURFACE PLANE

WAVLS IN A GLNERALIZLD VISCOELASTIC HALF—SPACL

Ph.D. Roger D. BorcherdT Engineering Geoscience

Chairman, Thesis Commééfee

ABSTRACT

The basic maThemaTical framework for problems of plane
monochromaTic wave propagaTion in a generalized isoTropic
linear viscoelasTic maTeriaI is presenTed. AnalyTic soluTions
for The reflecTion of a general (homogeneous or inhomogeneous)
plane SV wave and a Rayleigh—Type surface wave in a generalized
half-Space show many of The TheoreTical resulTs for anelasTic
media cannoT be predicTed from The corresponding resulTs for
elasTic media. The generalized resulTs are consisTenT wiTh
The resuITs of The low-loss approximaTion.

For elasTic media The only Type of inhomogeneous wave
which can propagaTe is one for which planes of consTanT phase
are perpendicular To planes of consTanT ampliTude. For
anelasTic media This is The only Type of inhomogeneous wave
which can 29: propagaTe.

An incidenT general SV wave is reflecTed from The free
surface of a viscoelasTic half-space as a general SV wave and

a general dilaTaTional wave. For The reflecTed waves, an

Z}?T"WT

exTended version of Snell's law is valid and The componenT of
aTTenuaTion along The surface is equal To ThaT of The incidenT
wave. For common anelasTic maTerials, an incidenT homogeneous
SV wave resulTs in an inhomogeneous dilaTaTional wave for all
angles of incidence. An incidenT homogeneous SV wave is ToTaIly
converTed To a dilaTaTional wave aT some incidenT angle only for solids
such ThaT Q;'= Q;'= le. Every angle of incidence can be considered
as a criTical angle for reflecTion of an inhomogeneous SV wave. The
ampliTude of The dilaTaTional wave reflecTed from a normally incidenT
inhomogeneous SV wave is 29:_zero.

For a Rayleigh—Type surface wave on an anelasTic half-space,
The propagaTion vecTors associaTed wiTh The displacemenT poTenTials
are inclined aT unequal angles To The surface, and The aTTenuaTion
vecTors are inclined aT unequal angles To The verTical. The
normalized parTicle—moTion ampliTude disTribuTions for anelasTic
solids exhibiT a superimposed sinusoidal dependence on depTh, which
is £9:_exhibiTed for elasTic solids. A parTicle describes an
ellipTical orbiT wiTh Time. The ellipse is generally reTrograde aT
The surface and changes To prograde wiThin a depTh of 0.2A . For
anelasTic solids The TilT usually rapidly increases away from The
surface To a maximum aT abouT O.|A . The velociTy and absorpTion
coefficienT can be compuTed from The low—loss assumpTion wiTh an
error less Than one percenT for solids wiTh 04$ O.|. The numerical
characTerisTics of a Rayleigh—Type surface wave are presenTed for
low-loss solids (Q-h£0.l) and moderaTe-loss solids (Q'lvO.l),

independenT of a parTicular viscoelasTic model.

CONTENTS

ABSTRACT

INTRODUCTION

I.|

References

CHARACTERIZATION OF LINEAR MATERIAL BEHAVIOR

2.|

2.2

2.3

2.4

2.5

General Consfifufive Relafion for a Linear Viscoelasfic (LV)
Maferial

Consfifufive Reiafion for an lsofropic Linear Viscoeiasfic
(ILV) Maferial

Addifional Paramefers for Describing The Sfeady—Sfafe
Response of an lLV Maferiai

One-Dimensional Models of Linear Viscoelasficify

References

PLANE MONOCHROMATIC WAVES IN AN INFINITE HOMOGENEOUS ISOTROPIC

LINEAR VISCOELASTIC (HILV) MEDIUM

3.I

3.2

3.3

3.4

3.5

Equafion of Mofion

Homogeneous Rofafionai and Diiafafional Waves
Inhomogeneous Waves

Two-Dimensional Solufions for Displacemenf Pofenfiais

References

ENERGY IN A STEADY-STATE VISCOELASTIC RADIATION FIELD

4.|

4.2

4.3

4.4

4.5

Energy Loss in Bulk and Qk“|

Energy Loss in Dilafafion and Qp-I

Energy Loss in Shear and QS-I
Reiafion of Q—I fo Logarifhmic Decremenf and Skin Depfh

Reference

RESTRICTIONS ON PLANE WAVE SOLUTIONS FOR A HILV HALF-SPACE
5.| Resfricfions Imposed by Boundary Condifions
5.2 Resfricfions Imposed for Reflecfion Problem
5.3 ResTricTions Imposed for Surface Wave Problem
REFLECTION OF A PLANE LNHOMOGENEOUS SV WAVE IN A HILV HALF-SPACE
6.I Specificafion of Incidenf General SV Wave
6.2 Propagafion and Affenuafion Vecfors for Reflecfed Waves
6.2.I Reflecfed SV Wave
6.2.2 Reflecfed P Wave
6.3 Amplifudes and Phases of Reflecfed Waves
6.4 References
THEORETICAL CHARACTERISTICS OF A RAYLEIGH—TYPE SURFACE WAVE ON
A HILV HALF—SPACE
7.I Formulafion of Surface Wave Problem
7.2 Comparison of Elasfic and Anelasfic Solufions
7.3 Analyfic Solufion of Complex Rayleigh Equafion
7.4 Properfies of a Rayleigh-Type Surface Wave for )\ = II
7.5 Properfies of a Rayleigh-Type Surface Wave for an Arbifrary
Viscoelasfic Media
7.6 References
GENERAL NUMERICAL CHARACTERISTICS OF A RAYLEIGH~TYPE SURFACE
WAVE ON A HILV HALF-SPACE
8.l Paramefer Analysis
8.2 Velocify and Absorpfion Coefficienf
8.2.I Hypofhefical Maferials
8.2.2 Low-Loss Maferials

8.2.3 Moderafe—Loss Maferials

III

8.3 ParTicle Mofion Characferisfics for Low—Loss and Moderafe-
Loss Maferials
8.3.I Parficle Mofion CharacTerisTics aT Ihe Surface
8.3.2 Parficle MoIion Characferisfics vs. DepTh

8.4 ParTIcle Mofion Characferisfics for a few Represenfafive
Maferials

8.5 References

CONCLUSIONS

ACKNOWLEDGMENTS

APPENDICIES

COMPUTER PROGRAM LISTINGS

NOTATION SUMMARY

l . Ill Il?()l)ll(ll'l ()il

The phenomenological Theory of linear viscoelasTiciTy daTes To
The nineTeenTh cenTury wiTh The applicaTion of The superposiTion
principle by BoIszann. In The IasT ThirTy years considerable
simplificaTion of The maThemaTical aspecTs of The Theory have
resulTed from The inTroducTion of inTegral Transform Techniques.
RecenTIy, several definiTive accounTs of The maThemaTicaI Theory of
viscoelasTiciTy have been published (e.g. Gross (I953), Bland (I960),
Fung (I965), FIUgge (I967), and a more rigorous formalism by GurTin
and STernberg (I962)). Gross (I953) suggesTs: "The Theory ... is
approaching compleTion. FurTher progress is likely To be made in
applicaTions raTher Than on fundamenTal principles.” HunTer (I960)
indicaTes ThaT The applicaTion of The general Theory of linear
viscoelasTiciTy To oTher Than one dimensional wave phenomena is
incompIeTe.

The purpose of This disserTaTion is To consider The applicaTion
of The general Theory of linear viscoelasTiciTy To The problem of
Two dimensional plane wave propagaTion in an isoTrOpic half—space.
BoTh The problem of reflecTion of plane waves and The problem of
a Rayleigh—Type surface wave are considered. The resuITs of boTh
of These problems for The special case of elasTic media are well
known (e.g. Ewing, JandeTzky, and Press (l957)). lT will be shown
ThaT many of The TheoreTical resulTs for general viscoelasTic media
are noT predicTabIe from The corresponding resuITs for elasTic
media. The basic reason is The difference in The naTure of

inhomogeneous waves which may propagaTe in The Two Types of solids.

The problem of an incidenT homogeneous SH wave was considered
by O'Neill (I949). The problems of incidenT homogeneous P and SV
waves have been considered by LockeTT (I962), Cooper and Reiss
(I966) and Cooper (I967). Their resulTs show ThaT in general an
incidenT homogeneous wave gives rise To inhomogeneous TransmiTTed
and reflecTed waves. Hence, To consider wave propagaTion in muITi-
layered viscoelasTic media one musT consider The problem of incidenT
inhomogeneous waves. The preceding auThors do noT consider This
case. The reflecTion of a general incidenT SV wave (i.e. eiTher
homogeneous or inhomogeneous) is considered in chapTer 6.

The problem of a surface wave,analogous To The surface wave
derived by Rayleigh for an elasTic half-space, has been considered
by Caloi (I948) for a resTricTed Type of viscoelasTic half-space.

He assumed The medium was a VoigT solid wiTh no dissipaTion in bulk
and ThaT The real parTs of The complex Lame'consTanTs were equal.
Press and Healy (I957) considered low—loss maTeriaIs. For such
maTerials They posTulaTed a relaTionship beTween The velociTy of a
Rayleigh—Type surface wave and The fracTionaI energy loss per cycle
( x 05' ) Then derived a relaTionship beTween Qg’for The posTu—
IaTed surface wave and og’and (fif'for homogeneous dilaTaTional and
shear waves, respecTiver. The TheoreTical resulTs of The low-loss
approximaTion are limiTed and inadequaTe for maTerials wiTh moderaTe
amounTs of absorpTion (e.g. soils). The soluTion for a Rayleigh—
Type surface wave on a general viscoelasTic half—space is derived in
ChapTer 7. No low—loss assumpTion is necessary for The soluTion.
The soluTion reduces To ThaT derived by Rayleigh for The special

case of elasTic media. The soluTion also reduces To ThaT derived

by Caloi for The special Type of VoigT solid he considered.

The disserTaTion is self-conTained in ThaT The basic
maThemaTical framework for characTerizaTion of a general linear
viscoelasTic maTerial is presenTed in chapTer 2. The derivaTion
of The soluTions for plane homogeneous and inhomogeneous waves
is presenTed in chapTer 3. A general expression for The ToTal
energy in a sTeady sTaTe viscoelasTic radiaTion field is derived
in chapTer 4. The resTricTions imposed on The general plane wave
soluTions for propagaTion in a half—space are discussed in
chapTer 5. The numerical characTerisTics of a Rayleigh—Type surface
wave are presenTed in chapTer 8.

The resulTs of This disserTaTion are applicable in The broad
field of solid mechanics. Examples of more specialized fields of
applicaTion are:

i) earThquake engineering, for predicTing The dynamic

response of soils and man made sTrucTures,

ii) seismology, for inTerpreTaTion of The inTernal sTrucTure

of The earTh,
iii) elecTral engineering, for analyses of delay lines,

iv) civil and sTrucTural engineering, for response analyses

of maTerials used in sTrucTures, and

v) applied mechanics and physics, for analysis of maTerial

response.

|.l References

Bland, D. R., The Theory of Linear ViscoelasTiciTy, Permagon Press,

 

I960.

Caioi, P., ComporTemenT des ondes de Rayleigh dans un mileu Tirmo-

,

élasTique indéfini, Publs. Bur. CenTral Séism. lnTern., l7

 

pp. 89-!08, I948.

Cooper, H. F., Jr., ReflecTion and Transmission of oblique plane
waves aT a plane inTerface beTween ViscoelasTic media,
J. AcousT. Soc. Am., fig, pp. lO64-69, I967.

Ewing, W. M., W. S. JardeTzky, and F. Press, ElasTic Waves in
Layered Media, McGraw—Hill, 1957.

FlUgge, W., ViscoelasTlciTy, Blaisdell, l967.

Fung, Y. C., FoundaTions of Solid Mechanics, PrenTice Hall, Inc.

,

 

I965.

Cross, 8., MaThemaTical STrucTure of The Theories of ViscoelasTiciTy,

 

Hermann eT Cie, Paris, l953.
GurTin, M. E. and E. STernberg, On The linear Theory of Viscoelas—

TiciTy, Archive of RaT. Mech. Analaysis, II, pp. 29l-356, I962.

 

HunTer, S. C., ViscoelasTic waves, Prog. in Solid Mech., I, pp. l—57,

 

I960.

LockeTT, F. J., The reflecTion and refracTion of waves aT an lnTer-

 

face beTween ViscoelasTic media, J. Mech. Phys. Solids, lg,
pp. 53-64, I962.

O'Neill, H. T., ReflecTion and refracTion of plane shear waves in
ViscoelasTic media, Phys. Rev., 12, pp. 928—35, I949.

Press, F. and J. H. Healy, AbsorpTion of Rayleigh waves in low-loss

media, J. Appl. Phys., E, pp. I323—25, l957.

2. CHARACTERIZATION OF LINEAR MATERIAL BEHAVIOR

The response of a maTerial To an applied load may be classified
as eiTher linear or non-linear. The linear response of a maTerial
may be characTerized maThemaTically by specifying a general sTress—
sTrain relaTionship based on The principal of superposiTion. In
general, such a relaTionship depends on Time, which is To say The
deformaTion of a corresponding maTerial depends upon iTs enTire
pasT hisTory of loading. Such a maTerial response is referred To
as viscoelasTic. Two special Types of viscoelasTic response are
elasTic and anelasTic. For elasTic response The sTress-sTrain
relaTion is independenT of Time. For anelasTic response The sTress-

sTrain relaTion is Time dependenT.

2.l General ConsTiTuTive ReIaTion for a Linear ViscoelasTic (LV)

MaTerial

A linear viscoelasTic maTerial is defined maThemaTically To be
one for which a causal TorTh order Tensorial TuncTion OJRI exisTs,
such ThaT The consTiTuTive relaTion for The maTerial may be expressed

as

f
pJ.J(r) :iwri‘j“(T-T) deul’t’), (I)

where,

p, (r) E componenTs of The 2nd order
'J Time dependenT sTress Tensor,
e -(r) E componenTs of The 2nd order

'J Time dependenT sTrain Tensor,

and fija, (T) is called The relaxaTion funcTion, characTerisTic
of The maTerial. if The Time derivaTive of The sTrain hisTory is
conTinuous, The Riemann-STielTJes inTegral in equaTion (l) reduces
To The common Riemann inTegral (Theorem 9-8, p. I97, AposTol (i960))

ThaT is,

,

pub” :j“r|.Jkl(ro1’)?—e_i—J-&'(T)d‘c. (2)
B'C

(A rigorous developmenT of linear viscoelasTiciTy using The Riemann-

STieTlJes inTegral is given by GurTin and STernberg (p. 29!, I962).)

The physical principal of causaliTy requires ThaT The relaxaTion

TuncTion is zero for negaTive Time, hence The consTiTuTive relaTion

may be wriTTen in The form;

N

gJLr): I“ anlu-«T) d8m(fl, (3;
or more compachy wiTh The definiTion of a Riemann-STielTjes convolu-
Tion operaTor as
psi = "in: 1: deal ‘ (4)
ProperTies of The Riemann-STielTJes convoluTion operaTor are lisTed
in appendix l.
AnoTher funcTion ofTen used To characTerize The behavior of a

LV maTeriai is The so-called Tensorial creep TuncTion Cija‘Lr),

which gives rise To The following sTress—sTrain relaTion

eiJ = CUM * d?

*I ' (5)

2.2 ConsTiTuTive RelaTion for an lsoTropic Linear ViscoelasTic

(ILV) MaTerial

An isoTropic linear viscoelasTic (ILV) maTerial is defined To
be one for which The Tensorial relaxaTion funcTion riJ h,kfl
is invarianT wiTh respecT To The roTaTion of CarTesian coordinaTes.
For such a maTeriallscalar TuncTions g and r1 exisTlsuch ThaT
=(r'

PM“ K' Q) 85“, 8' /5 T "6(52'11‘31 1» gnaw/a. (6)

For ILV maTerials, The consTiTuTive relaTion simplifies To

/ I . .
1D”: risaedeid' (li‘d)

(7)
fin : '1 * dean:

where The prime denoTes The deviaToric componenTs of sTress and

sTrain defined by

l
p” E “"ésu'fia

(8)

and

em" 5 an -453”. 9“

The funcTions Q and r; are The relaxaTion funcTions for
describing shear and bulk behavior, respecTively of ILV maTerials.
Similarly, creep funcTions cs and CK exisT for ILV maTerials

such ThaT

355 = C; *‘ dpgj (i#.ii
(9)
8“ C“ at an“ .

Since There is nofpriori relaTion beTween The behavior of an
lLV maTerial in shear wiTh ThaT in bulk, equaTions (7) and (9) show
ThaT The shear and bulk behavior may be specified independenle.
Hence, wiThouT loss of generaliTy, furTher discussion of The consTi-
TuTive relaTion will be resTricTed To one dimension wiTh omission of

subscrist, ThaT is,

11
u

rude
(IO)

Oi“

e =. c {- dp .

For The one dimensional case, The sTress may be considered as
acTing over a uniT area and hence be considered as force.

Similarly, The sTrainimay be considered as acTing over a uniT

lengTh and hence be considered as displacemenT.

Physically, The relaxaTion funcTion is defined To be The
sTress response To a uniT sTrain applied aT some Time T, say 7 = 0.
To show This physical definiTion is consisTenT wiTh The original

defianion in Terms of The Riemann-STielTJes convoiuTion, replace

e by The Heaviside funcTion,

H(Tl a
l Fan 1'20

in equaTion l0. Using properTy 5 lisTed in appendix l, The expres—

sion may be wriTTen as

?
?dl= mow NT) +.L+rw-1)bln$idf,
3%'

hence,

plT) : P(T) , (II)

The physical definiTion of The creep funcTion is ThaT iT is
The sTrain response To a uniT sTress applied aT + = 0. Similarly,
This definiTion may be shown To be consisTenT wiTh The original

definiTlon of The creep funcTion.

2.3 AddiTional ParameTers for Describing The STeady STaTe Response

of an lLV MaTerial

The form of The consTiTuTive relaTion simplifies for sTeady
sTaTe problems. To consider sTeady sTaTe behavior, assume
sufficienT Time has elapsed for The effecT of iniTial condiTions
To be negligible, and IeT 19 = Peiwfand e = E eiur where P and E
are in general complex consTanTs independenT of Time. HenceforTh,
whenever The complex noTaTion is being used, The acTual quanTiTies

of inTeresT will be represenTed by The real parT of The

corresponding complex expression. The resulTing consTiTuTive

relaTion may be wriTTen as

P = in) PLiiu) E
(l2)

Oi"

FT
ll

iw Cciw)P,

where Rliw) and C(icU) are The Fourier Transforms of rir) and
CLT) , respecTively.

The sTeady sTaTe response is ofTen characTerized by eiTher The
complex modulus Yiiuvor The complex compliance JXiIU) . They
are defined To be P/E and E/P, respecTively. The following rela-

Tions exisT beTween The various parameTers '

Q

Y(iw)= {w R(iw)

ll

JLiw) iw C(iw)

(l3)
YH'w) = (In w)“

Riitu) C(iuJ) = (i u1)'2.

AnoTher parameTer oTTen used To describe viscoelasTic
behavior is Tan 8, where g is The phase angle by which The sTrain

lags The sTress. Formally, iT is defined as

ran 5 5 ______I'"EY”“”3 , (l4)
ReEY(in

During each cycle of forced oscillaTion of a linear anelasTic
sysTem, energy is losT as well as Twice alTernaTely sTored and
reTurned. To accounT for The energy in such a sysTem, as charac—
Terized by a general sTress sTrain relaTion of The form (l2),
consider The sysTem under an applied sinusoidal sTress. Using The
complex noTaTion for sTeady sTaTe as previously defined, The
complex sTrain is given by

8=Ji" (IS)

The Time raTe of change of energy in The sysTem is given by

1: é , (l6)

where “.” denoTes differenTiaTion wiTh respecT To Time and The
subscripT “R " is used To denoTe The real parT of The expression
subscripTed. Solving equaTion (l5) for -p and Taking real parTs
shows

2
p=lJe+Je)/ia'l
R RR II ’ (l7)

where The subscripT "1” denoTes The imaginary parT. For sTeady
sTaTe

eI=—éR/w . (l8)
hence, The desired expression for The Time raTe of change of

energy in The sysTem is given by

° — B l J a l J'1' '2
P e ‘ -—(-———’i an ) ‘ -— - en '
R R M 21le w 1le (l9)

The firsT Term in equaTion (I9) represenTs The Time raTe of change

of poTenTial energy in The sysTem, ThaT is The raTe aT which energy
is alTernaTely sTored and reTurned. The second Term represenTs The
raTe aT which energy is dissipaTed. The second law of Thermo—
dynamics requires ThaT The ToTaI amounT of energy dissipaTed increase

wiTh Time, hence The second Term in equaTion (I9) shows

Jr s O .
(20)

A dimensionless parameTer, which is useful for describing The
amounT of energy dissipaTed, is The raTio of The energy dissipaTed
per cycle of forced oscillaTion ( ATE /cycLe) To The peak energy
sTored during The cycle ( EP ). lnTegraTing equaTion (l9) over
one cycle shows The energy dissipaTed per cycle is
AE /cycle = ~TT IPla Jr . The firsT Term in equaTion (I9)
shows ThaT The peak energy sTored during The cycle is

E? : lPl2 JR /2 , where ano. Hence,

AE/Cycle/EP = 2iT(‘J;)/J;?. (2|)

AnoTher dimensionless parameTer, which is a useful measure of

l3

. . . - . 'l ._
anelasTiCITy, IS 0’ defined by C) : ZLTT( AE/CYClE/Ep)= {EA};-
ExaminaTion of equaTion (l4) shows (Q'Ialso represenTs The TangenT
of The angle by which The sTrain lags The sTress, ThaT is

—' =
0 Tan 8. (22)

For The special case in which The sysTem is perfechy elasTic (T

11:0)

and equaTion (I9) shows There is no energy dissipaTed,in which case
Q—'= fa.n 8 == 0.

Expression (2|) for AE/Cycle/EP was derived for a one-
dimensional sysTem being submiTTed To forced oscillaTions. The
concepT of fracTional energy loss per cycle may also be exTended
To cerTain sysTems which have been submiTTed To some iniTial dis-
Turbance and have a free oscillaTion response. However, in
generaL The expression for The fracTional energy loss per cycle of
j£§g_oscillaTion is differenT from The expression for The fracTion—
al energy loss per cycle of forced oscillaTion, and iT is necessary
To disTinguish beTween The Two if The concepT of fracTional energy
loss per cycle is being used To define (3"_ The Two concest will
yield The same expression if The frequency of forced oscillaTion is
The naTural frequency of The sysTem and The amounT of absorpTion is

small.
2.4 One Dimensional Models of Linear ViscoelasTiciTy

ViscoelasTic behavior of solids under uni-axial sTress can be

represenTed by models consisTing of discreTe elasTic and viscous

elemenTs. An infiniTe number of models can be consTrucTed from
various configuraTions of The elemenTs in series or parallel. The
consTiTuTive relaTion associaTed wiTh each such model can be
wriTTen in The form ~p = r‘ at de.

The ideal elasTic elemenT is represenTed by a spring, whose
exTension is proporTional To The applied force. The ideal viscous
elemenT is represenTed by a dashpoT, whose raTe of exTension is
proporTional To The applied force. For elemenTs placed in series,
The sTrain response is The sum of The individual sTrain responses
and The sTress is The same in each elemenT. For elemenTs placed
in parallel, The sTress response is The sum of The individual
sTress responses, and The sTrain is The same in each of The ele-
menTs. The represenTaTion of viscoelasTic behavior by models
having more Than Two elemenTs is noT unique, so differenT confi—
guraTions of elemenTs may give rise To The same response.

The differenTial equaTion, sTrain response, sTress response,
complex modulus, complex compliance, and (9"are shown for some
of The simplesT combinaTions of springs and dashpoTs in Table l.
The corresponding derivaTions may be found in any elemenTary
Tebeook on linear viscoelasTiciTy ( 8.9. Bland (I967);

FlUgge, (l962); Fung, (l965)).

Bland (I960, p. 48) shows ThaT The mechanical properTies and
The sTored and dissipaTed energies of any linear viscoelasTic
maTerial can be represenTed eiTher by a seT 0T VoigT elemenTs in

series or a seT of Maxwell elemenTs in parallel.

TABLE I
LIneor Viscoelostlc Models

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I ' j" “m I l I
I Creep Function I Relaxation Funchon I Complex Compliance i Complex Modulus ‘ (3‘I
Name Modells) l vaferenhol Equohon ‘ p(t) if” m _. alt) l 9(1) ,I— m — pm ‘ Jliw)5 la: C(iul=Y6wI—I ' Y(‘w) 5 I“ Rllul= Jud-l 04:11; = IL
I | . Jn Ya
W__WW_,_ W _ ,WWWIL W. WWW W .W . 4
I P I _ I
5.05m: o—W—o I P = p e 7'; HI!) I‘ H“) I M ' I It 0
i I I
I I I
l ‘I I I
1; . . _ I .
Viscous o—«}—-—4 owe I ~EI-Hm I 178(1) I (IMI' I Im, I on
I I I
i I ‘ l
l I I
1 3 l l |
I I P {a p l l t l I I I . | | . I —I It
I Mame“ I WWWJWQ “7+? I (7+3)Hm I ,Ie' Hm I 7“"«7; (T‘W—v) fi
1 I I I l
I 1 i I I
f — VVVVV A p , 7 ~ — A“ 7 k if _ _ _ ’I’
l ‘ _:I I l l
i Voigl ——o D=#e+né ill-e " mm 3 namumn l (,niwq)" i pm, I “W”.
l 3,”... l I I I I "
E . ______ I W , _ ,l , W WWJI W. WW W _WW I .WW ,WW
I ”z ”I‘l‘z _ ,2 l o, J ' '
fl —,,‘2— ' .._1 I " l l , _ _ _| I -x r HI» " t
‘ I l 7) ' IL‘ + [ill-e '2 ))H(() UL}; (it. 9: *leHlt) 1‘ 7.1‘+ (#24‘W'Iz) ‘ (L. +(F2H-W'I2) ') E I m (#2 1,2) l
1 Slondard I z i‘Ie +I‘Ii‘z/‘W29 ‘ l ' Z 1 I 'I fi‘Re [U‘z*i'"”2)4]
' Linear -—~.—l‘_— ——l A— ___ I h » 7 WW ”7 VIL w » _ . . A, - if A l
'I , 4r *‘ I" I l ‘
', SolId ' 9+??? I ”2‘ I I I I I I I rm[(p'vZ-n:.'7)"]
'1 l I _ ~— I (flop: 9": )H(() I (y'+(-J-r-i—.—)- )- AL+(—r-l—-:)- . , _'
- "2 "2 (Imps + mug/nae 2 ' z I ' "2 “’2 "Z ”"2 W F“ (It'dn—a’l
I ‘ _W , , W W J I , , ,, W: WWW_.___..W
I . #3 "n I ' I 5‘: . -l
. . — - Im ( +m I ]
l - I1» 7] .fl. ‘. n _ n _ -1 .l u") x- [PK W
IGeneraIIzed I I 2 I , .K I_l___l.__ +2 + -I L _l_ 2 3 .
I o-vwv—c — —— -— + — - — (# l""I) +201 “0”) ) ) :
I V°'9' W 3.77;, $0 (“I ’12 3:3" (I e ))H(t) I I," "”12 l" x K F' wk “'3 K K I idlg Re[(#,(+i.wm()"] I
I l m I
I I — ~— A , ——————— I ~ ~ « 7 -~ ~ ~——»«————;
‘I ' ' ‘ ll ‘ W i _| I‘
‘. . . I -— n l ' w I
‘- Gene ol' ed : 5:”! P" I " 'x I " l_ _ _l_ WI. - .‘_I I _;!_"l$.
I M r I'zI “'3 LJ % __ _ Ifllnunnzsnnfihe Hm)‘ ,‘I+IW,Z+E’(H M2) ILI+qu+§3lH MK I #62 #4 I
I oxwe ‘ ’72 LrJ’ls ’7“ ‘ l ‘ | (a K I
IWW ,W ,I___W_WW$WWW,WLWWW L WW W , W W WW W WWLWW , WW, WWWW W , WWW , , l ., _ _W _W

 

a» Standard linear model is o specnol case of generalized Vang! model with n=3 and 71;: O.

 

S I

2.5 References

Aposlol, T. M., Malhemalical Analysis, Addison—Wesley, I960.

 

Bland, D. R., The Theory of Linear Viscoelaslicily, Permagon Press,

 

I960.

FlUgge, w., Viscoelaslicilx, Blaisdell, |967.

Fung, Y. C., FoundaTions of Solid Mechanics, Prenlice—Hall, |nc.,

 

l965.

Gross, B., Malhemalical STruclure of The Theories of Visco-

 

elaslicily, Hermann el Cie, Paris, l953.

GurTin, M. E. and E. Sfernberg, On The linear lheory of visco—

elasTiciTy, Archive of Raf. Mech. Analysis, ll, pp. 291—356,

 

I962.

Kolsky, H., Slress Waves in Solids, Clarendon Press, l958.

 

3. PLANE MONOCHROMATIC WAVES IN AN INFINITE HILV MEDIUM

3.| EquaTion of MoTion

The moTion of a maTerial conTinuum wiTh a specified seT of
boundary condiTions is governed by The conservaTion laws of mass
and momenTum TogeTher wiTh a consTiTuTive relaTion characTerisTic
of The maTerial. Using The Eulerian descripTion of The conTinuum,
The law of conservaTion of linear momenTum implies The following
equaTion of moTion;

Tfijsd

where
F; 2 body force per uniT volume,
f E mass densiTy,
and
u; E i’h componenT of The displacemenT

vecTor.

To linearize The relaTion beTween The componenTs of The
sTress Tensor and The componenTs of The displacemenT iT is
necessary To assume The displacemenTs are infiniTesimal. This

assumpTion yields The following linear relaTion for The sTrain

Tensor;

U __’2_(u;“j + LAJ,’-). <2)

FurTher simplificaTion of The maThemaTicaI model for The
conTinuum resulTs from assuming iT is "homogeneous." For such a
maTerial The relaxaTion funcTions are independenT of The spaTial
coordinaTes.

The equaTion of moTion (I) TogeTher wiTh The linearized
sTrain-displacemenT relaTion (2), and The consTiTuTive relaTion
(2.7) are sufficienT To describe The infiniTesimal moTion of an
infiniTe homogeneous isoTropic linear viscoelasTic (HILV) medium.

To find sTeady sTaTe soluTions of These equaTions, assume
sufficienT Time has elapsed for The effecT of iniTial condiTions

To be negligible and define

z'nuf
Id : EiJ 8 )
and I3)
iw‘l'
pld : faJ e ’
where [Ii, EN’ and Fad are in general complex TuncTions

independenT of Time.

CL

WiTh These definiTions The equaTions governing The sTeady

sTaTe moTion of an HILV medium in The absence of body forces are

P.,J,J+Pw"U;=’-O, (4)

E,.== %(u.. + U. .),

lJ VJ J}! (5)

and

I I

p,.=insE,J (i#J)

‘U
H

’ 6
H leKEH’ ()

where Rs and R denoTe The Fourier Transforms of The relaxa-

K

Tion funcTions characTerizing The shear and bulk behavior of The
medium, respecTiveiy.

The consTiTuTive reIaTion (6) may be wriTTen in oTher forms
familiar from The consTiTuTive reiaTion for an eiasTic maTerial by

defining
A 2—: iwLRK—sts, <7)
11 E {WRs/a’ (8)
and

K _=_ inK/a. <9)

20

Olher forms of lhe conslilulive relalion are

RJ :; 2 11 E3 (i ¢ J)
(IO)
Fifi z: 3 l< Ehl ,
p.. : 8.. A_E +. a 11 E.. ('I)
IJ u Ah u ’
and
Pi.) = 3iJ(K-%H)E&&+ZHEU. (12)

The equaTion of moTion (4) may be wrillen enlirely in Terms of
if; by subsiiluling equalion (12) info equalion (4) and simplifying

wilh equalion (5). The resulTing equalion in veclor nolalion is

lK+%H)V8 — Minvxiil + mu =0, (13>

where

The sleady slale equalion of molion (I3) for a HiLV solid
differs from Thai for a perfeclly elasiic solid only in Thai complex
modulii,which are in general funclions of frequency,have replaced
real modulii independenl of frequency. This faci is oflen used To

find solulions for viscoelasfic problems from knowing The solulion

2l

To The corresponding elasTic problem. Bland (l965, p. 67) sTaTes
This "correspondence principle" rigorously as; "If The elasTic
soluTion for any dependenT variable in a parTicular problem is of

The form F = RaEFE em“? and if The elasTic modulii in F

E are
replaced by The corresponding complex modulii To give F§E, Then

The viscoelasTic soluTion for ThaT variable in The corresponding

problem is given by F = Re[.F iwa.”

VEE However, in applying The

"correspondence principle” To solve a viscoelasTic problem, one musT
check ThaT The operaTions used To obTain The elasTic soluTion remain
valid for complex modulii. The resulTs of an elasTic problem are
a special case of Those for The corresponding viscoelasTic problem.
The physical inTerpreTaTion of The viscoelasTic resulTs are oTTen
broader in scope Than Those Tor The elasTic case. For This reason
iT is insTrucTive To develop each sTage of a problem independenT of
The correspondence principle. The correspondence principle may
Then be applied as a useful maThemaTical check on Tne derived re—

laTions.

3.2 Homogeneous DilaTaTional and RoTaTional Waves

Taking The divergence and curl of equaTion (l3) yields,
respecTively, The equaTions for waves of dilaTaTion and roTaTion.

NeglecTing body forces The equaTions are,

9 =0 (l4)

and

22

2. A Z a
Vivxuz + 9g(VXE) =0,
2 2

5‘ (l5)

where The complex velociTies a and ,8 are defined by

/
o< = p.v. < K +%/U')a (I6)
P
and
,8 E 1/?—

p.v. l {él ) .

y (l7)
( p.v.( 1;]: denoTes The principal value of The square rooT of The
complex number 2 and is defined as VE’ Exp {5 arg Z/Zi],

where ~fl‘< avg z 5 7? and f—— denoTes The posiTive square

rooT), Plane wave soluTions of equaTions (l4) and (IS) are given by

@=Aexp[:ti°<y IJ-XJ-J’ Llj=1) (I8)
and
.—I ._I h l
V: = 8 EXP[: 1%) nd XJJ ) (”J = 1) . ([9)

where sTeady sTaTe soluTlons of The equaTions governing The moTion

of The solid are

23

_ iwr
ReLoJ-Retae J, (20)
and
RevafiJ= RaLVXfi e‘mJ.
Z 2 (2|)

The planes of consTanT phase associaTed wiTh The soluTions (20)
and (2|) are parallel To The corresponding planes of consTanT ampli-
Tude. Waves for which The Two Types of planes are parallel are called
homogeneous waves. However, inhomogeneous waves, ThaT is waves for
which The planes of consTanT phase are 29: parallel To The planes
of consTanT ampliTude will also saTisTy equaTions (l4) and (IS).

Their properTies will be examined in The nexT secTion.

SoluTions (20) and (2|) sTaTe ThaT plane homogeneous waves of
dilaTaTion and roTaTion may propagaTe independenle in a HlLV solid
in The absence of body forces. The respecTive velociTies of The

homogeneous waves are

'l
(Retggll <22)

lil

VP
and

I “i
[RELEJ) . (27)

_)

m<
Ill

The respecTive absorpTion coefficienTs of The homogeneous waves are

III

“P “‘“Imli‘J (24)

and

_ L

Expressions (22) and (23) show ThaT boTh waves are dispersive in a
HlLV solid. Expressions (24)and (25) show ThaT The aTTenuaTions of
The waves in general are dependenT on frequency.

For IaTer reference, iT is convenienT aT This poinT To define
The complex waves numbers corresponding To homogeneous dilaTaTional
and roTaTional waves. Their relaTionship To The acTual velociTies
and aTTenuaTion of The corresponding wave is also given.

The complex wave number for a homogeneous dilaTaTional wave is

defined as

a? E LU/d ) (26)

and The one for a homogeneous roTaTional wave is defined as

as E “’45- (27)

From These definiTions iT is clear ThaT

24

25

a 1 9e +i§ : cud -iu)o< - u) —La
P P 9 J J. ‘ — P’ (25)
R t Id“ lq‘l VP
and
fiszasa+iasz= wfiz—iwfll= 54%. <29)
IBI lfll 5

ln Terms of The parameTers of The solid, equaTions (l6) and (I7)

imply
&;= m‘i KR+§LMR - ilK1+%MI))/IK+§.Mlz,(BO)
and
191:= szl/UR-illz)/l1llz. (3|)
lT will be shown in chapTer 4 ThaT The second law of Thermodynamics

implies KR ,JIR ’KI and ‘11} are nonnegaTive quanTiTiTes. Hence,
equaTions (30) and (3|) TogeTher wiTh definiTions (26), (27), (I6),
and (17) imply The real and imaginary parTs of The complex wave

numbers musT saTisTy The following inequaliTies;

2 2 Z 2
5p. ‘ Tip, 2 ° 1 Jls“, " £5; 3 O J (32)
%p ”>- 0 ) ’93 Z O J (33)

26

and

a, , z (34)

3.4 Inhomogeneous Plane Waves

An inhomogeneous plane wave is one for which planes of consTanT
phase are noT parallel To planes of consTanT ampliTude. In general,
an inhomogeneous wave propagaTes aT a velociTy less Than ThaT of The
corresponding homogeneous wave.

The use of inhomogeneous waves occurs frequenTiy in problems
concerned wiTh wave propagaTion in dissipaTive maTerials. For
example, The properTies of such waves are well known in The liTera-
Ture concerned wiTh elecTromagneTic wave propagaTion in conducTors
(e.g. STraTTon, l94l, p. 5l6-520). Presenle iT will be shown ThaT
The only inhomogeneous plane waves which can exisT in non-dissipaTive
maTerials are Those for which planes of consTanT phase are
perpendicular To planes of consTanT ampliTude. However, for dissi-
paTive maTerials These are The only kind of inhomogeneous plane
waves which can n91 exlsT.

An example of an inhomogeneous plane wave associaTed wiTh
elasTic maTerials is The dilaTaTional disTurbance generaTed aT a
boundary by a SV wave whose angle of incidence is greaTer Than
criTical. AnoTher example for elasTic media is a Rayleigh wave
on an elasTic half—space. For boTh examples The planes of
consTanT phase are perpendicular To planes of consTanT ampliTude.

Examples of inhomogeneous plane waves in viscoelasTic

27

maTerials will be considered in chapTers 6 and 7. As a preliminary
remark, iT will be shown ThaT The resulTs of several problems for
anelasTic maTerials are subsTanTially differenT from Those for
elasTic maTerials. A good example is The form of The P wave
reflecTed from a boundary exciTed by an incidenT homogeneous SV
wave, (secTion 6.2.2). For common anelasTic media, The reflecTed
P wave is in general iflhomogeneous for all angles of incidence, in
conTrasT To The siTuaTion Tor elasTic media.

To discuss The properTies of inhomogeneous plane waves in a
viscoelasTic maTerial iT is sufficienT To consider a general vecTor

Helmhosz equaTion of The form
(35)

where ‘k is The complex wave number, expressible in Terms of The
parameTers of The solid. Using separaTion of variables, one can
show a general soluTion of equaTion (35) for an assumed sTeady

sTaTe condiTion is a general plane wave of The form
G=G°expL-A-r‘18xp[-iP-r], (36)

where:

I) 5 represenTs The propagaTion vecTor and is perpendicular

.3—}

To planes of consTanT phase defined by P'l" = consTanT,

2) A represenTs The aTTenuaTion vecTor and is perpendicular

To planes of consTanT ampliTude defined by 53V: consTanT,

and

3) K and 5 musT be relaTed To The parameTers of The maTerial

by

(37)

The general soluTlon (36) includes boTh The cases of homogeneous

and inhomogeneous plane waves (This TormulaTion is similar To ThaT
given by Cowan, p. 225, l968). The angle beTween planes of consTanT
phase and planes of consTanT ampliTude of The general soluTion is
given by The angle beTween'5 and K. Taking The real and imaginary

parTs of equaTion (37) shows

5'5 ”A”? = £:- a: (38)
and
$5 = 4:16,; . <39)
FirsT consider The case of non-dissipaTive maTerials,where
k1 = O. EquaTions(38) and (39) Show ThaT
5‘5 ‘ A": = 9?: (40)
and

1%

1n
u
D

' (4|)

29

LquaTion (4|) implies ThaT The aTTenuaTion vecTor is eiTher zero,
in which case There is no aTTenuaTion, or iT is perpendicular To
The propagaTion vecTor. For The case in which if = 0, The soluTion

(56) represenTs a homogeneous wave whose velociTy is given by

v = w/ifv‘i = wz/ ,QRZ. (42)

For The case in which A # (7, The soluTion (36) represenTs an
.iflhomogeneous wave and The velociTy v' is given by
Z a

V’Z = 12.2 ‘ ““Jg"—‘ 43)
A “ .3 a ‘ ‘\
lF’l ‘

Comparing equaTions (42) and (43) showsThe velociTy of The
iflhomogeneous wave in non-dissipaTive maTerials is less Than The
velociTy of The homogeneous wave. For non-dissipaTive maTerials
when X 7: 0, 1K] and lfil can only be deTermined once The
boundary condiTions are applied. For example, in The case of The
soluTion for a dilaTaTional disTurbance,where if eand 5. are
perpendicular, un/lfid mighT represenT The velociTy of a Rayleigh
wave or in The case of an SV wave incidenT on a boundary aT an
angle greaTer Than criTical, u2/ lfiyl mighT represenT The
velociTy of The dilaTaTional surface wave generaTed.

Conversely, if The planes of consTanT phase are perpendicular
To The planes of consTanT ampliTude, Then equaTion (39) shows ThaT
The maTerial is non-dissipaTive.

The second case for consideraTion is ThaT of dissipaTive

30

maTerials for which RI ¢ C). For This case, equaTion (39) implies

The angle beTween K. and E is noT a rkfifl“ang|e and E. is noT

zero. EquaTions (38) and (39) expressed in Terms of The angle r

beTween K. and P are

R I (44)

and

lPl lAl cos I = - RR fix . (45)

EquaTions (44) and (45) solved simulTaneously give

 

 

1512 = Law’s—R: +Jlfil:-2:)z+ (zanxx/mrf) (46)
and
HM = mafia , <47)
cos 5 lPl
where The velociTy of The wave is given by tJ/'l5| . EquaTions

(46) and (47) show ThaT for a specified angle r , The velociTy
and aTTenuaTlon of a general plane wave in a specified anelasTic

medium are uniquely deTermined.

3|

For a homogeneous plane wave C()5 K = l , and equaTions (46)
.- Z 1 .—L
and (47) reduce To IP I = *2 and 1A l =-fit, respecTively.

Hence, for a homogeneous plane wave The velociTy V and The maxi‘

mum aTTenTaTion in The direcTion of propagaTion, 13! , are given

by
v2 ‘ ‘2} w:
lPI &R2 (48)
and
lil = - fizz , <49)

respecTively.

For an inhomogeneous plane wave cos r g 1. The velociTy
of such a wave is given by v; : w/ lSl , where [3| is
given by equaTion (46). From equaTion (46) iT is clear ThaT The

JP I for an inhomogeneous wave saTisfies

1%,, < “9’, (50)

hence The velociTy v/ : UJ/ I57 for The inhomogeneous wave is less
Than The velociTy v 1: uJ/ “a for The homogeneous wave. Also,
equaTions (47) and (50) show ThaT The maximum aTTenuaTion lKl

of The inhomogeneous wave is reIaTed To The maximum aTTenuaTion

lKW of The homogeneous wave by

32

lzll cos ‘4' < IR! . (5D

In parlicular, for viscoelaslic media The general plane wave

soluflons, which include Lghomogeneous waves, of equafions (l4) and

(I5) are given by

0 = A exp [nip-F] exp [*1 59°F]

 

 

(52)
and
VXZU : B'expE-AS-r‘] epr~zP,-r], (53)
where
15,32 2 Ja-(Rempzj +V9emjjz + Imlkezlz l, (54)
C052 KP

 

will; g-lnetkfil +Vnetészja + ImEJl’ssz } (55)

cos I;

33

SI
17
m

‘ ”P: H” I? ’ (56)

and

if

sl

- *5: /C05 65 , (57)

If The plane waves are homogeneous (ThaT is, 5;: 0 and 6;: 0 ),

The velociTies of The waves as given by equaTion (48) are

N

Z

— w
W .. “"i
J12
m
and
v2 - ”1
5"—I
z
k

sfl
respecTively. Using equaTions (28) and (29) one may show ThaT These
expressions for The velociTy agree wiTh Those derived earlier

(equaTions (22) and (23))for plane homogeneous roTaTionaI and dila—

TaTional waves.

3.4 Two Dimensional SoluTions for DisplacemenT PoTenTials.

The following developmenT for The displacemenT poTenTials is
enTirely analogous To ThaT for an elasTic maTerial. lT is included
here only for The sake of rigor in laTer argumenTs. The previous
commenTs on inhomogeneous plane waves will be equally applicable To

The derived displacemenT poTenTials.

A Theorem due To HelmhoITz implies There exisTs a scalar
funcTion 95 and a vecTor funcTion W such ThaT The vecTor dis—

placemenT field may be expressed as

Dizv¢+v><'v?, (58)

where

To consider sTeady sTaTe moTion, leT

and

Then, The equaTion of moTion may be wrlTTen in Terms of The

poTenTials as
vm‘vzé + wzafil + vxuszvzii + w2@)= 0. (59)

From equaTion (59) ET is clear ThaT if soluTions To The sTeady sTaTe

equaTions

V245 *§¢=0 (60)
and

V23*—;—’;¢‘0 (6i)

can be found Then They will also be soluTions To The equaTlon of

34

35

moTion (59). HT is TaciTIy assumed 53 ¢ 0 , which TogeTher
wiTh KR a 0 and Kr Z 0 implies «a # 0 . This assumpTion
prevenTs The applicaTion of The following resulTs To a hypoTheTical
fluid for which 11 : 0 .)

The vecTor wave equaTion (6|) may be reduced To Two scalar
wave equaTions by inTroducing a recTangular coordinaTe sysTem
(x|, Xz , X3 ) and requiring ThaT The parTicle moTions aT any
insTanT of Time along any line parallel To The xa coordinaTe axis
are equal. This resTricTion eliminaTes any componenT of prOpagaTion
or aTTenuaTion in The Xz direcTion. The resulTing scalar wave

equaTions are

2. 2
V95+_£22_QS=0, (62)

a z -
VW2+£2W2=0. (63)
6
and
z a
vuz+_z_u_zuz=o, (64)
p
where, henceforTh The subscripT on 9' will be omiTTed. The solu-

Tions To equaTions (62), (63) and (64) may be found using separaTion

of variables. For example, consider equaTion (62) and suppose
25 = X'(><,) Xscx3). (65)

SubsTiTuTing This expression inTo equaTion (62) and dividing by é
shows

+ .fli . (66)

 

36

Since each side of equaTion (66) depends on a differenT spaTial
coordinaTe, each side musT be independenT of The spaTial coordinaTes
and equal To some consTanT say - b. Hence, There resulTs Two

second order linear differenTial equaTions To be solved

)C + b 3C = (7

0:11 I (67)
and
x,,,. +(fé—blxs=o. (68)
The auxiliary equaTion corresponding To equaTion (67) IS
m2 + b = o. (69)
Hence, The soluTion To equaTion (67) is
X = D,e'"|" + 0 e5" (70)

where D, and 02 are consTanTs independenT of The spaTial

coordinaTes and n and r} are rooTs of The auxiliary equaTion (67).
For considering propagaTion in The +-x‘ direcTion iT is convenienT

To wriTe The rooTs in The form rh : {—l)"i &¢ (71: I,2,),where

2p 5 p.v. ( b)"2 . WriTing The rooTs in This form implies

&¢Rz O , hence, for propagaTion only in + Xl direcTion The soluTion

To equaTion (67) is

X = De" . <7”

Similarly, The soluTion To equaTion (68) is given by

.. 'W‘l -w *5
X — E e + £26 , (72)

where 1: r¢ are rooTs of The corresponding auxiliary equaTion

2 a z
m=k¢~£. (73)
a
a
The mosT convenienT way of wriTing The rooTs t r¢ , will depend on

The parTicular problem being considered. Combining soluTlons (7|)
and (72) gives The TundamenTal soluTion To The original Helmhosz

equaTion, namely
5 = A, exp [-L.k¢x, + V} x5]

+-A 8x ‘i/kX—Y'X]
2 FL ”' ¢3’ (74)
where A, and A2 are consTanTs independenT of The spaTial
coordinaTes.

Similarly, for propagaTion in The +—X‘ direcTion, The soluTions

To equaTions (63) and (64) are given by

Q 2 Bl expL-ifizyxl + rwx,J
+ 82 exp [—71 kW xI - r,” x3) (75)
and
U2 : cl expE-ikux‘ + r“ x‘]

+ Ca exp[—i ku>fi — “LX311 (76)

38

where

i) I, and .k¢ denoTe The principal values of The square

rooTs of minus one Times The corresponding separaTion

consTanT, —
. z z 2
2) t r? and I Pu are rooTs of The equaTions n1 : kw — 2%
: fl
' 7. Z
and, w} = km" 43 , respecTively,
)8 2
and
3) B- and C. are consTanTs independenT of The spaTiai

coordinaTes.
The componenTs of The vecTor displacemenT field expressed in Terms
of The soluTions for The displacemenT poTenTials are

T (77)

Read Ran-ins + ”(maul

and

, 'w‘l’
ReH-QQS - IMHe‘ i, (78)

H

Re[u5]

where ¢ and g? are given by equaTions (74) and (75).

3.5 References

Bland, D. R., The Theory of Linear Viscoelasficifx, Permagon Press,

I960.

Cowan, E. W., Basic EIecTromagneTism, Academic Press, 1968.

 

SfraTTon, J. A., ElecTromagneTic Theory, McGraw-Hill, l94l.

 

39

4O

4. ENERGY IN A STEADY STATE VISCOELASTIC RADIATION FIELD

Iinduay (p. I55, I900) gives Ihe derivaIion of a general
expression for energy associaIed wifh Ihe radiafion of eIainc
waves. The derivaTion is exfended in Ihe following To obTain a
general expression for energy associaied wiTh The sfeady sTaIe

radiafion of viscoelasiic waves.

The equaTion of moTion describing Ihe sIeady sTaTe response
of an arbiTrary viscoelasiic maferial being submiITed To forced

osciliafions is

ffi=IK+AIVe+MVu, (i)
5

where

a : L1 eta)?

and Ihe acIuaI displacemenf vecTor is represenied by Ihe real barf
of {I and denofed by Tin. The equafion of mofion in Ierms of Ihe

acTuaI displacemenf vecfor is

:- 2—5 a .2. r-
mR = (KR+,I.IR)VQR + ,uRv uR + fiuxggim‘? +j1tVZuRIM)
3 5

Forming Ihe doT producT of equafion (2) wiTh The velocify

gives an energy equaiion

:30 :3 _ La :4 2.;
PuR-LLR—(KR+&)uR-V9R+}1Rua'v L1R
3

’ .34 ‘ L 2;;
+$IIKI+LL££JLLR-V6R +lutuR~V uR). (3)
5

4 i

Using The vecTor idenTiTy
v-(M’): ¢V-A‘+V¢-A‘, (4)

for The firsT and Third Terms on The righT—hand side of equaTion (3)

and The idenTiTy,

>TL

\7 V§) =/‘i’-VB+A.B

in.) we: : (5)

for The second and TourTh Terms gives afTer rearrangemenT
h l I
u + — ~
27% R 2( Wu 2 11 u ,' Whiz}

'1 o -
{(HI*EL)9" ” 111“" - ““Jw}

HKI+,U[)éRfi‘R+flrifik-Vi‘RH}, (6)

Now, inTegraTing over an arbiTrary buT Tixed volume V’ wiTh ouTward
surface eiemenT a dA , and applying Gauss's Theorem To The righT—
hand side gives The desired expression for The Time raTe of change

of energy in V,

a 2
EM? +

n4—

3 i

(H + )9 + ——- u V
R 1:1 R ZflRuRmJ RJ!&}d

.2 a

.L (K 9

w{ 1*}:2)R +Mru‘nk’ju RJ,&}dV

 

: J'{<KR+&)9RIIR + finitR'Vl—IR)
A 3
+JJ((K,+_11_I)31’JR + fltifia-VLJRU}-RdA. <7)
3

42

For The special case of an elasTic solid )II = K,.= 0 and
expression (7) reduces To The expression given by Lindsay (I960,
p. l54).

The simplesT physical inTerpreTaTion of equaTion (7) resulTs
from considering a hypoTheTical fluid for which 11R =,11,=0. in This

case The equaTion reduces To

'8 L2 2 '2
fifvliaiun + The“ )dV + L H10“ /w dV

2

ll

_(jA’6RKR-.lrn'adA 1’ fA-éRfiRKZ/w.AdA' (8)

The Term {E'finz represenTs The kineTic energy densiTy, The Term 2% %:
represenTs The poTenTial energy densiTy and The Term éhaKr/uirepre—
senfs The energy dissipaTed per uniT volume per uniT Time. Hence, The
lefT-hand side of equaTion (8) represenTs The ToTal Time raTe of
change of energy in The volume V.

The righT-hand side of equaTion (8) may be inTerpreTed as The
raTe aT which energy is crossing The surface A. The firsT Term
represenTs The work which The maTeriaI inside (ouTside) is doing work
on The maTerial ouTside (inside), and The second Term represenTs The
raTe aT which energy is being absorbed by The maTeriaI which is mov—
ing across The surface A.

By analogy, The firsT Term in equaTion (7) represenTs The Time
raTe of change in The ToTal kineTic and poTenTial energy in The
volume V, The second Term represenTs ToTal energy dissipaTed per
uniT Time in The volume V, and The righT-hand side of equaTion (7)
represenTs The raTe aT which energy is crossing The surface A.

2. ' '

The quanTiTies d? and MR u . are always posiTive

43

and lhe second law of Thermodynamics requires lhaT The ToTal amounT
of energy dissipaTed increase wiTh Time. Hence equaTion (7) implies
KI and ll, are non—negaTive.

A useful measure of energy loss for a one dimensional sTeady-
sTaTe sysTem was defined in secTion 2.3 as The raTio of The loss in
energy per cycle of forced oscillaTion To The peak energy sTored
during The cycle. This definiTion is exTended To The presenT Three
dimensional problem, by considering energy densiTies. More explicile,
denoTe The loss in energy densiTy per cycle of forced oscillaTion by
zivv /cyc|e, and denoTe The peak energy densiTy sTored during The
cycle by VVP . HenceforTh, The sTaTemenT "fracTional loss in energy
densiTy per cycle" will mean

AMI/cycle . (9)
WP

The definiTion of 0" is exTended by defining

Q" E é-TriAW/cycie ). (l0)
WP

4.l Energy Loss in Bulk and 0‘4

To consider The energy loss in bulk and To define 0;, in bulk,
consider a hypoTheTical fluid for which 11::0 , and a sTeady sTaTe

compressional wave of The form

aR=ReraeiwL (H)

where Q is given by equaTion (3.52).

For such a disTurbance equaTion (8) implies The loss in energy

44

densiTy per cycle of forced oscillaTion is

2117(1)

W = K .2 = 2.
A /cycle 1) 739R d’r TTKIIQI . (,2)
The peak value of The energy densiTy sTored is
w = K 1912/2
P R ' (l3)

Hence, The fracTional loss in energy densiTy per cycle is given by

AW/czcle : 21TK,/KR , (W
W
P

and q" in bulk (denoTed by Q;' ) is

_ K
QKI= I :1’601 8“,

—" 5
KR (I )

where SK is defined To be The angle by which The bulk sTrain lags
The bulk sTress.

AnoTher measure of energy Ioss,which is applicable To sTeady
sTaTe plane homogeneous waves,is The raTio of The loss in The Time
averaged energy densiTy per wavelengTh To The mean Time averaged
energy densiTy in The space inTerval of one wavelengTh (Lindsay,

p. 35). IT will now be shown ThaT This measure of energy loss in
bulk is approximaTely The same as The "fracTional loss in energy
densiTy per cycle", provided The amounT oT absorpTion in The maTerial
is small. In general, however, The Two measures are noT equal.

Consider a plane homogeneous compressional wave propagaTing in
The posiTive direcTion along The coordinaTe x|. From equaTion

(3.l8) The complex represenTaTion of Such a wave is

a: Aexpf-aKxJ expLHwT-g‘ixfll, (I6)
where
a i-wImvai P )"‘°‘J
K K (l7)
and
VKE (Re[p.v.L£_)’/a)—I.
K
g(l8)

Choosing The volume V in equaTion (8) To be recTangular wiTh Two of
iTs faces perpendicular To The direcTion of propagaTion, The Time
average of The inTegrand on The righT-hand side of equaTion (8) when
divided by The velociTy represenTs The Time averaged energy flow
across The faces which are perpendicular To The direcTion of propaga-
Tion. Hence, The Time averaged energy densiTy Tlow . for The

assumed wave is

_ [(K a “I 9' o )
W—‘V sexual "' Tu— R“n, : (l9)

Using The relaTion ThaT a = l4

R and finding The appropriaTe

RH!

real parTs from equaTion (I6), The Time averaged energy densiTy may

be wriTTen in Terms of The wave parameTers as

:7 = We expl. -2.QK x.) ,
where
2
“5,2 21A12%%L0KK, +2.“).
(“+2.1 x u) VK (20)
Va

K
Hence, The loss in The Time averaged energy densiTy over one wave-

lengTh is

40

—- .. 10”

x w“, = w, e ‘(l- 6””) (21>

(I
l

and The mean averaged energy densiTy over a space inTerval of one

wave—lengTh is

x,+A __
W 2.).)- U em"x'dx =Yzeza“x‘(1—em*x)
A N ° I A E]; . (22)
K

The raTio of equaTions (2|) and (22) gives The desired measure of

energy loss

(Wx_ g,“ )/—'v‘-G = 2a A = 2.1T(?.O.KVK/w). (23)

1+) K

Using equaTions (l7) and (IS), Gfi' as defined by equaTion
(15) may be expressed in Terms of The parameTers of The homogeneous

wave as

o" = (2v a /w)/(l—.L_(2v o. /w)a).
" “ " 4 " " (24)

For solids wiTh a small amounT of absorpTion (ThaT is,vKQ“/qu

<< l ) This expression shows
-I 2V a
OK 8 ———-J) “ (25)
or
“’41 C; — ;7

==
MI

Hence, comparison of equaTion (26) wiTh equaTion (23) shows ThaT The
Two derived measures of energy absorpTion are approximaTely equal if

The amounT of absorpTion in The maTerial is small.

47

4.2 Lnergy Loss in DilaTaTion and (9;

For a homogeneous plane dilaTaTional wave propagaTing in The

+ x. , direcTion The lefT—hand side of equaTion (7) reduces To

Luigi}: +_'..(KR +_4_MR:)u )va'ijL’ (K+i“11)u: dV,

1 2 RH)

(27)

Olly
q

Hence, The fracTional loss in energy densiTy per cycle of forced

oscillaTion in dilaTaTion is

 

[SW/Cycle 3(‘K1*§, flfll/w)j1‘& U‘Rm dV 1= 2W(K'*ifi’u‘)
wP é-(Kgb + .511“) muiuRW] KR +%uR (28)

and 9-, fOF general dilaTaTion (denoTed by 0;' ) is

_ K v
69.3: “951“ =<L__P“P)/u———a ). <29)
‘41! w (Eli’—E a)
’3‘ R

Using an argumenT similar To The one in secTion 4.l, one may

show ThaT for a homogeneOus plane compressional wave

“6: "on-.\ = ZVPQP . (30)
$5 u)

Hence, for maTerials wiTh a small amounT of absorpTion
og' x ZVPOp/UD. (31)
4.3 Energy Loss in Shear and q;‘
For a homogeneous plane shear wave propagaTing in The + x!

direcTion and polarized in The X|X3 plane The lefT—hand side of

equaTion (7) reduces To

48

ally
4

J (.£.&2 -i}IR “2 )dV 1 ll '

v n —— R + ._1. u dV.
3 a 2 V w "3:! (32)

Hence, The fracTional loss in energy densiTy per cycle of forced

oscillaTion in shear is

T
Z
AW/C “6 =(fol-I—1u dT)/(1.L£_me[u 32):2nu:l33)
p ’ 2 31/ 11R

and 0" in shear (denoied by Q;' ) is given by

—l 111 (34)

= 3 = _ i 2 v a a
Os )1“ Tan 83 (szas/wvu 2,14%) l,

where 6;

is defined To be fhe angle by which ihe shear sTrain lags
shear sfress. Using an argumenf similar To The one given in seclion

4 i, one may show Thai for a homogeneous plane shear wave

(35)
77,
Hence, for maTerials wiih a small amounT of absorplion

95'25 szcts/w. (36)

4.4 Addil’ional Rela‘lionships for Q-’

The relaTion befween fig” , 9", and

QR" is easily derived
from equalions (l5), (29) and (34)

4 11 -l - (37)
(l + 1? ii) G5 3 .R— s .

u
4)
x
+
l
.5
E:
{D
l

AnoTher measure of absorpTion which is applicable To a sTeady~

 

stale homogeneous wave wiTh a spaTial decay of The form (3'4“x and
vuluuily v is lho logarithmic decremenT, l8 , defined as The
logarilhm of The raTio of successive ampliTudes one wavelengTh aparT.
ihaT is,
A
l8=ln(_l‘_):a(A=Tl'l?-°W) (38)
Axr)‘ ‘0

AnoTher quanTiTy applicable To such a wave is The skin depTh,

52? , defined as The disTance in which The wave decays by a TacTor

of e" . Hence

ll
9

$3 " (39)
and

88 = A/’|8 . (40)

The logariThmic decremenT for a homogeneous plane sTeady-sTaTe wave

is relaTed To The corresponding Q_'by

Q": (18/W)/(l«—4’—(is/n32). (4')
For maTerials wiTh a small amounT of absorpTion equaTion (4|) shows

Q” 2: lg/Tr. (42)

49

4.5 Reference

Lindsay, R. 8., Mechanical Radiafion, McGraw-Hill, I960.

 

50

5|

5. RESTRICTIONS ON PLANE WAVE SOLUTIONS

FOR A HILV HALF-SPACE

Each of The general soluTions (equaTions 3.74, 3.75, and 3.76)
involve Three arbiTrary complex parameTers, namely, Two complex
ampliTudes and a complex wave number. ResTricTions on The para-
meTers are imposed by The boundary condiTions of The problem under

consideraTion.

5.i ResTricTions Imposed by Boundary CondiTions

The boundary condiTions Tor an infiniTe half—space are ThaT
The sTresses on The Tree surface vanish. To seT up The problem,
consider a HILV half-space wiTh a recTangular coordinaTe sysTem
chosen such ThaT The space occupied by The solid is described by

x z o . For an assumed sTeady-sTaTe condiTion, The componenTs

3
of The sTress Tensor may be expressed in Terms of The displacemenT
poTenTials, using equaTions (3.58), (3.5) and (3.!2). The resulTing

reIaTions for The sTresses acTing on planes parallel To The surface

are
P31: Pflz(2¢'15+ IF)” T 9,35); (I)
P32 : P’dz up: I (2)

and
P33 : Potzlém + 43,55) + Zszifins— 95,”),(3)

52

The boundary condiTions aT The free surface are expressed by
equaTing equaTions (I), (2), and (3) equal To zero aT x3 = 0~

ApplicaTion ol The boundary condiTions To The general soluTions

(3.74), (3.75) and (3.76) places The following resTricTions on The

parameTers:
P __ _ 2 2.
35—0, (Z§¢-Jés)(/\|+Aa)=
2igk¢(8'-Bz)exp[-i(&w—k¢)x,] l4)
P3,=0; zir,n,iAz—Ali=
lZbe~fi:)lB‘+Bz)cxp£~i(k¢-k¢)x‘ (5)
F32 - 0 1 Ci:: Ca '
(5)

The derivaTion of The general soluTions (3.74), (3.75) and (3.76)
assumed The consTanTs Ai.) 8‘ and Cl (i = 1,2) were independenT
of The spaTial coordinaTes. This assumpTion implies The leTT—hand
sides of equaTions (4) and (5) are independenT of The spaTial coordi—

naTes. Hence, The righT-hand sides of equaTions (4) and (5) are also

spaTially independenT, which implies

2:2),

¢ W (7)

and The subscripT on k is no longer needed for problems in which
Cl :- C2::0. For problems involving plane wave propagaTion in a
HlLV half-space The parameTers of The general soluTions musT saTisTy

equaTions (4), (5) and (6) TogeTher wlTh derived relaTion (7). These

relaTions place cerTain resTricTions on The parameTers; however, The

53

Three equaTions involve seven complex parameTers, hence no unique
soluTions for The parameTers can be found from equaTions (4), (5),
and (6). To find a unique soluTion, addiTional resTricTions musT be
inTroduced. Physically, such resTricTions are moTivaTed by consider-
ing eiTher The problem of reflecTion of various incidenT wave Types

or The problem of a surface wave.

5.2 ResTricTions Imposed for ReflecTion Problems

To consider The reflecTion problem, a single incidenT wave is
assumed. This assumpTion is equivalenT To specifying R and requiring
ThaT Two of The incidenT ampliTudes are equal To zero. These addi-
Tional resTricTions on The parameTers TogeTher wiTh Those specified
by equaTions (4), (5), and (6) are sufficienT To express The para-
meTers of The reflecTed waves in Terms of Those for The assumed
incidenT wave. The reflecTion problem is TreaTed in deTail in

chapTer 6.

5.3 ResTricTions imposed for Surface Wave Problem

To consider The surface wave problem, The sTeadywsTaTe dis-
Turbance is assumed To be concenTraTed mainly along The free boundary.
This assumpTion implies Three addiTional resTricTions on The complex
parameTers, namely Al== B‘ = Cl: 0. These addiTional resTricTions
TogeTher wiTh Those specified by equaTions (4), (5), and (6) are Then
sufficienT To solve for .9 and one of The remaining complex ampli—

Tudes in Terms of The oTher.

For The surface wave problem The value of k is deTermined from

resTricTions placed on The oTher parameTers in The problem. This is

in conTrasT To The reflecTion problem in which The value of ‘fi is

assumed.

The surface wave problem is TreaTed in deTaiI

in chapTer 7.

54

55

6. REFLECTION OF A PLANE lNHOMOGENEOUS SV WAVE

FROM THE SURFACE OF A HILV HALF-SPACE

The problem of The reflecTion and refracTion of a plane homo-
geneous wave incidenT on a boundary beTween Two viscoelasTic media
has been considered by LockeTT (l962). His resulTs show ThaT The
reflecTed and refracTed waves are in general inhomogeneous. Hence,
To sTudy plane—wave propagaTion in maTerials wiTh more Than one
boundary iT is also necessary To derive The soluTion for an inci—
denT inhomogeneous wave. LockeTT does noT consider This case.

The purpose of This secTion is To consider The more general
problem which includes The case of an incidenT inhomogeneous wave.
The Technique for formulaTing This more general problem is mosT
easily illusTraTed by considering The special case in which The
boundary is a free surface. LockeTT's resulTs show ThaT, even in
The case of incidenT homogeneous waves, There are significanT dis»
TincTions beTween The reflecTion problem for anelasTic media and
The reflecTion problem for elasTic media. These disTincTions for
The reflecTed waves can be illusTraTed by considering The problem

of an incidenT SV wave.

6.l SpecificaTion of lncidenT General SV Wave

To consider The reflecTion problem iT is convenienT To specify
ThaT The firsT Terms in The general soluTions (3.74), (3.75), and
(3.76) represenT incidenT waves. To achieve This convenience, The

rooTs r

¢ , rw , and ru_ may be wriTTen in The form

56

id

.3
ll

¢ 0(7
' (l)
Vy=1dfl,
and
Pu:idu,
where
2 2 21/2
d‘=p.v.(w/°< ‘k¢) I
1 2 - 2 l/Z
d5: p.v.lw/fl k0.) .v (2)
and
z z 2 We
du: P.v.(w/fl ‘ A“) .
Using principal values in This manner implies ThaT d“ , d

Fa ’
and due are non-negaTive and hence, The TirsT Term in each of

The soluTions represenTsa wave propagaTing Toward The boundary.
For The problem of only incidenT SV, The consTanTs A’and C,
musT be zero. The boundary condiTion equaTion (5.6) Then implies
CZ‘: 0 ; hence, There can be no reflecTed SH waves.
lT is also convenienT To express The soluTions in Terms of

propagaTion and aTTenuaTion vecTors, where The propagaTion vecTors

are defined by

.a. n A
P». 5 RR”? + H) d“n x3
l3)
.5 A n A _
PW"; aRx' +- l~ll dfin x:5 (In-hi),

57

and The allcnuafion vecTors are defined by

-* __ A n“ A

A¢n : 311 X, + ("H d“r X5

-‘ = _ A _ n+l A _- (4)
AW" _ firx‘ + (l) dflzx3 (n-1,2).

Using .fi, =‘fifl The soluTions for only an lncidenT SV wave can now

be wriTTen as

f}
n

Az emf—Kg?) expL—i6¢z.?3

and

‘6
n

8, Exp [-K¢‘-FJ Exp [-ilgwl-FJ

(6)

I

+ B expt—A‘ ~FJ expL—ii'D‘ -F3
2 w: wk

.4
Where r‘ a X‘QI + x3 95 .

For The firsT Term in (6) To represenT a general (i.e., homo-
geneous or inhomogeneous) incidenT SV wave, & musT be appropriaTely
resTricTed. lnTuiTively, R musT be defined such ThaT uJ/.kR re-
presenTs The apparenT phase velociTy along The surface and *1
represenTs The apparenT aTTenuaTion. For a given angle of incidence

as.

I and a given angle 3;; beTween The aTTenuaTion vecTor and The

propagaTion vecTor, such a resTricTion on .9 implies

_ _. . —‘ . (7)
3 .. IF;J Sin égi - i lA,” l Sin lesi~ Bgil ,

where

58

 

a) lelz =._|_Re[Jks"J {I + Ji+lQ§‘/cos 65,)1},
‘ 3 ' (8)
.A I Z —-\
b) lAfll : T Imuls J /(lP,|l 605 T5”, (9)
and
c) The acuTe angles 53; and 6g; are chosen such
U' c ‘W
<.. .. 9 4 —— .
ThaT ligil 2 and 0 5i _ 2 (IO)

The preceding formulaTion does noT include The case of an
incidenT ighomogeneous wave in elasTic media. The velociTy of an
iflhomogeneous wave in elasTic media is noT uniquely deTermined by
specifying 5;; as in The case of anelasTic solids.

The condiTion ThaT Its“ (TT/Z insures ThaT a general plane

wave aTTenuaTes in The direcTion of propagaTion. For The special

case of an incidenT homogeneous wave K;.: 0 , and equaTions (8) and

(9) Show
l5. = .9
MI Sa’ (ll)
and
mm! : g5: ‘ (:2)

RelaTions (ll) and (l2) show ThaT for a homogeneous wave, The rela-

Tion for ‘R reduces To

a: £5 sin 95', - ('3)

The velociTy of The general incidenT SV wave in The direcTion

of propagaTion is %;:. é?_ . If Jgi: O , The incidenT wave is
w

59

homogeneous and The corresponding velociTy is V53 $3, which is
3n
greaTer Than or equal To The velociTy V; of The general wave.

(Refer To seCTion 3.4). The apparenT phase velociTy Va of The

general plane SV wave along The surface is

I

Va: —__“’ = _____Vs - W (I4)

[Efll Sin 95i sin 65; .&

 

R

which is less Than or equal To The apparenT velociTy of a correspond-
ing homogeneous wave.
The apparenT aTTenuaTion along The surface or The componenT of

aTTenuaTion in The + x'direcTion is

Hence, if 6;; is greaTer Than 653 , The incidenT inhomogeneous
wave increases in ampliTude along The surface. This apparenT siTua—
Tion does noT violaTe any physical principles because of The resTric—
Tion 3;; < fl’/2., which inSures The ampliTudes decrease in The
direcTion of propagaTion.

The above TormulaTion for The incidenT SV wave is general in
The sense ThaT The aTTenuaTion vecTor for The incidenT wave may
poinT in eiTher The +-x,direcTion or The “'M direcTion. The only
resTricTion on iTs direcTion is ThaT iT forms an angle less Than W/a

wiTh The propagaTion vecTor. The concepT of TiniTe ampliTude aT

infiniTy is noT applicable To This problem.

6.2 PropagaTion and ATTenuaTion VecTors for The ReflecTed Waves

The general form of The soluTions (3.74) and (3.75) and The

6O

relaTion ThaT' R¢zrflkdeTermines The direcTion of The propagaTion and
aTTenuaTion vocTors for The reflecTed waves in Terms of Those for

The incidenT wave.

6.2.l — ReflecTed Sv Wave - The second Term of squTion (6) represenTs
a reflecTed SV wave. The form of The soluTion (6) shows:
a) The componenTs of propagaTion in The +-x, direcTion for The
reflecTed SV wave is The same as ThaT assumed for The incidenT
SV wave,
b) The componenT of The aTTenuaTion in The a-xl direcTion for
The reflecTed SV wave is The same as ThaT assumed for The
incidenT SV wave, and
c) The propagaTion and aTTenuaTion vecTors for The reTlecTed
wave are in The same plane as Those assumed for The incidenT
SV wave.
These properTies may be expressed in more familiar forms by wrlTing
The propagaTion and aTTenuaTion vecTors in Terms of The angles They
form wiTh The x axis. The propagaTion and aTTenuaTion vecTors for

3

The incidenT wave wriTTen in Terms of The angles are

5;, = 13w]: (sin 9,; Q, — {cos 65; Q5)
——b _ *‘A _ A -
Aw, — ”‘10, l (sm (65'.~ 2);...) x, - Laos (65“ (5;) X3) ,

respecTively.

For The reflecTed SV wave denoTe The angle of reflecTion by

6 and The angle beTween The propagaTion and aTTenuaTion vecTors
r .

by X . In Terms of These angles The propagaTion and aTTenuaTion

bl

voclors ior the ieflecTed waves are

—‘ __ - i A A
pyz" ”341} (5m (95er + 1.505 esr x5)

and (17)

A ._~b
; ’ . 6 _ A - - A

The preceding resulTs formulaTed in more familiar Terminology are:
a) Snell's law is valid, ThaT is, The angle of reflecTion esr
equals The angle of incidence 6g; , and fizz and F2! are

in The same plane,

5) leII sin 95.. : [Pwal sin 95" , (I8)
or in Terms of The apparenT phase velociTy along The
surface

Va v; vs

c) The maximum aTTenuaTion of The reflecTed SV wave is The same

as ThaT for The incidenT SV wave, ThaT is

and

d) The apparenT aTTenuaTion along The surface of The reflecTed

SV wave is The same as ThaT for The incidenT wave, ThaT is

l'Aw‘l siniesf~ {5;} = lszlsinwsh— B’sr) . (20)

From equaTion (20) iT follows ThaT if Egi= 0 , Then ggrzcx

ln oTher words, if The incidenT SV wave is homogeneous, Then The

reflecTed SV wave is homogeneous.

C
r\‘

The direcTion of The aTTenuaTion vecTor for The reflecTed wave
relaTive To The surface is opposiTe The direcTion of The aTTenuaTion
veclor for The incidenT wave. For example, if The aTTenuaTion of
The incidenT 5V wave is Toward The surface, The aTTenuaTion of The
refiecTed SV wave is away from The surface. The direcTion of aTTenua-
Tion relaTive To The Surface, ThaT is, The X5 componenf of
aTTenuaTion depends upon The sign of d5: . The sign of sz is

expressible in Terms of The angles 55-

and as, . To derive This
| l

reIaTion, recall ThaT d, was defined as The principal value of The
square rooT of a complex number. The real and imaginary parTs of

The principal value of a complex number 2 are given by

whiz)”z :|/i£'_z*z_n + isignlzzilfiLé'iL', (2|)

where

signfzrj: (Kreysia ,p 55!),l9621

Using equaTion (2|), The algebraic sign of dp is given by
I

sign [rm [(1,211 = sign [ Z( «”535, - ham]

Sines.
I

= Sign [Elisnkstu- simesi—rsim

cos J'Si

= sign [Zfisnfisl ( l — sinzesi + Tan X's; CoseslsinQfill ,

Now, since RsR—Qs E 0 (inequaliTies 3.33 and 3.34), The desired
!

reIaTion follows, ThaT is, dp ;> 0 if and only if
I

63

a
sh 9,-1
“in Us. <.__n___i___
I

= —l/T1m as,“ (22)
cos 6%; sn16g;

6.2.2 - ReflecTed P Wave — The general soluTion (5) represenTs a
reflecTed P wave. The aTTenuaTion and propagaTion vecTors for The
reflecTed P wave are relaTed To Those of The lncidenT SV wave by The

relaTion Jb¢ =~-% (equaTion 5.7). This relaTion implies The com—

y
ponenTs of propogaTion and aTTenuaTion in The + xldirecTion of The
reflecTed P wave are The same as Those for The incldenT SV wave.
DenoTing The angle of reflechon of The P wave by epr and The
angle beTween iTs propagaTlon vecTor and aTTenuaTion vecTor by 6?”,

The propagaTion vecTor and aTTenuaTion vecTor expressed in Terms of

These angles are

P4; = lP¢2l ( sm 9,,” x, + cos 9?” x3)
and (23)
_.§ uh . A
Ag : lA¢zl (5m (epn— In”) 9' + cos (99,,“ 25“,") x3 )’

respecTively.
Since The componenTs of propagaTion and aTTenuaTion in The +-x,direc—
Tion for The reflecTed waves are The same as Those for The incidenT

waves, equaTions (l8) and (20) may be exTended To

“3W sin 95'. 2 Iszl sin 63” : lP¢z| sin 9,2!"

and

64

IA I siniesi— ‘5.) : lezl sin(e‘r—Ifsr)

(25)

J

: "l ‘nLB —
lA¢a Sl Pr Upr)

respecTively.
In Terms of The apparenT phase velociTy Vh along The surface,

The exTension of equaTion (l9) is

_l- : Sin 953 : SlY'l 9” = sin 99y» ’ (26)
Va v’ v’ v’
s s p

 

where V; a U)/‘E%3l is The velociTy of The general reflecTed P wave
in The direcTion of propagaTion.

EquaTion (26) is similar To The corresponding equaTion which is
derived for elasTic media (e.g., equaTion (l), p. lOl, Bullen, l965)
wiTh The excepTion ThaT The velociTies in equaTion (26) are Those of
general waves, which includes The case of inhomogeneous waves.

For elasTic media, The reflecTed P wave is inhomogeneous only
for angles of incidence greaTer Than some criTical angle. We shall
now show ThaT The siTuaTion is quiTe differenT Tor anelasTic media.
FirsT, iT will be shown ThaT even if The incidenT SV wave is homo—
geneous, The reflecTed P wave is commonly inhomogeneous for all
angles of incidence. Second, iT will be shown ThaT for a given angle

6;; There is noT always a criTical angle for reflecTion.

The firsT properTy sTaTed more precisely is

THEOREM i: If The incidenT SV wave is homogeneous and noT

normally incidenT, Then The reflecTed P wave is

homogeneous if and only if sin2 95 s

05" : QS' ’ (5.8. Tan 85 = Tan 8“).

65

The proof of Theorem I is given in appendix 2.

The conTraposiTive of Theorem I shows ThaT if Q§I:# 054, Then
The reflecTed P wave is inhomogeneous for all angles of incidence.
This properTy is one of The basic disTincTions beTween The reflecTion
problem for anelasTic media and The reflecTion problem for elasTic
media.

The criTical angle for reflecTion may be defined as ThaT angle

 

of incidence for which The reflecTed P wave propagaTes in a direcTion
parallel To The undisTurbed free surface, ThaT is fig: is parallel
To The boundary. IT will be shown ThaT The role played by The cri-
Tical angle for anelasTic media is enTirely differenT Than iTs role
in The case of elasTic media. To sTudy The role of The criTical
angle for reflecTion, iT is convenienT To TirsT esTablish

THEOREM 2: A criTical angle 6& exisTs if and only if

i
‘ "9 lb
sm ,3 — (92,,a ,I/fizsa fist)

sn165_ C05 63,
l l

Tan ifs, =
I

(95i¢0,1%). (27)

 

Theorem 2 is proved in appendix 3, where iT is also shown ThaT

relaTion (27) is equivalenT To

sin 95_ cosiesi- 3;!) = COS Yrsi (ThPR hp: / flsxkszi , (28)

To invesTigaTe The exisTence of a criTical angle for reflecTion
in The case of anelasTic media, firsT consider The case in which The
incidenT SV wave is homogeneous. For This case, a resulT firsT
derived by (LockeTT, I962) and reTormulaTed here is

THEOREM 3: If The incidenT SV wave is homogeneous in an

anelasTic solid such ThaT 0;’ < ng , Then no

criTical angle for reflecTion exisTs.

66

Theorem 3 is proved in appendix 4. ExperimenTal evidence shows ThaT
for many anelasTic solids, 0g” 4 Q;', and hence, for These anelas-
liu solids no crilical angle for The reflecfion of a homogeneous SV
wave exisTs. The resulT of Theorem 3 for anelasTic solids in in
conTrasT To The resulT for elasTic solids. For elasTic solids, a

criTical angle for reflecTion exisTs for every solid and is given by

 

z .&2
Slh 6g : PR : 15; , (29)
' a
ksa KR + ‘g‘llk

For The second case, in which The incidenT SV wave is Lflhomo—
geneous, each angle of incidence 9% nay be regarded as a criTical
angle for reflecTion in an anelasTic half—space. More specifically,

THEOREM 4: For every angle of incidence < 55;: 0 and 9S; fig. )

and every anelasTic solid There exisTs some F5;
associaTed wiTh an incidenT general wave such ThaT

9a

is a criTical angle for reflecTion.

The proof of Theorem (4) follows immediaTely from The facT ThaT for

each 93; ( Eg.#£7and 6gi¢ fl72) and for each anelasTic solid, There
I

exisTs some Kg,

such ThaT —%< 7;“sz and relaTion (27) of Theorem
l l

2 is saTisfied.
Theorem 4 shows ThaT for each angle of incidence 6g_, iT is
l

possible To choose 3;, such ThaT égiis a criTicaI angle for reflec—
l

Tion in an anelasTic half—space. However, each 6;; is noT
necessarily associaTed wiTh a criTical angle. More specifically,
THEOREM 5: If 3;. and The parameTers of The anelasTic solid
l

are such ThaT

(hp. hPx/ksa 95:) C05 31;} > T , (30)

Then no criTical angle for reflecTion exisTs.

67

For The special case in which z;i==0 and G§' < Gg‘ , Theorem 3 shows
63; is noT associaTed wiTh a criTical angle for reflecTion. The
proof of Theorem (5) follows immediaTely from relaTions (30) and
(28).

AnoTher characTerisTic (menTioned by LockeTT (l962)) of The
reflecTed P wave which exisTs only for anelasTic media can be seen
by considering The angle aT which The P wave is reflecTed. This

angle is given by

 

 

Tan 6&‘= 3% =(V?T RR)/JIQ:-Qal + ReiQ:-Pfi

“R

(5i)

EquaTions (7), (8), and (9) show ThaT .& depends on (95i , 6;_,
I

and Q5, which for anelasTic media, is in general aiuncTion of
frequency. if 63; and ugidre hold fixed Then equaTion (3|) shows
ThaT The angle aT which The P wave is reflecTed also depends on
frequency. Hence, an incidenT SV wave composed of several differenT
frequencies in anelasTic media will produce a fan of reflecTed P
waves. This is noT The case for elasTic media, since The angle aT
which The P wave is reflecTed does noT depend on frequency.

The direcTion of The aTTenuaTion vecTor relaTive To The surface

for The reflecTed P wave depends on The sign of d“: . Recalling

ThaT dd was defined as a principal value, equaTion (2|) implies

Sign [ d‘I] : Sign]: 2(QPRRPI _ 33,, 2’51— 5m 95. sin(esi— $391.62)

I
cos I;

SimplificaTion of equaTion (32) shows dd 3>0 , ThaT is, The aTTenua-
1

Tion is Toward The surface if and only if

Tan {5. 4 (swabs; — 2'93 13?; )/(Sin65_ COS 05,).
' as k ' '

as;

(33)

lnequaliTy (33) may be combined wiTh The corresponding inequaliTy
for The incidenT SV wave (22) To show for which anelasTic solids
and for which orienTaTions of The incidenT aTTenuaTion vecTor, The

aTTenuaTion of The reflecTed P wave is Toward The surface.
6.3 AmpliTudes and Phases of ReflecTed Waves

The ampliTudes and phases of The reflecTed waves can be
expressed in Terms of Those for The assumed incidenT wave by using
The boundary condiTion equaTions (5.4) and (5.5). The boundary con—

diTion equaTions for only an incidenT SV wave are

(aka-(kph =—2dpR(B‘-Ba) (34)

and
Z 2.
Edda/la =—(2fi—Qs)(8,+8a), (35)

where Q is given by equaTion (7).

For an incidenT SV wave 3. ¢ 0 , The complex ampliTudes of
The reflecTed waves in Terms of The incidenT ampliTude BI are
given by

Z 2.
A —4dflk(2l§ was) (36>

2 ..
8. 4dqdfl k2 + (ZRZ~R:)Z

 

and

69

 

Ba _ 4dqdpRzZ—(Z&Z~Q:)z
-'— _ z z z 2 ’ (37)
3. +ddd61Q +£2k—fis)
where
gkk) 2 4dddfi k2 + (2222— Rf)” 7&0. (38)

If g<&):o , Then equaTions <34) and <35) impiy 8; : 0 . 1+ will
be shown in chapTer 7 ThaT The rooT of The equaTion g( E )=0is The
complex wave number for a Rayleigh’Type surface wave on a visco—
elasTic half—space. Hence, The resulT " 8,710 implies 9(Q)¢0 may
be sTaTed as:

"An incidenT SV wave upon inTeracTing wiTh The surface does noT

generaTe a Rayleigh—Type surface wave.H

The ampliTude raTios of The acTual displacemenT poTenTiaIs Reii¢51

and Re[_¢31 are given by
lAai / i8I l (39)

and

1le /lB,i. (40)

The phases of The reflecTed displacemenT poTenTials relaTive To

ThaT of The incidenT wave are given by

Tan—'(Imtkl/Re[f_z.]) (4H

8‘ B.

and

7O

ran'liImLELJ/ReLEJ. (42)
B, B,

The form of The relaTions (36) and (37) are enTirer similar To
Those derived for elasTic media. However, The inTerpreTaTion of The
formulas for anelasTic media is subsTanTially differenT. For example,
in The case of elasTic media, a normally incidenT homogeneous SV wave
is ToTally reflecTed as an SV wave wiTh no corresponding reflecTed
diIaTaTional disTurbance. |T will now be shown ThaT a normally
incidenT inhomogeneous SV wave in anelasTic media is noT ToTally
reTlecTed as an SV wave.

Consider a normally incidenT SV wave, ThaT is, 65i='0. EquaTion

(7) shows ThaT for such a wave
4% = llAwllsin {5i . (43)

For There To be no reflecTed di laTaTional disTurbance [All /l}\:[ : O

and equaTion (36) implies eiTher

a) dp = O ,
b) R = 0 ,
or (44}
c) 2 a: — Q: == C).
if condiTion (a) is True, (i.e.,d5::O ), Then by equaTion (43)
Im Ldj] = deflclpz = 25351 2 0,

which implies ks - O and The maTerial is elasTic in shear.

7|

ll condiTion (b) is True (i.e., .& = O ), Then equaTion (43) implies

2 2 -‘ 2. _ 2
d5 : «Q : —IA¢'I Sin Ti; ,
which implies
d : : 0
pa apt ask a: ,
and —& :: 0 , so The maTerial is elasTic in shear. in oTher

5:

words, iT has been shown ThaT if The normally incidenT SV wave in
general anelasTic media is inhomogeneous, Then The ampliTude of The
reflecTed P wave is noT zero.

in The case ThaT The normally incidenT wave is homogeneous,

Q = as sin Q” and equaTion (36) shows There is no reflecTed dila-
TaTional disTurbance.

For elasTic solids angles of incidence exisT such ThaT an
incidenT SV wave is enTirely reflecTed as a dilaTaTional wave.
However, This resulT is True for only a resTricTed class of visco—
elasTic solids. More specifically,

THEOREM 6: If a non-normally incidenT SV wave is homogeneous

and There is no reflecTed SV wave Then The solid
is such ThaT Os" = 03' = 0x -

The proof of Theorem 5 follows. For an incidenT homogeneous SV wave

fie : 2:: sin2 653 (equaTion l3), hence
2 z a
33 = 915 - 3? = _ l , (45)
a 17" Tanzes’,

For There To be no reflecTed SV wave,equaTion (37) shows

N

a 2
l6 EE_.§1 :: (2 — sgi )4 . (46)
£2 2 k2

The lefT-hand side of equalion (46) is real and equaTion (45)

2 z . a z
dp /JZ IS real, hence, dd /& musT be a real number and

2

2
Im[%] = Imlfl1/sin295f: o.
k x:

Equaiion (47) shows Q; /Q: is a real number and Lemma l

(appendix 2) implies
-I _ —l —l
95 — 0 p : K .

EquaTion (48) concludes The proof of Theorem 6.

ln addiTion, equalions (45) and (46) show

2. Z
:12 = __T_Ji____ _.l s 0,
3{ Jés 51111951.

from which if follows Thai

7.
Sinz'ég; Z .fg : 1L; .
2
£5 KR + €%.LQ

72

shows

(47)

(48)

(49)

(50)

If equaliTy holds in (50) Theorem (I) shows lhe reflecled P wave

is also homogeneous.

73

6.4 References

Bullen, K. E., An InTroducTion +0 The Theory of Seismology, Camb.

Univ. Press, Ihird ed., I963.

Kreysig, E., Advanced Engineering Malhemalics, John Wiley and Sons,

 

second ed., I967.

Lockell, F. J., The refleclion and refraclion of waves aT an infer—

face beTween viscoelasfic media, J. Mech. Phys. Solids, 19,

 

pp. 53-64, I962.

AddiTional papers on Ihe refleclion problem are:

Cooper, H. F., Jr., Reflecfion and Transmission of oblique plane
waves al a plane inferface befween viscoelaslic media,

J. Acousf. Soc. Am., fig, pp. I064-69, I967.

Cooper, H. F., Jr., and E. L. Reiss, ReflecTion of plane visco-
eIasTic waves from plane boundaries, J. Acousf. Soc. Am. 22,

pp. II33-38, I966.

O'Neill, H. T., Reflecfion and refracfion of plane shear waves in

viscoelasfic media, Phys. Rev., 12, pp. 928—35, I949.

74

7. RAYLEIGH-TYPE SURFACE WAVE ON A HILV HALF-SPACE

The purpose of This secTion is To derive The analyTic soluTion
for a plain Rayleigh—Type surface wave on an arbiTrary viscoelasTic
half-space. The soluTion will be developed for maTerials charac—
Terized by a general consTiTuTive relaTion of The form (2.7) for
which no resTricTion on The amounT of absorpTion is necessary. The
soluTion reduces To ThaT derived by Rayleigh in The case ThaT The

maTerial is elasTic.

7.l FormulaTion of Surface Wave Problem.

To consider The surface wave problem iT is convenienT To
specify ThaT The second Terms in The general soluTions (3.74),
(3.75), (3.76) represenT plane waves wiTh aTTenuaTion direcTed inTo
The solid. To achieve This convenience The rooTs r¢ , r¢ and

r“ may be wriTTen as

r¢ 1: ba ,
(I)
r¢ = bk ,
and
ru 2 bu ,
where
_ Z _ Luz i/z
bq "' p~v.(ޢ :1) ,
_ a 2 1/2
hp :: p.v.(§y ‘ $5.2) ; (2)

75

and
_ z z i/&
bu=p.v.(ku-}B_z) .
B
Using principal values in This manner implies ThaT' b“a’ be and
a

bun are non—negaTive, and hence, The aTTenuaTion of each of The
second Terms in The general soluTions is inTo The solid. One can
reTain The form in which The rooTs were wriTTen for The reflecTion
problem (equaTion 6.|). However, This jusT leads To a number of
unnecessary cases for consideraTion.

For The disTurbance represenTed by The soluTions To be essen-
Tially a surface one, The ampliTudes of The disTurbance musT
decrease wiTh increasing disTance inTo The solid. WiTh The rooTs
r¢ , my and fi* wriTTen in The form (l) The aTTenuaTions asso-
ciaTed wiTh The firsT Terms in The general soluTions are direcTed
Toward The surface of The solid, hence for a surface disTurbance,
The ampliTude of These Terms musT be zero, i.e., Al 2 Bl : C. : 0_
The condiTion Cl::0 TogeTher wiTh boundary condiTion equaTion (5.6)
shows There are no surface SH waves in a viscoelasTic half—space.

For The posTuIaTed Surface wave disTurbance, The boundary con—

diTion equaTions (5.4) and (5.5) reduce To

2. z .
(2R‘fislAz=—21bfiflz82
and
. _ z a
thqkAz— (25k -Rs)Bz. (3)

EquaTions (3) involve Three complex unknowns .KJAZ, and Ba , Hence,
There is a unique soluTion for Two of The parameTers in Terms of

The Third. Assuming 81¢ 0, solving The firsT equaTion for Az/Ba 7

76

and subsTiTuTing inTo The second equaTion yields
2 2 z 2
4bpbgfi " (24% -RS) . (4)
For h To represenT The complex apparenT wave number of The
posTulaTed Rayleigh-Type surface wave, k musT saTisfy equaTion 4.
For & ¢ 0 The rooTs of equaTion (4) are The same as The rooTs

of The raTionalized equaTion

2 Z 2
4ii—faiwu—fi)“=(2~£=)2, <5)
2 £2 A:

where The square rooTs of each facTor are To be inTerpreTed in The
same sense i.e., (if one facTor is inTerpreTed as a principal value
Then The oTher TacTor musT also be inTerpreTed as a principal value).

Squaring each side of equaTion (5) and simplifying yields

2’2 3 92 a k2 W 3?:
(_LJ - 8(_£ ) + (Z4 -i6..£)(_3 ) ’ l6(l"_f i: 0.

2 2 z z 2 (6)
a *5 1» Des

EquaTion (6) may be wriTTen in Terms of The complex velociTies by
defining c E u:/.& and using definiTions (3.26) and 3.27). The
resulTing equaTion is

z z 2 fl1 2 z
c 3 .. c — __ __c_ - -5 = .
EquaTion (7) differs from The equaTion for elasTic media de-
rived by Rayleigh (l885, Eq. 24, p. 7) in ThaT real velociTies are
replaced by complex velociTies. WiTh complex velociTies The solu—
Tion To equaTion (7) is more complicaTed Than in The case of real

velociTies. FurTher implicaTions of The complex velociTies are

noTed in The nexT secTion.

/.7 Comparison ul

ilauiiu und Anelasiic Soiulions

To compare The form of The soluTions (3.74) and (3.75) in The

case of a surface wave on an aneiasTic half—space wiTh Their form

on a elasTic half—space, suppose for The momenT ThaT a complex R

can be found which saTisTies equaTion (4).

is convenienT

To express The soluTions in Terms of The propagaTion and aTTenuaTion

vecTors defined by

and

The soiuTions can now be wriTTen as

¢ =

and

Q :

AV,2

exp DA}:- F] exp [-i

8x? 1—K, - r1
2

HI

exp his“, ~71 .

(8)

(9)

For elasTic media The oniy Type of inhomogeneous wave which

can propagaTe is one for which The aTTenuaTion vecTor is perpendi—

cular To The propagaTion vecTor (refer To secTion 3.3).

For

anelasTic media This is The only kind of inhomogeneous wave which

78

can no: propagaTe. This disTincTion for The Two Types of solids is
a basic resulT. IT gives rise To a number of properTies for
Rayleigh-Type surface waves on anelasTic solids which are noT pre—
dicTed by The corresponding prOperTies of Rayleigh waves on elasTic
solids.

For example, in The case of elasTic media The soluTion for .k
is real which implies bq and b3 are real. In This case The pro-
pagaTion vecTors are parallel To The surface and The aTTenuaTion
vecTors are perpendicular To The surface. IT is shown in appendix 5,
ThaT no soluTion of The complex Rayleigh equaTion (4) for anelasTic
solids exisTs in which boTh of The propagaTion vecTors are parallel
To The surface. ln general, The resulTs of secTion (3.3) and
appendix 5 show ThaT if There is a soluTion of The complex Rayleigh
equaTion for anelasTic solids Then The propagaTion vecTors 5;: and
5kg are inclined wiTh respecT To The boundary and form acuTe
angles less Than TWZ wiTh respecT To The corresponding aTTenuaTion
vecTors Rt: and 754,2 ,

The boundary condiTions for a viscoelasTic half—space implied
.W¢==-fip (equaTion 5.7). As for The reflecTion problem (chapTer 6),
This relaTion implies The apparenT phase velociTy along The Surface
is The same for each of The displacemenT poTenTials. Also, The
componenT oT aTTenuaTion parallel To The surface u)/ fin is The
same for each of The displacemenT poTenTials. These relaTions are

valid for boTh elasTic and anelasTic media.

7.3 AnalyTic SoluTion of Complex Rayleigh EquaTion

The complex Rayleigh equaTion (7) is a cubic polynomial wiTh

79

coefficienTs in The complex field. Hence, The fundamenTal Theorem
of algebra implies equaTion (7) has Three complex rooTs, all of
which may noT be disTincT. The one of These rooTs which saTisfies
The original unsquared equaTion (5) will yield The soluTion for The
posTulaTed surface wave.

The Technique for finding The soluTion of a cubic polynomial
wiTh complex coefficienTs is well known in The field of pure maThe~
maTics (e.g., Birkoff and MacLane, p. Il2, I953). The same
Technique is applied here To solve The complex Rayleigh equaTion (7).

For noTaTional convenience, denoTe The complex coefficienTs of

equaTion (7) as follows:

0'
"I
N
-¢.

I
3

lb
"‘ N

and ((0)

z
-l6(l- 5 ).

‘—

0(2.

p
lll

WiTh This noTaTion The resulTing Cubic polynomial equaTion To be

solved is

y3 + Qya + by + c : 0 (ll)

1

where
Z 2
yr: C/P .
FirsT, by an appropriaTe TransformaTion equaTion (ll) may be

puT in a reduced form in which The resulTing equaTion conTains no

second order Term. The TransformaTion is

y = 2 - LL.
3 (12)

This change of variables applied To equaTion (II) yields The reduced

equaTion
3
Z+PZ+$=0, (l3)
where
_ 2
P = — £L. + b (l4)
3
and
12.2.0559; + c.
27 3 (IS)

A second TransformaTion may now be applied To The reduced
equaTion (I3) To yield a quadraTic equaTion. This second change of

variables, known as VieTa's subsTiTuTion, is

Z:‘2aJ-_f..
31d (l6)
Applying VieTa's subsTiTuTion To equaTion (I3) yields a quadraTic

equaTion in 205 , ThaT is

\w3)2-+ $13 —(§43 = O . “7)

The Third and obvious TransformaTion is 1): h)3 . The rooTs

of equaTion (17) in Terms of U are

_ I, 3
7J+ :—._§' 4» kg); + (a?) (I8)
and
= q, ‘i, a P 5
u , —__. — ___ __ .

(I9)

80

8|

The rooTs To The original equaTion (ll) are given by Tracing
The rooTs (l8) and (I9) back Through The Three successive Transfor-
maTions. The rooTs (I8) and (I9) TogeTher wiTh The subsTiTuTion
1/ = «u;3 show There are six rooTs for 1d and in Turn six rooTs To
The reduced equaTion (I3). IT will be shown ThaT aT mosT Three of
The six rooTs of The reduced equaTion are disTincT.

To express The six rooTs of The reduced equaTion, iT is con-
venienT To inTroduce The concepT of The principal value of The cube

rooT of a complex number 2 , defined as

1/3

E V-—- - (20)
p.V.(Z ) lzl exptiarg321]

where

-iT<argz.eiT.

Using principal values, The six rooTs of The reduced equaTion can be

expressed as

2; 2 wjun” — P (n: 1,2,3) (2')
Elwiun‘l
and
_ _ _ n-l
2n .: we u - ___F’___n__ ln=l,2,3), (22)
3 15 u
where
l
w; = p.v.(‘U+) ’3 (23>

and

‘0

.v. (4)-)”. (24)

Using The idenTiTy

v, 2/.

ll

3
~(p/3) , (25)

iT can be shown (appendix (6) ThaT each of The rooTs z; (r): l,2,3)
is equal To one of The rooTs 2;. (n = h 2, 3) .

Hence, The Three
rooTs of The original equaTion (II) are

m
N
l

y. ' %? , (26)
YZEZZ"%: <27)
and
ysi 2;-%. (28)

. z
The rooTs of The original equaTion (7) expressed in Terms of B /o<Z
are

2 3 VB n—l
yn=p.v.i‘&2—‘l’(%) +(—;—)) U.

“(P/H3p.v.(‘_g'z_’ (‘1’): P

__ , (29)
5
where
P E- 2 — [6.41.2
5 0(2
and

(30)

83

N

(la—Z7 -3195.
27 3 «2
IT is shown in appendix (7) ThaT The parTicular rooT given by

equaTion (29) for which lc2|/|lecl saTisfies The original equaTion

(4) for The posTulaTed surface wave.
7.4 ProperTies of a Rayleigh—Type Surface Wave for" J\ = 11

The assumpTion ‘An 2 11k is ofTen made for elasTiciTy pro—
blems, especially Those relaTed To geophysics. The assumpTion ofTen
gives rise To a considerable simplificaTion in formulae, and is a
good firsT approximaTion for The elasTic response of many earTh
maTerials.

An exTension of The assumpTion for viscoelasTic problems is
wa= 11R and }‘r = 11, . This assumpTion implies The following

relaTions beTween The parameTers of The solid:

K = .3.” , (3|)
2

32.; : l/5 , (32)
q

vp = Y’i‘vs, (33)
aP= (ls/[3", (34)
-l__ —i _ —l

Os — Op — OK , (35)

Tom 6K = Tan 85 , (36)

84

and
2
;&f : mkpg £5 = _L_.
z 5 (37)
1725 a,“ fist

SubsTiTuTion of The relaTion Bé/dihg inTo The complex Rayleigh
equaTion shows The coefficienTs of The equaTion are real. In facT,
The resulTing equaTion is exacTIy The same as The one for an elasTic
half-space wiTh The assumpTion JKR =,IIR . Hence, The rooTs for
Ci/fiz for .K.=JI should be The same in The general viscoelasTic case

as in The elasTic case. The rooTs in The elasTic case are 4-,

2. _2.
2 + 7%; , 2 f?‘ (e.g., Bullen, I965, p. 90). As a check on The
general soluTion, The resulT §L==% implies
s 5 a a 4 . - 3/2 —in74
=_a :3; P3 Q, _-2 :'Z_ [+1 Uzi e.
P -7; 1 Q 3, I EEJ T F5) -‘33 ) U— ‘;5( )1 o 3
(380
SubsTiTuTion of (38) inTo (29) shows
Yz=4-0 I (39)
= 2 + 3L
Y2 [5— I (40)
and
=2—_é_ .
Y3 v.3— (4|)

Hence, The rooTs derived from The general formulae for The visco-
elasTic case when }(=:}I are indeed The same as in The elasTic
case.

SubsTiTuTion of The Three rooTs (39-4l) inTo The original
equaTion (5) shows ThaT The only rooT which saTisfies The original

2
equaTion is )3 = 2 _Y%" Hence, .9 for The posTulaTed surface

85

LY'JA -
r3— 5 (42)

wave in The case )L =,LI is R2 : ( Z —
A Rayleigh-Type surface wave is an inhomogeneous wave. Hence,
we would expecT iTs properTies for an anelasTic half—space To be
differenT from Those for an elasTic half-space (refer To secTion 3.3).
Before deriving These disTincTions for a general anelasTic solid, iT
is helpful To consider firsT The disTincTions for anelasTic maTerials
which saTisfy A = ,u.
Recalling ThaT .W was defined as a principal value and using

relaTion (6.2l), iT follows from relaTion (42) ThaT

4% :: l-0877 “RS . (43)

The complex wave number .R for a homogeneous plane 3 wave is

relaTed To The velociTy v and The absorpTion coefficienT Q5 for

s
such a wave by QS::%%-ifl3 (equaTion (3.29). Hence, The velociTy
S

of The Rayleigh—Type surface wave along The Surface expressed in

Terms of The velociTy of a homogeneous shear wave is

v _=. w/RR: 0.9/94 v5 .

R (44)

The componenT of absorpTion of The Rayleigh—Type surface wave along
The surface, expressed in Terms of ThaT for a homogeneous shear wave
is
Q E —& : L0877 a ,
R ‘ s (45)

lT can be shown (appendix 8) ThaT relaTions (6.21) and (43)

imply

b = 0.92l8 225

°‘ (46)

86

and

b = 0.4278 Rs.

5 (47)

SubsTiTuTing (46) and (47) inTo The definiTions (8) shows The
propagaTion and aTTenuaTion vecTors expressed in Terms of The para-

meTers for a homogeneous shear wave are

5952 = l.0877 45‘? Q. + 0.9218 925: $2, , (48)
54,2 2 1.0877 ks“ 3?. + 0.4275 155123 , (49)
34,1: —l.0877 4513?, 1' 0.4218 Lisa)?“ (50)
74);: 4.0877 2513?, + 0.4278 125an _ (5|)

Several inTeresTing properTies are apparenT from equaTions (48) - (5|).
FirsT, consider The propagaTion vecTors E; and 5; given by
2. Z
equaTions (48) and (49). They are noT parallel To The Surface unless

% = k? = 0 , ThaT is, unless The maTerial is elasTic. In

51. I

addiTlon &s

I

x

< 0 and Rs z_0 (equaTions 3.33 and 3.34) imply The
R

propagaTion vecTors are inclined Toward The surface. The acuTe

angles which The propagaTion vecTors make wiTh respecT To The posi—

Tive x, axis are given by

Tom ep 2 0.8475 (RSI/1125“) (52)

and

fan as a 0.3933 (fir/925R). (53)

87

From equaTions (52) and (53) iT follows ThaT

lesl < l epl (54)
for anelasTic solids and
as = 6P = 0 (55)
for elasTic solids. From relaTion (3.32) iT follows ThaT
R’s: / 3’s: s '- (56)
Hence, The angles 8? and es are bounded by
~4o.2a°= Tan“'(-o.8475) s e? s o (57)
and
— 2i.47°= ran"<—~o.3q35)s as e 0. (58)

in oTher words, relaTions (57) and (58) show ThaT for anelasTic

.4

solids such ThaT .A.= ll The propagaTion vecTors 5; and sz
z

cannoT be inclined Toward The surface aT angles greaTer Than 40.3

and 2|.5 degrees, respecTively.

A

Second, consider The aTTenuaTion vecTors ‘A¢E and l§¢ given
2
by equaTions (50) and (Si). Clearly, They are noT perpendicular To

The surface unless Q 2 g
SI %

Again, Q5 5() and 25 2C)imply They are inclined away from The sur—
1: R

= 0,i.e., unless The solid is elasTic.

Tace inTo The solid. The angles which The aTTenuaTion vecTors make

wiTh respecT To The X3 axis are given by

Tan 6P 2: 1.1800 (" R731 / 95R) (59)

and

Tan 95 : 2.5425 ('ksr /R)sn)'

(60)
From relaTions (59) and (60), iT follows ThaT
8p < 85 (6|)
for anelasTic solids, and
99‘ ‘95: O (62)

for elasTic solids. RelaTion (56) implies The angles are resTricTed

by

O
M

9,; s mn"ii.ieoo> = 49.72" (63)

and

o .4. 95 s mn*‘(2.5425) = 68.53". (54)

RelaTions (63) and (64) show ThaT The aTTenuaTion vecTors 2i¢2 and

h.‘

f‘wz cannoT be inclined wiTh respecT To The >8 axis aT angles more

Than 49.7 and 68.5 degrees, respecTively.

The angles beTween The corresponding propagaTion and aTTenuaTion

vecTors are given by

D]; = iT/Z ~ 6,, + lap! (65)
and

B; = TT/Z '* GE 1- lesl J (66)
where 6%, ep, 95 ) and as are given by equaTions (52), (53), (59),

88

8 0

and (00), rospecfively. lnequalilies (54) and (6i)

imply
6’s 4 tip (67)
for all anelasfic solids, and
X5 : KP = TF/Z (68)
for elasTic solids.
The angles Up and U; are also given by
C05 d A z — = ..
a"? l P¢al lA¢al Jig,“ )5! Jean lbs: / 3 (69)
and

Expressions (69) and (70) are consisTenT wiTh equaTion (3.39) derived

for a general plane wave. As before, expressions (69) and (70) show

7? 4 ma <71)
and
65 < TT/Z (72)
for anelaslic solids and Thai
Zip: T5 = lT/Z (73)

for elasTic solids.

The angles §,and 5; associaTed wiTh The corresponding

iflhomogeneOUS waves cannol approach zero.

The minimum values They

90

can assume follow from The inequalifies (63), (64), (7|), (72) and

(73), Thaf is

O

40.280 < zrp s 90

and (74}

al.26" < a, s qo"
The parTicle displacemenfs associafed wiTh a Rayleigh-Type
surface wave on a viscoelasfic haif—space for which IL=_II can be
derived from expressions (48) - (5|) and expressions (3.77) and
(3.78). The parTicle displacemenfs expressed in Terms of fhe para~

meTers for a homogeneous shear wave in The given solid are

Reiu.) = L48“? ”5118;! ei.0911&5tx.{é.0215&,nx3

sin (u)+ —L0877 ks‘x, —.<22181b$‘x3 +<1rg Res + 5)

~ 0.5775 2‘4278 3%.":

sin (a)? —- L087? R51: 34' - 0.4278fiszx3 + Org 5&5 + 5 7}

(75‘)
and

1.0517 “ks: x, e_.9al8 ks“ x5

news] = 1.4319 {ksllBai e {—3475

cos(wf -{.0877fle x,—.4218& x +ar J? + )
5“ SI 3 g s g

—.7
+ L4679 e 42 8 khx5

C05 00+ ~ [.0877 ksnx, - 0.4278 92,: x3 + arg ks +g)},
{76)

9!

where

52 Gr c-ia ) g ran"isz-iB,J/Ret—i8 1).
9 a z

EquaTions (75) and (76) show:

a) The Two sinusoidal Terms of each componenT of parTicle dis-
placemenT have a posiTive componenT of propagaTion in The
—X3 direcTion, unless The solid is elasTic,

b) each of The parTicle displacemenTs decay aT The same exponen—
Tial raTe along The surface, wiTh an apparenT absorpTlon
coefficienT of l.0877CLs unless The solid is elasTic,

c) each Term for each parTicle displacemenT decays exponen-
Tially away from The surface aT a differenT raTe, and

d) The componenTs of propagaTion in each of The sinusoidal
Terms cause The ampliTude disTribuTion for anelasTic solids
To display differenT characTerisTics Than when The compo-
nenTs of propagaTion are zero as for elasTic solids (see
secTion (7.5) for more deTails).

AT The surface, X3: 0 , and The expressions (75) and (76)

reduce To

Re: u.1 = .5265 lel lBal em" ks2"‘sm (wT-l.0877 ksax, + my res + g)
(77)
and
, k
Rams] : ,9lq4 lfisl {Bil 9‘0577 szxi cos (Lo-1' ~I.0877&5Rx, + (”9 R314).

(78)

92

Expressions (77) and (78) show ThaT aT The Surface,
a) The parTicles describe a reTrograde ellipTical orbiT wiTh
Time,
b) raTio of major To minor axis is l.4679, and
c) The verTical axis of The ellipse is parallel To The X3 axis,
i.e., There is no TilT of The ellipse.
These resulTs are True for boTh elasTic and anelasTic solids and
agree wiTh Those derived in The liTeraTure for elasTic solids (e.g.,
Ewing, JardeTzky, and Press, l957, p. 33). General expressions for
The TilT of The major axis are derived in secTions (7.5). These
expressions show ThaT for The case }\=:,L[ The TilT of The major
axis varies wiTh depTh Tor anelasTic solids, buT remains zero aT all
depThs for elasTic solids.
The expressions for The parTicle displacemenTs in Terms of The
apparenT wave number along The surface .RR and The apparenT

absorpTion coefficienT — fix are

a .—.8 75.k
Rem.) = :3525 lfillaale ‘x' {e 4 R x

3

51.n(w+« fink, — .8475 fig x3 + arg «925 1- 5)

- . (79)
—0.5773 (2'3933 ”2“?) ,

wn(w1 - kRm ~.3933 filxs +-arg Q5 T'f)
and
—. k
Rams] = L3625 m £821 e“! x'{—.8475€ 8‘75 “‘3
cos(w+-—&R&—a8475U&x5+<ng &s+-§) (8m

—.3933 k x
+ ,,467q e R s COS(wT dim—39551255 + are} 3. +51],

93

For an elasTic half—space £1: arg % = o , and expressions (77)
and (78) reduce To Those given in The liTeraTure. (e.g., Ewing,
JardeTzky, and Press, I957, p. 33).

The preceding resulTs for anelasTic solids are independenT of
The amounT of absorpTion specified for The solid. For The case
}\=,L[ and The assumpTion of low—loss, The resulTs of Press and

Healy (I957) imply

aR : 1.0877 as. ‘ (8i)

EquaTion (45) shows This resulT is True independenT of The low—loss
assumpTion provided (1.5 denoTes The arbiTrary absorpTion coefficienT

for a homogeneous shear wave.

7.5 ProperTies of a Rayleigh—Type Surface Wave for an ArbiTrary

ViscoelasTic Media

The velociTy Vk and absorpTion coefficienT an of The posTu—
IaTed surface wave expressed in Terms of The rooT c/A =‘Q3/J% of

equaTion (5) are

va=vs(F?e[-§s—] + F(0£'lImE%1)-' (82)

3

and

£3
:5
ll

(15(RcC—fi—s] - FLOS'IJImE-é;1) , (83)

respecTively, where

94

—l _ + O — l
F<Qsl=———3—-——-— , (84)

l+05 +l
Vs == velociTy of homogeneous shear wave, and as: absorpTion

coefficienT of a homogeneous shear wave.

The displacemenTs of a poinT in The medium wiTh respecT To The
axes of a fixed coordinaTe sysTem are expressed in Terms of The dis—
placemenT poTenTials by equaTions (3.77) and (3.78). SubsTiTuTion

of The soluTions (9) inTo These equaTions and simplifying wiTh

equaTion (4) yields

a . -
ReEu.] = lAal We "‘ { abs": Simw’r dim -b..,><3 “"9 5b ”"9 Ag)

 

15M”;
‘L.:._l.ebp“x 3 sinin— kalx —bp ”x + ar'gik 2"]
l1: l 4&2
+ Org «Q +— arg A2 )} (85)

and

_.

Retu31=jA2llfile§;Xi{_‘L’§-L e— b'laxs SinUU‘r—Rux. -b‘2x3 +4r‘9 b“ fora 91 r.)

~—

:
lfi+fiHl_%x

#- ._______ Q R 5lh(wf— kaxl- b 5323 +arngJ+ar9 5:4...)
Zl§,fil }

(86)
where & saTisfies equaTion (4).

To describe The naTure of The moTion defined by equaTions (85)
and (86) iT is desirable To inTroduce noTaTion such ThaT The equa—

Tions may be wriTTen in a simpler form. EquaTions (85) and (86)

are of The form

R: [11"] = D{F’n sinhuf + an + dm)

95

+6" sm(w1' + an + d5n’} (n:!,3),

where The noTaTion infroduced is

o.-—. IAZIIM 3""

I

 

-b‘ X b -bu
E E e a a, F; E _iénl 8 X3 ’
a 2 2
6,5 11% +6)?! e—b‘nx3) G E lfi+bpzléb,ax5 ,
mm 5 lab”:

QIE"I&RX’+0.P3%+QY‘9A2, (132091-121;

Equafion (87) may be rewriffen as (appendix 9)

(87)

(88)

96

ReLunJ: DH,1 sinth+an+6,.), (89)

where

Hn '—:. (Fnfi 3f + 2F" 6,. cos (d.,, — awn“
and

a : m“-.[r,, sin dm + (3,, Sin aw]
,, _

Fr, cos c1m + an cos :1an

The parameTers H" , an, and 9h depend only on The spaTial
variables. Hence, equaTion (89) is consisTenT wiTh The sTeadyesTaTe
assumpTion ThaT each parTicle exhibiTs a simple sinusoidal Time
dependence.

From equaTion (89) The normalized ampliTude disTribuTions aT a
fixed Time T , when The corresponding ampliTude aT The surface is

a maximum, are given by

.51 C05 (6n— 9n ) for n: l,5, (90}
H“ °

0
where The subscripT "0" indicaTes The corresponding subscripTed
quanTiTy is evaluaTed aT x3 = 0.
For an elasTic solid The normalized ampliTude disTribuTions

given by equaTion (90) reduce To

F G
filcos(9n—en) _"_+_£ . ‘9')

H" ° fi1-r G"

0

ll

97

For an anelasTic solid 9" depends on depTh. Hence, equaTion
(90) shows The ampliTude disTribuTions wiTh depTh for a fixed Time
also exhibiT a sinusoidal dependence on depTh. This sinusoidal
dependence does noT exisT for elasTic solids. This resuIT can be
seen inTuiTively by considering The angles which planes of consTanT
phase and consTanT ampliTude associaTed wiTh The displacemenT poTen-
Tials make wiTh respecT To The Surface for The Two Types of solids.

The maximum ampliTudes during a cycle of oscillaTion normalized

by The corresponding maximum aT The surface are given by
Hn/Hn for n: l,3. (92)

For elasTic solids This ampliTude disTribuTion is equivalenT To (90).
The equaTions (89) describing The posiTion of a given parTicle

are of parameTric form wiTh parameTer fifidefined by

Jlf) E inf + aI + 9, . (93)

The equaTions(89) wriTTen in Terms of The parameTer f , which is a

linear funcTion of Time, are

Re[u,]

ll

DH, smifcri)

(94)
and

Rellu31= DH: cos (fir) + S) ,

where 5 depends only on The spaTial coordinaTes and is defined by

(95)

98

lT is shown in appendix IO ThaT The parameTric equaTions (94) aT a
fixed poinT in space represenT The equaTion of an ellipse as a
TuncTion of Time. The following properTies of The ellipTical moTion
are also derived in appendix IO:
a) The direcTion in which The parTicle describes The ellipse
wiTh Time is eiTher,
i) counTerclockwise (i.e. reTrograde) if cos [5) > 0;
ii) clockwise (i.e., prograde)if C0$(.5)<Q or
iii) in a sTraighT line if cos<S) :0:
b) The TilT of The ellipse K defined as The acuTe angle Through
which The coordinaTe axis musT be roTaTed in order To coin-

cide wiTh a principal axis is

2 SinlS) if H, ¢ H3

_H_5-H.

Hl H5

Tan 2 X : , (96)
fl’ If H = H
.— I 3
4
where negaTive values of 3’ are illusTraTed.

 

99

c) The ellipTiciTy of The ellipse defined as The raTio of The

lengThs of The principal axes of The ellipse is given by

A/c ,
(97)
where
A]: Acosz U + Bsnwt cost + Csmg I
(98)
C = A snwz X ‘ 8 sin 6 cos I +— 6 cos2 r
and
_ a
As (DchoslSH
B = 25’n(5)(DzH H cosaisn"
— ' v 5 (99)
_ -2
For an elasTic solid
dmn = 0 (wwzi,3 ; n=l,5 ),
0 For Fn+Gh>0
9n: ] ,
n For Fn+ G" < 0
(I00)
S: 9 —9 6:0

|OO

and

Hence, for an elasTic solid
a) There is no TilT of The ellipse aT any depTh,
b) The ellipTiciTy of The ellipse is given by
*3 /Tfi = 'Fsi'Gbl/ IF,+ G.l , and
c) The parTicle moTion is reTrograde aT The surface down To a
depTh aT which Fl+ GR = 0 and prograde below This depTh.
For an anelasTic solid, S depends on depTh and equaTion (96)
shows The TilT depends on depTh.
For solids such ThaT q": 03 = 0, , Lemma l (appendix 2)

K

shows The coefficienTs of The complex Rayleigh equaTion are real.
LeTTing p)(§:) denoTe The lefT-hand side of equaTion (7), iT follows
ThaT p(0) is negaTive and p(l) is posiTive. Hence, equaTion (7) has
a real rooth/flz such ThaT o<’§:<l . From which if follows ThaT

The original equaTion (5) has a real rooT. More specifically, we

have esTablished The

THEOREM: lf qu = 654 Then The rooT c775a associaTed wiTh
2

C

a Rayleigh~Type surface wave is real and O< .3 < L
)5

AT The surface of solids for which 6§4 : Gk?! , The preceding

Theorem implies

5; n=|,3) (IOI)

lOl
and

Fn + G" > 0 (n = l,3 ).
(l02)
Relalions (IOI) and (l02) Togelher wllh deflnllions (89) and (95)

show Thaf for such solids lhe fill of The parlicle molion ellipse

aT The surface is zero.

lOZ

7.7 References

Birkoff, G. and S. MacLane, A Survey of Modern Algebra, Macmillan

 

“)55.

Bullen, K. E., An lnlroducfion lo +he Theory of Seismology, Camb.

Univ. Press, Third ed., I963.

Caloi, P., Comporfemenf des ondes de Rayleigh dans un mileu firme-

élasfique lndéfini, Publs. Bur. Ceniral Séism. lnTern., ll,

 

pp. 89—!08, I948.

Ewing, W. M., W. S. Jardelzky, and F. Press, ElasTic Waves in

Layered Media, McGraw—Hill, I957.

 

Rayleigh, Lord, On waves propagaled along The plane surface of an

elasfic solid, Proc. Roy. Malh. Soc., ll, pp. 4-ll, I885.

 

VlcTorov, l. A. (lrans. by W. P. Mason), Rayleigh and Lamb Waves,

 

Plenium Press, I967.

IOB

8. GENERAL NUMERICAL CHARACTERISTICS OF A RAYLEIGH-TYPE SURFACE

WAVE ON A HILV HALF-SPACE

The numerical compuTaTion of The properTies of a Rayleigh—Type
surface wave on a HILV half-space depends upon The compuTaTion of The
Three rooTs of equaTion (7.7). The Three rooTs can be readily compuTed
by a digiTal compuTer using The analyTic soluTion (7.29). The
parTicular rooT cz/fi2 which saTlsfies 0 < ICZ/ 52I< I (appendix 7)
can Then be used To compuTe The properTies of The desired surface wave,

To compuTe general numerical characTerisTics, iT is desirable To
perform The calculaTions as a funcTion of parameTers which are
independenT of a parTicular viscoelasTic model. IT is also desirable

ThaT The chosen parameTers have an immediaTe physical inTerpreTaTion.

8.l ParameTer Analysis

The parameTer in The complex Rayleigh equaTion (7.7) which

characTerizes The maTerial is fizi/cxz which wriTTen in Terms of The

complex bulk and shear modulii is
2
A =5.
a2 K (I)
Defining
“Vi—211R <2)

ZISKR +LCR)

[04

and using equaTions (4.|5) and (4.34) equaTion (i) may be wriTTen as

“43;: (l+i05")/(—§—(.:_:’_%E_)(l+i0;')+.§_(l+o;')}, (3)
EquaTion (3) suggesTs compuTaTion of The rooTs cz/fia of equaTion
(7.7) as a funcTion of The Three parameTers G:', 0;, and U .

Physically, GE' and o;' are proporTional To The fracTional
loss in energy denslTy per cycle of forced oscillaTion in bulk and
shear, respecTively. U'(definiTion (2)) is a funcTion of KR and
IIR. The parameTers HR and 14,are proporTional To The peak energy
sTored during a cycle in bulk and shear, respecTively (equaTions
4.l3 and 4.33). Hence, physically U depends only on The capaciTy
of The maTerial To sTore energy. 0’ as defined for viscoelasTic
media (definiTion (2)) is equivalenT To The Poisson's raTio as
defined for elasTic media.

Numerical calculaTions as a funcTion of The Three parameTers
le 1 Q;' and U are general in The sense ThaT They are applicable
To any viscoelasTic model. For consideraTion of a parTicular visco-
elasTic model only The frequency dependence of each of The parameTers
need be specified (e.g. see. Table l, secTion 2.4).

For general numerical consideraTions of viscoelasTic behavior,
There is no a priori reason To assume ThaT The behavior in bulk is
relaTed To The behavior in shear. Hence, for numerical analysis of
anelasTic response each of The parameTers may independenle assume
any value in Their respecTive numerical ranges. The numerical range

for each of The Three parameTers Q;', <?;'

and a‘follows from The
range esTablished in secTion (4) for each of The parameTers KR, LI,

KI and c . The range for each of These parameTers is The half—

IOS

closed inferval [0,00] . Hence, The range for 0;' and Q;' is
[0,ad and lhe range for G'ls L-l ,l/ZZJ. Love (I927, p. I04)
remarks +ha+ for elaslic solids "Negafive values of G‘are nol
excluded by condilions of sfabili+y, bul such values have no+ been
found for any isofropic malerial." Experimenfs show ThaT for many

elasfic solids G'is near 0.25.

8.2 Velocily and Absorpllon Coefficienf

Wilh +he appropriafe roof R/ksof equafion (7.5) compuled as a
funcfion of The +hree paramelers 0;”; Q;' and U, The velociTy
and absorpfion coefficien+ of a Rayleigh—Type surface wave can be
compufed as a funclion of fhe corresponding velocify and absorpfion

coefficienf of a homogeneous shear wave from

 
    

,Vg, = m .8: + NO") Im .43. )"
V5 4125] s [*5] <4)
and
a ._ I: _ -' i
71-“ - (Re [7;] H05 ”Mg“ ’ (5)
5 S
where
I -2
HOS"): 295—; . (6)
l|+q§2 r I
For solids such Thal 6§4 = Q;' +he Theorem in seclion 7.5

shows +he approprial‘e roo+ 31/215 is real and depends only on a” .

Hence,for such solids equalions (4) and (5) show vR/g and uR/03

lOo

depend only on a' and are independenT of The amounT of absorpTion.

8.2.l HypoTheTical MaTerials. Before considering curves of vR/\g
and (la/aS which are useful for realisfic maTerials, iT is
insTrucTive To consider curves compuTed over The enTire range of
each of The parameTers. A family of curves for vR/vg wiTh each
curve corresponding To a parTicular value of Q;' has been compuTed
as a funcTion of G'and a fixed value of Q;'(figures l-S). As

shown analyfically in The previous paragraph The curve corresponding
To 0;. = 0;, is The same regardless of The amounT of absorpTion.
The curves for small amounTs of absorpTion (G§'< O-l and 0;|< 0J3
are essenTially The same as The curve for di :2 0;]. ln addiTion
iT is apparenT (figures l-5) ThaT The velociTy of a Rayleigh-Type
surface wave for The range of parameTers considered is less Than

The velociTy of a homogeneous shear wave in The same maTerial.

A similar family of curves has been compuTed for aR/£k(figures
6-l2). Again, as shown analyTically, The curve for qu :: 0;' is
The same regardless of The amounT of absorpTion. The large values
of The raTio QK/(g(figures 7-ll) correspond To hypoTheTical solids
for which q;' < 6§4_ For cerTain hypoTheTical combinaTions of
The parameTers 0-, G§4 and (2;, The absorpTion coefficienT for
a Rayleigh—Type surface wave can be less Than ThaT for a homogeneous

shear ane (figures 6—9).

8.2.2 Low Loss MaTerials. The amounT of absorpTion for a large
number of common maTerials is small. In facT, The majoriTy of The

presenT TheoreTical liTeraTure is based on This assumpTion (e.g.

VR/Vs

 

LOO I

0.9 0

0.70

1*TrT

 

co
IIIIITVIVTIIVV'Vrrilvilr'vvrv'vvivlvrvi

 

0.60 1

 

 

 

lll‘lL‘LLJAL‘LlJAJ_IIAJAAIIJJA‘L4AAIIAIA

 

 

. . . J
'l.0 '05 0 0.5

FIG. I. RaTio of +he velocify of a Rayleigh—Type surface wave

To The velocify of a homogeneous shear wave for hypofhefical
maTerials wi’rh o;' = 0. '

LO!

LOO

0.9 O

(I!

§ 0.80

0.70

0.60

 

r
-<
.
.1
4
—‘
4
.4
4
.4
4
-—1
4
q
.
.1
.1
.1
..
-q
.
.—
...
4
.1
.4
.1
..

 

 

 

 

I'TTTW’VYTFI’I’VTTTYYY‘I’YYY' [TvaIYfYrrf‘rf
LLLLLLLLJLJLLLLL‘ LA LAALLJL4411111AJ LALLL

 

 

 

FlG. 2. Rafio of The velocify of a Rayleigh—Type surface wave
To The velocify of homogeneous shear wave for hypofhefical
maferials wifh Q§’ = IO'I .

80!

LOO

 

0.90

O. 70

0.60

_
j
_.
4
4
.4
_
_.
.1
1
.,
L
j
J
_
L
..
..
L

 

 

 

 

frrf‘rrv‘frr‘rfvv l v vrv l‘rffrr1fv’rl'1ffrrT1’1'T
LL‘LLLLALJ‘LIAALLIA ALLLLLAJILLLLLLLLJILALLL

 

 

.-
.
.
.—
.
.—
.
‘—
—
.
>-
r
y—
,
.—
.
n—
P
b-
y.
.—
.-
.-
..
.—
p
h-

 

4
o
I
9
0|
0
o
U!

FIG. 3. Rafio of The velocify of a Rayleigh—Type surface wave
To The velocify of homogeneous shear wave for hypofhefical
ma+erials wi+h Gfi" = I.

60!

LOO

090

u,080
3:.

070

050

 

 

 

 

 

 

 

 

. I r I I I I 7 ’17, I I I I I . T ti ‘1'" I I I I r 1 I I l J
" 4
L 4 4
. °K"O .
:_ oglfl 1
. ID" A 4
» 0 ”50* 4
r IO 0“: .
P— —!
. 4
L 4
r- 4
l- <
>— -I
L- 4
. 4
I. .
L . *
10 -
L 4
r- 4
> 1
~ 4
p 4
b 4
. 2 4
l
L 3 4
h '0 <
—’ 10‘ «
b I
- 1
1. 4
-- —<
b 4
1. 4
L 4
L 4
I .1 . 1 1 .1 . l. 1 .1 . 1 .1 . l. 1 .1 . 1. 1 LI
‘1 O “0.5 0' O O 5

FIG. 4. RaTio of fhe velocify of a Rayleigh—fype surface wave
To fhe veloci+y of homogeneous shear wave for hypoThefical
maferials wiTh 0.2' = IO.

Oll

 

LOO

090

070

 

 

 

 

 

 

rTTTfV—TrTrrrfT rfol I I lTTI’YrTITTrTI‘rrY v

 

LLLLLLLLJLALLLLLLLLJ‘LLLJLJALJILLLI‘LLL

 

050

y.—
.
.—
y.
,—
y.
.-
—.
.—
v
.—
h
.—
l-
p.
.
-
.—
.
,-
.
y-
.—

 

i
O
I
9
u
0
O
u

FIG. 5. Rafio of The veIOCITy of a Rayleigh—Type surface wave
To fhe velocify of homogeneous shear wave for hypoTheTical
maferials wiTh O;' = IO .

Ill

 

 

 

 

 

 

 

I I 1 V l I l I l I l 1 I fl I ‘ i 1 I I ‘ V I 1 l I r I l I l
L l
o" = 0
, K 1
IOO ‘— —1
I 1
: 1
‘ T
. 1
1
at
Q
C > 1
‘ 1
: 1
‘ 1
‘ 1
Q§LL=102
m° Id fi:::;%
1— -
L IO'S. [0-4 ..... .104 1
> 1
1 1
‘ 1
y. 4
I .1 .1 .J .J .J .1 .1 .1 .1 .l .1 .4 .1 .1 .J
-|.O “0.5 O 0.5
cr
FIG. 6. Ra+io of The absorpTion coefficienf for

a Rayleigh—+ype Surface wave +0 +he absorpfion
coefficienf for a homogeneous ehear wave for
hypofhe+ica| maferials wi+h Q<' = O.

 

 

 

 

 

 

IVIVII'V‘IIIIWI‘l'l'IIlIlIl'I'l
o-s' =10" a: = :04
" .4
IOO—- _
IO’a
c3“
\IO— _
It. .4
O ,. -
r .
' Io* ‘
—_______ IO"
| M
T“ o“
r '0" ?
Li,n.1.1.lln.z.1.1.1.1.:.LJ:.I
-Lo -05 o as
0'

FIG. 7. Rafio of +he absorpfion coefficienf for
a Rayleigh-fype surface wave +0 The absorpfion
coefficienf for a homogeneous shear wave for
hypofhefical maferials wi+h Q:' = IO‘3.

 

 

 

 

 

 

 

'.11‘.,..rl.,.,.,.,.'.l.,‘.v‘rl
03:.0-5 0:510‘2
F 4
IOOP— _
v- 4
p 4
no“
as
0
\IO— _.
m.
o . 3
. .1
IO‘3 -
‘ IO‘
.‘2
\~\1QO \
'I‘ no" ‘\/‘.
v- d
. .4
d
1.141.].1.lj141.1.1.1.111.1LI.1
“L0 ‘05 0 (15
a-

FIG. 8. Ra+io of fhe absorpfion coefficien+ for
a Rayleigh-+ype surface wave +0 +he absorpfion
coefficienf for a homogeneous shear wave for
hypofhefical maferials wiflw O;' = IO—a.

114

 

 

 

 

 

 

'.,.11,—.,VI.I.T1,.1.T.,.].,..1l
* -|. -I -4
Q§'=IO" °x "0
'00— w
A
d
10”
* .
U)
‘\|O—- -—
a: ~ ‘
O > 4
IO’2
02
0‘ IO' \
‘—-—‘
I:- I00 I'4
r .
L. .
l.n.|.1.|.l11.1.1,1.l.111.1.14J
-O.l *0.5 O 0.5
0‘

FIG. 9. Rafio of The absorpfion coefficien+ for
a Rayleigh-+ype surface wave +0 +he absorp+ion
coefficienf for a homogeneous shear wave for
hypofhe+ical maferials wi+h Qfi" = 10".

 

lvli‘wjvlv—lv‘vlwlv‘vlvxv‘u'vrfi

-1 _ O .1
0K ~10

Q§=IO‘3

 

 

 

411111

 

 

[41.111.1.l414111.11141.11111.

”DJ “0.5 O 0.5
0'

 

FIG. IO. Rafio of +he absorpfion coefficienf for
a Rayleigh-fype surface wave +0 The absorpfion
coefficien+ for a homogeneous shear wave for
hypofhe+ica| maferials wi+h Q;'= I.

Hf)

 

 

 

 

 

 

[.1.‘.,.1YI.I.,.,.,.lj,.1.,.ljl
-l l
OKcIO
IOO— -.
- -1
r -
-u- —z ‘
. Qs—IO -
U!
Q
«'0.— 1
o r -
' I0"I ‘
P -
r a
. 3 .
o I
IO 10‘ _¥ 07
. [0m
|’"‘ —-4
r .
p 4
111:1.111.141.J.1.1.1414111‘14J

 

-O.l '05 0 0.5
0’

FIG. II. Rafio of The absorpfion coefficien+ for
a Rayleigh-Type Surface wave +0 The absorpfion
coefficienf for a homogeneous shear wave for
hypofhefical maferials wiTh 0;’= IO .

|l7

 

IT'I'VII'VIVTVIIIIrTIYlIlI'r'v‘!

0:3:qu

I00

7 fifijff[

03/05
6
I

11111]

p

L

y- ..I, -l
+ 05 IO
’-

 

 

 

I.1411141.1L1.1.1.1.l.141.44|.l
"OJ 0.5 0 0.5
0'

 

FIG. [2. Rafio of fhe absorp+ion coefficienf for
a Rayleigh-Type surface wave +0 The absorp+ion
coefficienf for a homogeneous shear wave for
hypofheflcal mafer‘ials th OK": 102.

l19

Kolsky, I954). The assumpTion is ofTen noT sTaTed explicile and
confusion resulTs regarding The correcT expression for Q". For an
arbiTrary amounT of absorpTion, The correcT expressions for Q;' and
9;! are (4.24) and (4.34), respecTively. For maTeriaIs saTisfying
The low—loss assumpTion These expressions are approximaTely equal
To (4.25) and (4.36), respecTively.

From laboraTory measuremenTs, a liTeraTure survey by Knopoff
(I964) shows values cfi Q"'ranging from essenTially 0 To 0.027 for
various rock Types and Types of glass. He reporTs a larger value
of 0.I43 for celluloid. VikTorov (I967) reporTs laboraTory
mea5uremenTs aT I megacycle of

" 4.| x IQ-3 for a meTal (Dural),

i) 05 = 3.3 x 10'3, og‘ =
ii) q;' = 5.3 x l0-3, 0;” = l0.2 x I0.3 for a glass (mirror
glass), and
... —l —3 —l -3 .
Ill) Cg = 3l.0 x I0 , Qp = I2.9 x IQ for a plasTIc

(polysTyrene).
From in—siTu measuremenTs for The lower crusT and manTle of
The earTh, a recenT IiTeraTure survey by Jackson and Anderson (I970)

reporTs values for le ranging from 5 x IO-4 To I x IO—Z, and

values for Qfid ranging from 2 x IO-4 To 3 x [0-2.

From in—siTu measuremenTs for surface earTh maTerials, Knopoff's
survey (I964) shows values of Q5' ranging from 0.0I79 for magneTiTe—
hemoTiTe To O.l43 for PoTTsville sandsTone. A value of Q;'= O.I is
reporTed for Pierre Shale.

HamilTon eT. al. (I970) wiTh The assumpTion of no absorpTion in

bulk reporTs values of q§* ranging from 0.555 To 2 for ocean boTTom

IZO

sedimenfs. (The characferisfics of such sedimenfs are similar fo
fhose of "Younger Bay Mud" found af The surface along The margins
of San Francisco Bay, California).

All of fhe preceding measuremenfs cfi‘ Q'lexcepf for celluloid,
Poffsville sandsfone, Pierre Shale, and The ocean boffom sedimenfs
are less fhan O.|. The curves for Vkl’vs of secfion 8.2.l suggesf
fhaf for such maferials fhe deviafions from fhe low loss approxima-
Tion are small. The analyfic solufion (7.30) for a Rayleigh—fype
surface wave allows an exacf evaluafion of fhe low—loss
approximafion.

Before considering The low-loss approximafion for a surface
wave, we shall derive fhe resulfs of The assumpfion for body waves.
The general expression for fhe velocify of a homogeneous shear

. —).
wave expressed In ferms of 05 IS

 

H —l
Vs: ._R. 6(05), (7)
P
where
(8)
For elasfic maferials or maferials for which 0;‘<< l (i.e. The low—

loss approximafion is valid) G(.Q§!) x>l and expression (7) reduces

+0

v aiLTR/P. (9)

Sn

Similarly, The absorpfion coefficienf for a homogeneous shear wave

may be wriffen as

l2l

u _ -
as: ($47; Gtos’l) ', (10)

and for IOeross reduces To

05“- - (zl’BPE/wos“)“, (ll)

From expressions (7) and (9), The expression for The percenTage by

H

which The general expression for The velociTy deviaTes from The low—

loss expression is given by

(V, " VSLL) '02

V%

H

(i— (mog'n‘moz, (:2)

where G(.Q;U is given by equaTion (8). Similarly for The absorp-

Tion coefficienT, expressions (l0, and II) imply
z 0" -‘ z
5 5 i : 5 _
(_u.____) 0 (E. GLOS) l) 10. (l3)

The expressions anologous To (I2) and (IS) for a diIaTaTional wave
are The same as (l2) and (IS) excepT ThaT a "p" subscripT replaces
The "s" subscripT.

The low—loss error percenTages (equaTions (l2) and (l3)) were
compuTed as a funcTion of GK“ (figures l3 and i4). For small
amounTs of absorpTion ( Q'I< 0.05, figure l3), The error percenTage
for The velociTy of a homogeneous body wave is less Than O.l percenT,
and The error percenTage for The corresponding absorpTion coeffici-
enT is less Than O.l6 percenT. The errors increase rapidly To much

larger values for Q") O.l. For example, for solids wiTh Grl> 0.6

l22

 

 

 

 

 

FIG. l3. Low«loss error percenfages
for The velocify and absorpfion
coefficienf of a homogeneous shear

wave wifh 0.005 s Q; s 0.05.

 

O
8 9
,1 A
.J
6” —:
I “m
.l >>
4 I
6’ i”
v

 

 

 

 

FiG. l4. Low—loss error percenfages for
+he velocify and absorpfion coefficienf
of a homogeneous shear wave wlfh

a675's~ Q"‘5 L0.

|23

i24

The error for The velociTy is greaTer Than IO percenT, and The
error for The absorpTion coefficienT is greaTer Than 20 percenT.
For many experimenTs such large errors are noT permissible.

Press and Healy (T957) were The firsT To derive resuITs for a
Rayleigh—Type surface wave on low—loss maTerials. They derived a
relaTion beTween The absorpTion coefficienf for a Rayleigh—Type
surface wave and The absorpTion coefficienTs for dilaTaTional and
shear waves. in addiTion, Their analyses showed ThaT The raTio of
The velociTies vR/vg is The same as for elasTic solids. Their low-
Ioss reIaTionship for The absorpTion coefficienT as laTer simplified

by MacDonald (I959) is

 

_ v _ v
aaLL—méap +(lm).\;s;as, (l4)

where

m:_: ace-bicl-b) (15)

ail-b)(l-b) - bll-ailZ-Sb)
z

0.5 ivR/vp) , (l6)

and
2
b 35 (wk / vs) . (l7)

The percenTage by which The velociTy “h for a Rayleigh—Type
surface wave deviaTes from The velociTy Va” for low-loss maTeriaIs

is

(Lin)102= ll—_\_/“_u.)l02. (l8)

V
VR R

l25

To evaluaTe (l8) as a funcTion of The parameTers 0;]; Q;'and

0' v / v can be wriTTen
’ “u. R

v v v v
__a_“ :. _iu. .3?“ _._‘__. ’ (lg)
V“ VSLL s V“
where
vR“_/ \GLL is The same funcTion of G as for elasTic
maTerials,
- —l —l
VSLL/VS .— (Gth J) ,
and
v3/ VR is compuTed from The general analyTic soluTion

as a TuncTion of Cg“ , Q;‘ and 6‘.

Expression (l8) was compuTed as a funcTion of Q§'for various

values of Q;' and for (T = 0.0, 0.01, 0.02, --—, 0.5. The curves
corresponding To (T=O.25 are represenTaTive (figure l5). More
specifically, The error percenTages for variOus combinaTions of

Q5” and Q;' are TabulaTed in Tables l and 2 for cr = 0.25.

The amounT of Tolerable error depends on The parTicular experimenT;
however, for many of The preceding measuremenTs an error of | percenT
can be significanT. Errors l percenT or greaTer are encounTered for
Q—

errors of less Than I percenT are significanT, Table I can be used

' greaTer Than O.l5 (see Table 2). For experimenTs in which

To esTimaTe The error in using The low—loss approximaTion for Wv/V .
The percenTage by which The absorpTion coefficienT OR for a

Rayleigh—Type surface wave deviaTes from The expression for The

30———

 

CT'= 0.255

 

 

l26

 

 

o k 0.5
og'

FIG. l5. Low—loss error percenfages for fhe velocify of a
Rayleigh—fype surface wave.

LO

Low~Loss Error Percenfages for +he Velocify

TABLE I

[27

of a Rayleigh—Type Surface

 

 

 

Wave, i.e.
(“R ‘ Va») .04 wifh o- = 0.25.
a
0.."
10a
qgl‘o 0.0 0.5 [.0 [.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
.5 0.09 0.09 0.|2 0.[6 0.22 0.29 0.39 0.50 0.63 0.78 0.95
[.0 0.36 0.36 0.37 0.4I 0.47 0.54 0.63 0.74 0.87 [.0[ |.|7
[.5 0.80 0.80 0.8l 0.84 0.89 0.96 [.05 [.[6 [.27 [.42 [.58
2.0 [.42 [.4[ [.42 [.45 [.50 [.56 [.65 [.75 [.87 2.00 2.[6
2.5 2.22 2.2I 2.2! 2.24 2.28 2.34 2.42 2.25 2.63 2.77 2.92
3.0 3.20 3.[8 3.[8 3.20 3.24 3.30 3.37 3.47 3.58 3.70 3.85
3.5 4.35 4.33 4.33 4.34 4.38 4.43 4.50 4.59 4.70 4.82 4.96
4.0 .5.68 5.65 5.65 5.66 5.69 5.74 5.8! 5,89 5.99 6.[| 6.25
4.5 7.[8 7.[6 7.l5 7.[5 7.|8 7.23 7.29 7.37 7.47 7.58 7.72
5.0 8.87 8.83 8.82 8.82 8.84 8.89 8.95 9.02 9.[2 9.23 9.36

 

Low-Loss Error Percenfages for The Velocify of a Rayleigh-Type Surface

TABLE 2

128

 

 

 

 

 

Wave, 1.e.
(v9 - VR“,)102’ for 0' = 0.25.
Va
o;'

9;! 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.05 0.09 0.11 0.21 0.37 0.56 0.79 1.03 1.27 1.50 1.72 1.91
0.10 0.35 0.37 0.46 0.61 0.80 1.02 1.25 1.49 1.72 1.94 2.14
0.15 0.79 0.80 0.88 1.02 1.20 1.42 1.65 1.89 2.11 2.33 2.53
0.20 1.38 1.38 1.46 1.59 1.77 1.98 2.21 2.44 2.67 2.88 3.08
0.25 2.12 2.12 2.18 2.31 2.49 2.69 2.92 3.15 3.37 3.58 3.78
0.30 3.00 2.99 3.05 3.17 3.34 3.54 3.76 3.99 4.21 4.42 4.62
0.35 4.00 3.98 4.03 4.14 4.31 4.51 4.72 4.95 5 17 5.38 5.58
0.40 5.09 5.07 5.11 5.22 5.38 5.58 5.79 6.01 6.23 6.45 6.65
0.45 6.27 6.24 6.28 6.38 6.54 6.73 6.94 7.16 7.39 7.58 7.80
0.50 7.52 7.48 7.52 7.62 7.77 7.96 8.17 8.39 8.61 8.81 9.02
0.55 8.83 8.79 8.81 8.90 9.05 9.23 9.44 9.66 9.85 10.09 10.29
0.60 10.17 10.11 10.13 10.22 10.37 10.55 10.76 10.98 11.20 11.41 11.61
0.65 11.53 11.48 11.49 11.57 11.71 11.90 12.10 12.31 12.54 12.75 12.95
0.70 12.91 12.84 12.86 12.94 13.08 13.25 13.46 13.68 13.89 14.11 14.31
0.75 14.30 14.22 14.23 14.31 14.44 14.62 14.82 15.04 15.26 15.47 15.67
0.80 15.67 15.59 15.59 15.67 15.80 15.98 16.18 16.40 16.61 16.83 17.03
0.85 17.03 16.95 16.95 17.02 17.15 17.33 17.53 17.74 17.96 18.18 18.38
0.90 18.38 18.29 18.28 18.35 18.48 18.66 18.86 18.07 19.29 19.51 19.72
0.95 19.71 19.61 19.60 19.66 19.79 19.96 20.16 20.38 20.60 20.82 21.03
1.00 21.00 20.88 20.88 20.95 21.07 21.25 21.45 21.66 21.88 22.10 22.31

I29

absorpTion coefficienT for low—loss maTeriaIs is

(l— QM) 102. (20)
QR

(-
P
D
V
ES
H

To evaluaTe (20) as a funcTion of The parameTers Qg', ;’ and U,

(LRLL/aR can be wriTTen as

52“.. an as“ _0_s_ , ' (2:)
an QSLL as R
where
a v 0"
i.:__s_i_ilm(_il~l)+i), (22)
5n. VRLL OS
-1 —)
9g,=(i+2(g+o~i)" fi‘+2<1+0‘)), (23)
Qs l - 2.0' s l ’ 2 O—
0 _ _ (24)
3:; = OS’G(05!)/2,
as
and
05/ QR is compuTed from The general analyTic
soluTion as a TuncTion of le, OJ‘ and q;

The error percenTages for QR were compuTed similar To Those
for v“. The curve for V”: 0.25 is represenTaTive (figure l6).
The low—loss error percenTages were also TabulaTed (Tables 3 and
4). Errors greaTer Than l percenT are apparenT for Q—lgreaTer
Than O.| (Table 4). For experimenTs in which errors less Than
l percenT are significanT, Table 3 can be used To esTimaTe The

error.

As a general sTaTemenT, The preceding resulTs show ThaT The

 

 

 

 

 

o 0.5 ‘ LO
05"

FIG. l6. Low—loss error percenfages for The absorp+ion
coefficien+ of a Rayleigh—fype surface wave.

TABLE 3

131

Low-Loss Error Percenfages for fhe Absorpfion Coefficienf of a Rayleigh—

Type Surface Wave, i.e.

 

 

 

 

 

( Ru. R) 10 w1+h 0‘ = 0.25.
QR
og’z

QEaOJO 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.5 0.16 0.16 0.24 0.47 0.93 1.63 3.11 3.94 5.60 7.60 9.98
1.0 0.64 0.61 0.62 0.72 0.94 1.32 1.89 2.68 3.70 4.98 6.53
1.5 1.45 1.40 1.38 1.41 1.52 1.75 2.12 2.65 3.35 4.26 5.38
2.0 2.58 2.51 2.46 2.44 2.50 2.64 2.89 3.26 3.37 4.45 5.30
2.5 4.03 3.94 3.87 3.82 3.83 3.91 4.17 4.33 4.72 5.23 5.90
3.0 5.81 5.70 5.60 5.53 5.50 5.52 5.62 5.81 6.09 6.49 7.02
3.5 7.90 7.78 7.66 7.56 7.50 7.49 7.53 7.65 7.86 8.17 8.58
4.0 10.32 10.18 10.04 9.92 9.83 9.78 9.78 9.85 10.00 10.23 10.56
4.5 13.06 12.91 12.75 12.60 12.49 12.41 12.37 12.39 12.48 12.65 12.91
5.0 16.13 15.95 15.78 15.61 15.47 15.36 15.29 15.27 15.32 15.43 15.62

 

TABLE 4

|32

Low—-Loss Error Percenfage for fhe Absorp+ion Coefficienf of a Rayleigh—

Type Surface Wave,

i.e.

 

 

 

 

 

(QRLL ' a" )102 for o- = 0.25.
O'R
Oil
ogl 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
0.05 0.16 0.23 0.91 2.50 5.17 8.91 13.71 19.49 26.17 33.66 41.83
0.10 0.64 0.62 0.93 1.81 3.40 5.71 8.74 12.43 16.72 21.52 26.77
0.15 1.45 1.37 1.52 2.08 3.15 4.79 6.96 9.64 12.77 16.29 20.13
0.20 2.57 2.44 2.48 2.85 3.63 4.86 6.54 8.62 11.07 13.84 16.86
0.25 4.00 3.83 3.79 4.02 4.60 5.57 6.91 8.60 10.61 12.88 15.37
0.30 5.73 5.52 5.42 5.55 5.98 6.75 7.85 9.27 10.96 12.88 15.01
0.35 7.77 7.51 7.35 7.39 7.71 8.33 9.25 10.45 11.91 13.58 15.43
0.40 10.10 9.80 9.58 9.54 9.76 10.26 11.03 12.08 13.35 14.82 16.47
0.45 12.72 12.37 12.09 11.98 12.11 12.51 13.17 14.07 15.20 16.52 18.00
0.50 15.61 15.21 14.87 14.70 14.75 15.06 15.61 16.41 17.42 18.61 19.95
0.55 18.77 18.31 17.92 17.68 17.66 17.88 18.35 19.05 19.96 21.04 22.27
0.60 22.19 21.68 21.22 20.92 20.83 20.98 21.37 21.98 22.80 23.79 24.93
0.65 25.87 25.30 24.78 24.41 24.25 24.32 24.64 25.18 25.92 26.83 27.89
0.70 29.79 29.15 28.57 28.13 27.90 27.91 28.15 28.62 29.29 30.13 31.12
0.75 33.95 33.24 32.59 32.08 31.78 31.72 31.90 32.30 32.91 33.68 34.61
0.80 38.33 37.55 36.83 36.25 35.88 35.75 35.87 36.21 36.75 37.47 38.33
0.85 42.95 42.09 41.28 40.63 40.19 39.99 40.04 40.33 40.81 47.48 42.29
0.90 47.77 46.83 45.94 45.21 44.70 44.44 44.42 44.64 45.08 45.69 46.46
0.95 52.81 51.78 50.81 49.99 49.40 49.07 48.99 49.16 49.54 50.10 50.82
1.00 58.06 56.93 55.86 54.96 54.29 53.89 53.75 53.86 54.18 54.70 55.38

133

low—loss approximaTion is adequaTe aT The presenT sTaTe of The arT
for compuTing The velociTy and absorpTion coefficienT of a Rayleigh—
Type surface wave on maTerials similar To Those in The crusT and
manTle of The earTh wiTh GQ' 4 10‘2,
For such maTerials, The relaTionship beTween Vn"% and G‘
is given by any of The curves for which q;' =- 0;. (figures l-5).
EquaTions (22) and (23) show ThaT for low—loss maTerials The raTio
aR/as for a fixed value of 0' is a linear funcTion of The raTio
Q;'/ Q;‘. The raTio (IR/aS as a funcTion of U'has been compuTed
for low—loss maTerials for various values of Q['/ cgJ (figure I7).
This family of curves (figure l7) may be used in place of equaTion
(l4) and published curves for "m" To compuTe O.R for low—loss
maTerials.

For maTerials wiTh lO-2< Q"é IO-l, The low-loss approximaTion

\

is adequaTe for experimenTs in which errors of up To one percenT are

lé Q"é IO (hereaTTer refered To

Tolerable. For maTerials wiTh IO—
as moderaTe—loss maTerials), The low—loss approximaTion is noT
adequaTe for many experimenTs. The velociTy and absorpTion

coefficienT of a Rayleigh-Type surface wave on moderaTe—loss maTerials

are compuTed in The nexT secTion.

8.2.3 ModeraTe—Loss MaTerials. Examples of moderaTe—loss maTerials

 

menTioned in The previous secTion are The waTer saTuraTed sedimenTs
( 9,? = 0.0, 95" = o.5-2., 0' = O.494—O.498). An example,which
is on The border of moderaTe—loss,is Pierre Shale ( QZI= IO-Z,
QQJ = lO—I 0‘: 0.4l8). For The Pierre Shale The error in using

I

The low~loss approximaTion is approximaTely 0.5 percenT for vR and

L0

03

08

354

 

[V'I'l'IIYVIVTVUTYII'ITVTYTTVVIII1II'I’YVV‘IY'V'I’IYTT-J

1

LOW-LOSS

lhTTj‘IYV

0.2703 =
ID

£0

Vr11Tjj1TIIVIIIITfITTYWTTTTTl'IYI

/

 

ubLme

‘11ll]111111444114]lllLLLllllJJuLlllllJJJJlillLlIIIIJJLALJJLLJ

VI'TIT'YYTTTjT‘Y‘VTllVIVI

1111

 

 

lIIlllll‘llllillllllllullllllllIIIlALIIlIllIlAIIIl

 

0 OJ 02 03 04' 05
0'

FIG. I7. Rafio of +he absorpfion coefficienf for a Rayleigh—Type
surface wave +0 +he absorpfion coefficienf for a homogeneous shear
wave for Iow~|oss maferials. This figure fogefher wi+h +able l2
simplifies fhe compufafion of aR/c% for low—loss maferials.

approximafely 0.6 percenT for QR . Such error percenTages would
require a very carefully defined field experimenT To be significanT.
However, for The wafer saTuraTed sedimenTs The error in using The
low—loss approximafion ranges from approximaTely 9 To 43 percenT
for VR and from l8 To I84 percenT for QR . BoTh of These error
percenTages are sufficienle large as To be significanT in a field
experimenT.
Using The general analyTic soluTion, a family of curves for

VR/’v3 as a funcTion of O" has been compuTed (figures l8, l9, and
20). Each curve wiThin a family corresponds To a differenT value
of 05". Each family corresponds To one of The Three chosen
values for 0;! = 0.0, O.l, l-O. The Two families of curves
corresponding To 6&4 = O and 0;“ = O.l are very similar. A
subsTanTial change in The family of curves is apparenT for a larger
amounT of absorpTion in bulk, le = |.O (figure 20). AlThough The
errors in compuTing vR from The low—loss expression for The wafer
saTuraTed sedimenTs are large, The curves (figure l8) for VR/ vs
show ThaT The error for The velociTy raTios is much smaller. More

specifically for The wafer saTuraTed sedimenTs

(Va—V32“) l0a y, Q To 43 Percenf; (25)

Vk
however,

VR _ VR

7 v 2

( s s._._ ) lO 25 —/ percenf. (26)
.3.
V

A family of curves similar To ThaT for vR/\g has also been

compuTed for‘ aRz’as using The general analyTic soluTion (figures 2|,

LOO

 

'I‘VTTYYYIITI'I'TTII‘TWV’VII'II'ITfiIITVTTYIIV‘YYTII

q

1

q

, od=o

/ 1
d

.l

0.9 O

A

J J A4 A

J

VR/VS
o
(D
o

 

 

 

.6 IJLLIllllJlllLLillLllqulllILJIIlllLALllllllLljlll
O O O 0.1 0.2 0.3 0.4 0.5
0-

FIG. l8. Veloci+y raTio for low—loss and moderaTe—loss maferials
wi+h OR" = o.

 

LOO

 

'IYTIFIVITITIIV’YV‘TIY‘Y'IY1TTlIVI"IYTTITTI’VITTT'I’

. q
Q"=OI ‘

. K - q

. q

>— —

r- -1

r- q

r- -1

0.90~ 1

r 1

» 1

. ,

?‘ -1

r— -«

F 1

> 4

. 4

>0) - ‘
\ 080*- -1
>m " "
.- 1

L

 

p -1
-- _l
> 1
P 1
_ 1
’ -1
0.70 -- . —
.. 1
L <
F 1
g —
L 1
L- ,

 

 

0.60 llll‘LLllllllLJlILlllLLLLlJ—LlLlllllljlllllLJllllll
0 0.! 0.2 0.3 0.4 0.5
0"

FIG. l9. Veloci+y ra+io for low—loss and moderaTe—Ioss maferials
wiTh q:' = O.L

 

LOO

 

0.90 —

 

0.70—

 

O.60 [AlllLLLAlllJlllllLLlLLLlllllLljllllJLLJLLiilJ‘ll

[IFITIIYTTIYTY‘TrTI[YIUYIV'TfTT‘TTrTIYVI"TTITITTrTT

fi
1
i

138

 

 

0 0| 0.2 0.3 0.4
0"

FIG. 20.
wi‘rh 9K" = I.O.

0.5

Velocify rafio for low~loss and moderafe—loss maferials

OR/OS

LE

09

0.8

 

PI'TTII"ITII"I'l’TIITII'TI'IIIIIII'V‘YI‘T‘TYrY—ITITTr“

02:0

T‘TY'IVTT'TITIITV

ITTTT1T1111771‘1111‘IIYV1‘IIT1

0
L
In

 

Lilli]ILJ‘llJJllJlJJ4JAJllJlll‘JJJJJJlllJJIIlljlljjljjlllll‘llAllll

IUTIITUYTITIIIITIV

 

 

p
1LJL‘JL‘IAJJALILJJLIALIIJJL‘IIAAJA‘LALLIIILALJJ‘JI

 

0 0| 0.2 0.3 0.4 0.5
0‘

FIG. 2!. Absorpfion coefficienf rafio for low—loss and moderafe‘
loss maferials wi‘rh OK" = O.

I40

22, and 23). The curves show fhaf for small amounfs of absorpfion
in bulk ( 0;, = O, O.l), fhe absorpfion coefficienf for a Rayleigh-
fype surface wave can eifher be greafer fhan or less fhan fhe
absorpfion coefficienf of a homogeneous shear wave, depending on
fhe magnifude of 0‘ . The percenf deviafion of aRLL from an is

large for fhe wafer safurafed sedimenfs, buf The percenf deviafion

in The rafios aR L/ 03 L and GR /as is small. More specifically,
L L

for fhe wafer safurafed sedimenfs

0. - Cl
(Lu—1) IOO as IS TO I84 perceni‘; (27)
R
however,
a a
a! _ YT:
(fig—.i) IOO x l percenf. (28)
9.3
as

8.3 Parficle Mofion Characferisfics for Low—Loss and Moderafe'Loss

Maferials

If was shown in chapfer 7 fhaf The fheorefical characferisfics
of a Rayleigh—fype surface wave on an anelasfic half-space could nof
be predicfed from fhose for a Rayleigh wave on an elasfic half—space.
For anelasfic solids, The propagafion vecfors E11 and S;a are
inclined af unequal angles wifh respecf fo fhe surface. The
affenuafion vecfors 5:2 and aka are inclined af unequal angles
wifh respecf f0 fhe verfical and form angles less fhan 90° wifh

respecf fo fhe corresponding propagafion vecfor. The amplifude

disfribufionsfor a fixed fime show a superimposed sinusoidal depen-

l.3

0.9

0.8

l4!

 

TTTTTITTT‘IT1111‘1

T111IIYIY‘

L'I'TYTI'ITYITITTTI[TITI"I’TTTITYV’I'ITTYTITTI'TTT'I'I‘TTTVI

og' =04

IIJLJJALJIALJLIALALJJIJLJLI‘LllllL|LliLLJ_IJLJ

 

1111111‘r11l11711111

 

YIITIYjTTj7111T111
‘ ’-
—
.

 

. 4
llLlllL‘LIILgl‘lllLJlLlLllllll‘lllLlLLlllllLLLllLl

lJLlJLlllllLlJllAlJ]AIJJJI

 

 

0

FIG. 22.

. .3 .4 0.5
0.! O 2 0_ O O

Absorpfion coefficienJr raflo for lowfiloss and modera+e*

ioss ma’rerials wi‘rh OK“ = O.l.

0.9

0.8

 

rTI’I’ ITI' 'Yrr [TVr r
Pr r l r r'IV'rY‘T'VVrv'IT'T'r'I'ITTTTI.

0.2' = LO

0
o m;
II

()1 b m a) .0

JJJJJJlJlJJlJLJJJLllllJJJJLJ4JJJJILLJ411JJJJJJJJIJJIJJIJLIALILALJ

 

TTTTT1fiTTIVTTTTWT‘1ITTTTTTTTYIT1‘rT117171TTT7]VYTTI‘IT‘ITTTYTYIYTTI‘ITY

JJJJJ

——

AllllLlLllllLLlLlllllllllellllllILLlLlllllllull

 

 

Q
U!

0.1 0.2 0.3 0.4
0—

FIG. 23. Absorp+ion coefficienf r‘aflo for low—loss and moderafei
loss maTerials wi‘l'h 9K" = 1.0..

O

dence wiTh depTh. The ellipTical orbiT defined by a parTicle wiTh
Time is TilTed wiTh respecT To The verTicaI and The amounT of TilT
is dependenT on depTh. These TheoreTical properTies for anelasTic
solids are noT predicTed by The corresponding resulTs for elasTic
solids nor are They predicTed by The work of Press and Healy (I957),
which is based on The low—loss assumpTion. The numerical magniTude
of These characTerisTics will be compuTed in This secTion for values
of The parameTers cg", GM:’ and 0' corresponding To arbiTrary
low—loss and moderaTe-loss maTerials. CompuTaTions for a few
specific represenTaTive maTerials will be presenTed in secTion 8.4.
The angles which The propagaTion and aTTenuaTion vecTors make
wiTh respecT To The surface and The verTical respecTively, are
TabulaTed for a wide range of parameTers (Tables 5-8). For IOWv
loss maTerials The angles associaTed wiTh The propagaTion vecTors
are less Than 2.9°, and The angles associaTed wiTh The aTTenuaTion
vecTors are less Than 9.6°. For moderaTe—loss maTerials The angles
are considerably larger. The relaTive magniTudes of The angles are

consisTenle relaTed as illusTraTed.

P':

The angles beTween The propagaTion vecTors and corresponding

aTTenuaTion vecTors are TabulaTed (Tables 9, l0). The properTies
of a Rayleigh-Type surface wave as derived in secTion 7.5 may be
compuTed as a TuncTion of depTh in uniTs of fracTions of a wave-
lengTh. CompuTaTion of The properTies depends on compuTaTion of

The quanTiTies defined by (7.88). The basic quanTiTy needed for

TABLE 5

 

 

lnclinafion of Propagafion Vecfor fig: wifh Respec+ To Surface
0K": 0 q;' = 0.: 0;'=1.o
Q5" 0: .25 0-: .35 0': .45 0-: .5 <r= .25 <r= .35 0': .45 cr=. <r= .25 cr= .35 cr= .45 er: 5
O O. O. O. O.
.l —2.7 -2.9 —2.9 —2.9 -2.4 -2.5 -2.7 —2.
.5 ~|2.3 —l3.2 -l3.5 -|3.3 -|2| —|2.9 -l3.3 -l3.
I. -20.0 -22.| -23.0 ~22.5 —19 8 ~2|.7 -22.7 -22. -|9.3 —20.3 -2|.6 -22.5
2. -24.3 -29.4 —32.4 —3|.7 -24.5 -29.0 ~32.l —3!. -26.8 -28.6 -30.7 -3l.7
5. -2|.O -25.7 -38.9 —39.3 -22.5 —26.7 -38.4 -39. -29.8 -32.3 -37.l -39.3
IO. -26.5 -22.l -36.2 -42.I -27.| -23.6 -35.9 —42. -30.9 -3|.4 -37.3 -42.|

 

 

 

 

 

 

 

VP!

|nclina+ion of Propagafion Vecfor

TABLE 6

B
a

wifh Respecf To Surface

 

 

 

 

 

 

_ —: -|
K1:0 QK =04 (,7K =I.O
O;' cr=.25 c=.35 c=.45 a: . a=.25 0:.35 0:.45 ¢=.5 0:.25 v=.35 0:.45 6:.5
o o. o. o. o.
.l -.6 -.7 -.8 - —l.| —I.O -.9 -.8
.5 -2.6 -3.| -3.6 —4. —3.2 -3.5 -3.8 -4.0
I. -4.2 -5.| -6.2 -7. -5.0 —5.5 -6.4 —7.0 -9.2 -8.4 —7.4 -7.0
2. -4.9 -6.2 -8.7 -lO -6.2 -7.0 -9.0 -lO.4 -12.6 -ll.6 -IO.8 -l0.4
5. —|o.3 -5.0 -9.2 -|3 -ll 6 —7.0 -9.8 -!3.6 -|7.6 -15.5 -|4.0 -l3.6
l0. -2o 7 -|2.7 -6.8 —15 -2!.O —l3.8 -8.| -15.0 -22.9 —19.2 —15.5 -15.0

 

 

 

St!

TABLE 7

lnclinafion of Affenuaflon Vec‘ror‘ A¢2 wiTh Respecf +0 Ver‘flca!

 

 

 

 

 

 

 

 

~l -| .-
QK = O 0K : 0.. K| : 1.0
q;' 0= .25 a: .35 a: .45 a: .5 a: .25 a: .35 a: .45 a: .5 o-= .25 a: .35 c: .45 o: .5
0 0 0. 0 0

I 3 I 3 I 2 9 2 8 3.4 3 2 3 0 2 8

5 I4 6 I4 3 I3 7 I3.3 I4 8 I4 5 I3 8 I3 3
I. 25.2 24.7 23.4 22.5 25.4 24.8 23.4 22.5 26.0 24.9 23.4 22.5
2. 35.9 36.4 33.7 3I.7 35.9 36.2 33.8 3I.7 36.7 35.6 33.4 3I.7
5. 37.4 44.0 45.0 39.3 38.6 43.8 44.8 39.3 44.2 44.9 43.I 39.3
I0. 40.5 38.8 5I.4 42.I 4I.2 40.I 50.4 42.I 45.8 46.6 47.6 42.I

 

917i

TABLE 8

Incl inaflon of AHenua’rion Vecfor Kw: wH-h RespemL To Verfical

 

 

 

 

 

 

0;” = 0 g‘ = o 1 g' = 1 o

— I
as a =.25 a =.55 6==.45 a: .5 a: .25 aw=.55 a: .45 a: .5 6: .25 c: .55 a: .45 c: .5

0 0 0. o 0

1 6 7 7 7 8 9 9.6 7 2 8 o 9 0 9 6

.5 28.1 51.8 56.2 58.6 28.5 52.1 56.5 58.6

1. 40.5 45.7 51.4 54.5 40.7 45.8 51.4 54.5 46.5 49.4 52.7 54.5
2. 40.5 52.2 60.4 64.4 45.1 52.5 60.5 64.4 - 52.7 56.8 61.7 64.4
5. 58.2 45.5 62.2 70.1 40.1 46.7 62.1 70.1 51.4 56.5 64.6 70.1
10. 41.7 58.9 56.9 71.9 42.8 40.8 56.9 71.9 49.9 52.5 62.1 71.9

 

 

 

L171

TABLE 9

 

 

 

 

 

 

 

 

Angle b’p Be+ween P¢2 and A¢z
I '= 0 OJ’ = 0-! o;’ = 1.0

QE' 9= .25 c==.35 9: .45 a: .5 a: .25 9: .35 a: .45 a: .5 a: .25 a: .35 c: .45 c: 5

0 90 00 90.00 90.00 90.00

.I 89.58 89.80 89.97 90.00 89.05 89.36 89.76 90.00

.5 87.72 88.9! 89.83 90.00 87.24 88.45 89.60 90.00

i. 84.8I 87.36 89.58 90.00 84.46 86.92 89.30 90.00 83.30 85.44 88.27 90.00
2 78.36 83.00 88.72 90.00 78.59 82.77 88.34 90.00 80.06 82.92 87.25 90.00
5 73.60 71.7: 83.95 90.00 73.87 72.90 83.63 90.00 75.53 77.41 83.96 90.00
IO 76.0I 73.37 74.79 90.00 75.87 73.50 75.44 90 00 75.|4 74.8I 79.68 90.00

 

RVI

TABLE IO

 

 

 

Angle 21’s Be+ween SW and Kw
2 2
-I —| _
K - 0 ox = o 1 QK' = 1.0
o;' c: .25 o==.35 a: 45 c: .5 w: .25 6: 35 a: .45 a: 5 a: .25 or .35 v: .45 o: .5
0 90.00 90.00 90.00 90.00
.1 89.90 83.01 81.92 81.27 83.90 83.01 81.92 81.27
.5 64.56 61.23 57.46 55.38 64.72 61.38 57.53 55.38
1. 53.88 49.38 44.79 42.49 54.28 49.76 44.97 42.49 52.77 48.92 44.73 42.49
2. 50.48 43.98 38.31 35.91 51.13 44.70 38.66 35.91 49.93 44.86 39.08 35.91
5. 62.15 49.45 36.91 33.45 61.55 50.35 37.71 33.45 56.16 49.17 39.41 33.45
10. 68.93 63.82 39.84 33.06 68.30 63.04 41.22 33 06 63.01 56.84 43.39 33.06

 

 

 

 

 

 

6171

I50

compuTaTion of The quanTiTies in definiTion (7.88) is k/LkR which

expressed in Terms of The raTios aR/as and Vh/‘G is

J: =1-iEKfiF-‘(0g'), <29)
*2 as V3

where

 
 

Jl+ Q;2 ~ I

———_____.__._

l+ q; + l

(30)

 

For example, The quanTiTy Gl in (7.88) wriTTen in Terms of (29)

and fracTions of The wavelengTh A : Elf along The surface is

R
#3: + b2
k
G,—- “ e expl-f’zx ZTT 1‘2), (3')
I2- 4:2! in ”
4%n
where
_ fit _ c2 1/2
bB/&R —p.v.(_§zl.l _;3)) . (32)
ll

The oTher quanTiTies in definiTion (7.88) may be wriTTen in Terms

of &/&R similarly.
SeTTing X3 = O, The properTies aT The surface can be compuTed

wiTh The preceding formulaTion as a funcTion of qg", 6%:’ and T .

8.3.l ParTicle MoTion CharacTerisTics aT The Surface. IT was

 

esTablished in secTion 7.5 ThaT The orbiT described by a parTicle
wiTh Time is ellipTical wiTh The major axis of The ellipse TilTed
wiTh respecT To The verTical. ParameTers of The parTicle moTion

ellipse as defined in secTion 7.5 are shown (figure 24).

 

 

 

FIG. 24. Parficle mofion diagram
illusfrafing The paramefers defined
in secTion 7.5 of +he fexf.

J I

The TirT J of The ellipse is given by equaTion (7.96). The
TilT as a funcTion of 6' has been compuTed for q;' = O, O.l and
Q;' = O, O.l, 0.2, ... l.O (figures 25, and 26). For These
amounTs of absorpTlon The TilT increases wiTh an increase in o;‘,
in oTher words, The TilT decreases monoTonically To zero for a
fixed value of '0' as Q;’ approaches Qf' (figures 25 and 26).
(This resulT is consisTenT wiTh The analyTic resulT derived in
secTion 7.5 which shows ThaT The TilT aT The surface is zero for
solids such ThaT G§" = GE” ). Also for fixed GE’ and 0;)
The TilT decreases monoTonically To zero as 0‘ increases To 0.5.

For larger amounTs of absorpTion in shear ( l 5 0;] 5 l0)
(figures 27, 28, and 29) The TilT no longer increases monofonically
wiTh 0‘ for fixed cg‘land 6&4 . InsTead for a given GE] and

Q;' The largesT TilTs occur aT some inTermediaTe value of 0‘ .
The TilTs decrease irregularly To zero as Gg'l approaches 0;} .

Figures 25-29 are useful for esTimaTing The amounT of TilT
associaTed wiTh various Types of maTerials.

The raTio of The maximum horizonTal displacemenT HM To The
maximum verTical displacemenT Hfl,aT The surface is given by
(7.88) and (7.89). Families of curves for The raTio lfio/ H5° as
a funcTion of 0' were compuTed (figures 30, 3|, and 32). For The
range of absorpTion considered The maximum horizonTal amplifude

aT The surface is less Than The maximum verfical. For each pair

of fixed values of

TILT (DEGREES)

 

' 014:0

'TTTITIIVIIIIV'IIIVTTIIT'I‘II‘TYII'TTTTITY'IllIIV

og'=o

 

I T'T 1

7

I

, , 1.0

1

r'I

1

'-

 

lLlllLllllLLlllllLlllLlLLlLLIllLlLlLLLIlAALlLlIl

1

LL LLLJ

l

1141

1

1

—1
4

 

 

O 0.! 0.2

FIG. 25. TilT of fhe major axis of The parficle mofion

0.4

\f

LN

0.5

eilipse for low”lcss maferials and moderafe-Ioss maferials

wi+h 0K" ~—- 0.

TILT (DEGREES)

 

 

.Jm_,L_~1_..L._J._0Lu.lw.Janu1_.wL._l_..L_~J_‘~Lm“J..»L._J._.Lm_J._.I ’i

“L1“

h

1 I 1
4~_L~;“J“.L~J.J_HLWJ.ML«JMwl~4~uLm4

 

luv.»

-|9 I‘L‘lLLLlllllLlLlllllLLllllllllllllllllll‘lllLLlL

C) (1| C12 C13 C14
C7"

.0
Lfi

FIG. 26. TilT of +he major axis of The DarTicle mofion e'iipjg

for low loss maferiai: and moderafe floss maTerials wifh QK -| : O.1_

TILT (DEGREES)

 

  

 

 

Or—
—I ..
: 0K - o
-5 ,_
C 3; q
-lo —~
, -'- ,
QS - 4
b I -+
“i5 *"
r- -4
I ‘ I
-20 -*
P2 .4
’25 - .4
, 4
. 3 4
L q
'30 ” \\ 4
P 4 \\\ :
: \\\\\ :
-{35.— 6 ‘Qa‘ ’“
.- 7 8“ -1
r- \ -<
9 -1
P lo q
‘40 kl I I I I I I LL11 I I Ll L LI I [I I I III I I I l L L L11 I IAI l I II I [I I I i
. .2 .3 0.4 0.5
O O l O 0_ 0
FIG. 27. Til? of fhe major axis of +he parTicle moTion

ellipse for Q;’

= 59 23 --~, IO and Q; = O.

TILT (DEGREES)

“l5

 

"II"'"'I|I’TTYI"rI'ITYT'III‘erTIl'II|'I“rr'rrF—‘
1

  

 

'-
’ i
r “I: =5
. QK O.I I
~ 4“,
. “G 4
r- I' h“;
+— -l -]
v- Q5: 4 §
, I ”i
’ fl
1
. 1
-4
1
1
1
w
1
~q
-1
q
4
q
4
.1
4
LLLLlllllllllLllllllLIllIIJILLIllAlLLlllllllllllL

 

 

0 0.! 0.2 03 0.4 0.5

67'

FIG. 28. Tilf of The major axis of The par+icle mofion
ellipse for Q;' = 044.

TILT (DEGREES)

 

TTTYITIIIIYYYTrTIII‘ITTr'rTrTIIT‘T'IIII'I‘ITIITTT

 

 

p d
F _
E 1
~20— .q
’- -1
_ ~1
~ 0: = LO fi
" 1
~25; .M
P s
b
,. d
'- -
‘30 "" -1
>- 1
’- 1
P -1
" q

 

 

-40 LlllleLLLLllllllllLALlllllllllllllllllllllllIIL
0 DJ 0.2 0.3 0.4 (3.5:
0"

 

FIG. 29. "NH of The major axis of +he parficie rnoflon
ellipse for 0““ = 5.0.

L0

0.9

C18

H*o/H3o

C17

CXG

CL5

138

 

I'TI—I'IITUIII'TI'IfrlIII'UIITUII'IIV'IT'ITTT‘IITIrIrTY

Oo“=o

    

IVTW

1
1
1
.1
1
'1
1
.1
1

i

1

 

'VITTVYTVIUTT‘IIIVVTIYTYT‘VUVI

 

L141 ALllJ—ALAALILILIJALALJJIlllljjJulelllli‘Ll:
0 OJ CLZ (13 (14 0.5
CT"

 

FlG. 50. Rafio of The horizonfal ampliTude To The verTical
amplifude aT +he surface for low—loss and moderafe—Ioss
maferials wifh 0;, = O.

IIO‘rrIr Trir IIrI’ 'TVT rrrI Illl I‘TT TTYT TTTY 'T"
I f r T l [ l I l

 

J...

 

 
 

0:3 =o.s j
3
09 3
0.8_ l}
.3 : j
{ L
159 C 3
0.7C- .1
I _ 3
cxsE— E
E
0.5P11A11A11LL1L11141IILLLLLllllllA‘ llJlLlLll l I l l I ll Lj

 

 

0 OJ C12 C13 C14 (15
7

F96. 3!. Ra+io of fhe horizonfal ampliTude +0 +he verfical
amplifude af The surface for low-loss and moderafe—Ioss
maferiaIs wi+h Q;' = 0.1.

\O

6

C19

()0

 

rTTT'IIY'rTITI'V‘I’FI'ITTT'TFYTIIllllrIVVITTTIrITF

o;'=u.o

1

 

 

I
y-
OBE-
«9 I
I r-
\ >—
9 p
z: I
0.7:-
E
L
l-
0.6 E— .V
; ~.
,
0.5 '
o 0.: 0.2 0.3 0.4
0—

FIG. 32. Rafio of The horizonfa! ampliTude +0 The verfical
amplifude af The surface for low-loss and moderaTe—loss
maferials wifh Q;’ = I.O.

g[LlIlll‘lilllllilifijLLIJIIIL¢JAIIILLLILJLl

 

C15

l6l

_l _
05 and (,7M

approaches 0.5.

The raTio Pho/ H3 approaches 0.5437 as (T
O
For solids wiTh o;' = C[" The raTio H.0/H3 is in—
O
dependenT of The amounT of absorpTion (compare figures 30, 3|, and
32). This resulT is also easily esTablished analyTically. From
secTion 7.5 The raTio H,/H3 aT The surface for solids such ThaT

cps": ox" is

2
c
2——

szligslz 71:414—— aa _‘[“‘.-_c:

H30 IF}: 4- G!" A 2 I‘_C__ (X7.
p2

As shown in secTion 7.5 boTh 52/53 and (at/(3Z depend only on G:

(33)

 

 

Hence, H,,/ H3 depends only on 0' for solids such ThaT Q;l== O;{
0 0

Figures 30, 3|, and 32 are useful for predicTing The raTio
H%./ H5° as mighT be measured on The surface of various Types of
maTerials.

The raTio of The lengTh of The semi-minor axis of The ellipse
To The semi—major axis ( {3777E7 ) is given by The reciprocal of The
ellipTiciTy (equaTion 7.97). Three families of curves for The axis
raTio aT The surface have been compuTed (figures 33, 34, and 35). For
solids such ThaT qg" : 0;‘ There is no TilT of The parTicle
moTion ellipse and The curve for The axis raTio as a funcTion of 0'
coincides wiTh The curve for lfl / H5. The families of curves for The
axis raTio (figures 33, 34, and 35) are somewhaT similar in shape
To Those for The raTio Phox'fgo(figures 30, 3|, and 32); however,

-I —i

The numerical values for ng > 0.] wiTh Gk 5 5% are signifi-

canle differenT.

AXIS RATIO

1.0'1-1‘1'1‘

(19

Y

()8

x

07’

0/6

 

T'Wj‘j’TIIIYW'YTYTYVWIITYYIVYY‘

 
    

JJLJIJIJIJLJIJIJJJIJALIJLJLJJLIJJJLLJJIJJI

%;
r C.-
J

 

JLJ

‘llIILLllLllllllLIllLllJllllLlllle‘IL11L1_LIL

 

0.51Lll

C)

0.! C12 0.3 (14 C15
(7

FIG. 33. Rafio of +he major axis +0 +he minor axis of +he
parficle mo+ion ellipse aT +he_surface for low—loss and
modera+e~loss maferials wifh QK = 0.

I62

AXIS RATIO

L0

(19

0.8

057

C16

(15

“)3

 

rli'll‘I‘I'I'TIIIIIll'I'T'I‘Ir]IIITTVTVIIIIIIl"11

J_

-l -
0K ' O.l

 
   

 

.AAJJLILLJAL‘LJLJJLAIIJJLILJlLlLLlLZlLIAllIJJLi

I‘V‘l‘IYWT‘IYVVTjj—VjIUVVY]I‘Yl

 

II‘LLILIILLIIIlllll'lJLllLlllll‘llLLLlllllllLLLL

0 OJ C12 C13 (14 (15
(7-

FIG. 34. Ra+io of #he major axis +0 +he minor axis of ihe
parficle mofion ellipsq aT fhe surface for IOW"iOSS and moderafe’
ioss ma+erials wiTh Q; =O.|.

AMS RAUO

 

“11111:14l11L11IJJJLJAlijLAJJILAJJl-11I11J;L

 

OsrllllLllllllllllLllllALll‘llLlillllllllllll‘l‘LllAr

0 OJ C12 (13 C14 C15
(7‘

F56. 35. Raiio of +he major axis +0 The minor axis of ihe
parficle mofion ellipse a? The surface for tow—loss and
moderaTe—Ioss maferiais wiTh ng = |.0.

I65

8.3.2. ParTicle MoTion CharacTerisTics vs. DepTh. To sTudy The
behavior of various parTicle moTion characTerisTics as a funcTion of
depTh, families of curves have been compuTed. Each curve wiThin a
family has been compuTed as a funcTion of depTh and corresponds To
a fixed value of cg” ( 0 s q§" f [0 ). Each family of curves
corresponds To a fixed value of Gfi'l = 0, 0.l, or l.0 and To a fixed
value of U" = .25, .35, .45, or .5. The Technique for compuTing The
various parTicle moTion characTerisTics as a funcTion of depTh is
described aT The beginning of secTion 8.3.

The normalized ampliTude disTribuTions as a funcTion of depTh
aT a fixed Time, when The corresponding ampliTude componenf aT The
surface is a maximum, are given by equaTion (7.90). Families of
curves for The normalized ampliTude disTribuTions corresponding To
G§nl = O, 0.! and Gg-l = 0, .l, .5, l.O wiTh 0;) e ng have
been compuTed (figures 36 - 43).

The following cenclusions are apparenT from figures 36 — 43 for
The range of parameTers considered:

I) The verTical ampliTude firsT increases away from The surface
Then decays rapidly To zero,

2) The horizonTal ampIiTude decays rapidly away from The sur—
face,

3) The majoriTy of boTh The verTical and horizonTal disTur—
bances associaTed wiTh a Rayleigh-Type surface wave is
concenTraTed wiThin one wavelengTh of The surface,

4) for fixed values of 0‘ and CH4 , The effecT of increasing
The amounT of absorpTion in shear ( G§4) is a Tendency

for The disTurbance To become more concenTraTed Toward The

 

 

  
   
   
 
  

 

 

 

 

I 1 I' I’ I I t I T r T T l’ I T T T I
D— .4
, og‘ =0
_ 0' =0.25 «w
LO .4
J
.4
4
u] .4
3 0.5 .4
[.—
3 "
0- a
E
4 ~
53 VERTICAL
12‘ o" =
_J _ 3
<1 .
E
Z r-
0 .—
" HORIZONTAL
- 0-! _ 4
S -
“ O .l "
__ .5 | .,
-O.5 — —
4 L J_ 1 i l J I l l A l A L l I A 4 A
O 0.5 LO US 2.0
DEPTH {X3/X)
FIG. 36. Normal?zed ampfiifude versus depTh for Iow—ioss and

. . -1
moderafe'loss ma+erials WlTh QK =

O, and 0': 0.25.

!67

 

    
   
   
 

 

 

 

 

I II .1
Q; 3C) .
CT—=(DJ35 d
|.O '-
.l
u: _
CD a
:3
20.5 .4
.J
0. -
E _
<1 _ 4
Q _ VERTICAL ‘
U] -
'2' ~ Qsl ‘
.J
<1 _
E ,.
x F
C)
Z r-
O h
.. HORIZONTAL 4
0-1-
— s - —I
u- _5 | q
’05 '— _.
L l I 1 L l I L l I 1 l L L l L l K A
O 0.5 |.O [.5 2.0

QEPTH {xa/Jx)

FlO. 57. Normalized ampiifude xersus depTh for low—loss and
moderafe—loss maTeriais wifh Q; = O, and 0': 0.35.

HJH

 

  
    
 
    
   
 

 

 

 

 

 

P T T V f I r r r v I r I“ I v I I I “J
-l-
QK ‘ 0 d
0‘ = 0.45
LO ..J
{ *1
.4
m I
O
:3 —-1
t 005
a" .-
2 —{
<1 4
Q VERTICAL .4
UJ Q" :
:1 ' S
_l .-
4
2 ,.
a:
O P'
Z P
O .—
:_ \ HOR!ZONTAL
. og' = q
:- 0. .I _
~ I. q
-O.5 — —-4
1 i I 4L 1 L L l l l L L L A l l L
O 0.5 I.O l.5 2.0
DEPTH (x3/x)
FfiG. 38. Normafifized ampfiifude versus depfh for low—loss and

' . ~ ’, .
moderafe—ioss ma+ereals wzrh QK‘ = O, and a“ = 0.45.

NORMALIZED AMPLITUDE

169

 

T I V 1 I Y T Y T I Y Y 1 ‘7 I T T T T
v— x Q’;|=O -
r \ 0- =0.50
i.0~— n.
- J
"' --¢

 

 

0.5 P" "
—— -v+
P VERTICAL 4
» (3g': 4
.. O, J
_ ,l a
" .5 “
P I. * ’ , . "‘

(y.—
L
r-

 
 
   

 

 

 

 

h HORIZONTAL
o" =
_ S q
-05; .. _«
. J J_ l 1 l I I I 4 L I l L L l L 1 I 1
o 0.5 LO 1.5 20

DEPTH (x3/X)

FiG. 39¢ Normaiized ampiifude_versus depfh for low—loss and
moderafe—loss maferials WIfh QK = O, and c = 0.50.

NORMALIZED AMPLITUDE

I70

 

 

 

 

 

 

 

 

Y 1 I I l’ I I r T I T Y I V I ‘ V I j
_ o;'= 0.: 4
LC \ —
\~ -
' \. 1
r <
" '1
0.5 - —
r- q
p _
L. ..
L .4
_ VERTWCAL q
-l-
>— OS ' -.
, .l _.
p. l. ._
L
C)P'
* HORMONTAL “
. (3g: -
P c) _ —
, v _
_ .5 l __
-()£5r—
L L A L J L I l L l L L l L J L L J L _—!
0 0.5 1.0 l.5 2.0

DEPTH (Kg/k)

FIG. 40. Normalized amplifude versus depTh for low—loss and
moderaTe—ioss maTerials wirh Q;' : 0.1, and 0-: 0.25.

NORMALIZED AMPLITUDE

i7l

 

0.5

 

r 7 V Y I r Y 1' V l I r T V r t T

Q": 0.: ‘H

V ‘ 0- ' 0.35 .4

I > "

LO ..
\

\. .

.\ d

.1

«A

.J

.4

4

.4

 

 

     
 

VERTICAL
..|_

OS '

.I

HORIZONTAL

 

 

 

-l-
QS -
4
. *
‘0.5 __
4 A A l 1 A A 1 l A l l l 1 1 1
O 0.5 I.O LS 2.0
DEPTH (X3/A)
FIG. 4|. Normalized amplifude versus depfh for low—loss and

modera+e~loss maferials wi+h Q;' = O.l, and U = 0.35.

NORMALIZED AMPLITUDE

I72

 

LO

0.5

 

IV~T f‘l’ T Y

  
   

 

 

  

*I' I 1 Y I T

VERTICAL

I V 1
ofi=0J
o- = 0.45

    
 

LgLL

L

 

 

h- "' ._.
QS
L J
()_—
' HORIZONTAL ‘“
y- -l- d
Q8 - -l
$.— .a
} L ‘
’().5r- —~
1‘ A A L L I 1 L A l I L L l l A J
O 0.5 |.O l.5 2.0
DEPTH (x3/x)
FIG» 42. Normalized ampliTude versus depfh for low—loss and

moderaTe—ioss maferials wifh Q;3 = 0.1, and 0': 0.45.

 

 

 

     
 

 

 

 

[.0 -—
1
.J
LU F‘ -1
C3 . i
:3
': O.5- _
-J -4
Q r-
:E _ -
<t VERTICAL
. Q" = .
o __ s _1
“J .l
t. >- 4
z; p . 5 q
E . ° ‘4
a
CD - d
z I-
Dr—
_ HORIZONTAL .4
. 0;: a
"" "'1
4
' I. __
'05 —-
L J L L l A L A L l L 4 L L l l L l l
O 0.5 I 0 I5 2.0

oepTé(x3/x)

FIG. 45. Normalized amplifude Versus depfh for low—los: and
moderafe—ioss maferiais wiTh q: = O.l, and 0‘= 0.50.

l74

surface,
5) for fixed values of U’The ampliTude disTribuTions for low—
—l

loss solids (O s ;" g 6% = O.|) are essenfially

idenTical wiTh Those for elasTic solids (figures 36 — 39).

The effecT of changing U‘for 6&4 = O, Q; I: 0.! and 0;! =
l.O is shown (figures 44 and 45). The maximum ampliTudes aT depfh
decrease as G'decreases for fixed amounTs of absorpTion. For
0' = O, QJI= O, G§'|= 0.l, The maximum verTical ampliTude
occurs aT The surface.

The superimposed sinusoidal dependence on depTh indicaTed by
The TheoreTical considerafions of secTion 7.5 is noT readily
apparenT for The smaller amounTs of absorpTion O 5 ng £3] and
Ofi G§" f 0-! . Normalized verTical ampliTude disTribuTions
were compuTed for larger amounTs of absorpTion in shear ( cg” = I,
2, 3, 4, 5, l0) (figures 45 - 57). For The larger amounTs of
absorpTion, especially 65" = l (figures 54 ~ 57), The normalized
verTical ampliTudes show a Tendency To osciilaTe abouT zero. For
sufficienle large amounTs of absorpTion and 0‘ # 0.5, The
maximum verTical ampliTude occurs aT The surface (figures 46 — 57).

Normalized horizonTal ampliTude disTribuTions were also
compuTed for larger amounTs of absorpTion in shear (figures 58 — 69).
For These larger amounTs of absorpTion The normalized horizonTal
ampliTude disTribuTions also show a slighT Tendency To oscillaTe
abouT zero as The ampliTude decays To zero wiTh increasing depTh.

For elasTic solids, The depTh aT which The normalized horizon-
Tal ampliTude becomes negaTive is The depTh aT which The parTicle

moTion changes from reTrograde To prograde. For anelasTic solids,

 

      
  

NORM ALIZ E D AMPLI TU DE

 

 

 

r T T T I r ' ‘ I I r I’ ' r I r T T Y

.I 4
0K =0 4
0‘3 = 0.: <
y .
LO --
I -1
+ VERTICAL 4
,. 0-: H
0.5 "‘
$- —<
P i ‘
r— k ..
,. 4
h \E
O L— HORlZONTAL “fl
’- 0- : ' fl.
_ ' ‘ 0 1
~ i
" .25 1

P .35

f .45

r-

‘ 1
"O.5 1 l L 1 1 A 4 A L l L L 4 L L 4 L L L I
2.

o 0.5 1.0 l.5
DEPTH (X3/M

FIG. 44. Normalized amplifude versus dep‘rh for low—loss
maferials wifh 0”: (L0 — 0.5.;

I76

 

 

 

 

 

 

r Y V f r T T T Y T Y r r T r r 1 I
-l _ 4
0K - 0 ~
-| _ q
Qs-l o
! 4
u ___‘
m _'
0 1-3 .4
3
E L; VERHCAL '
o. a: .
E 05~fi 5r= ._
q r-
O -1
E P -<
3 L 4
<
E " "
o L 4
z __ _1
‘ y -
oL— ' HORMONTAL #_ W; 4~
b- 0—3 / "' ~
_ Q ~
, .25 _
F” —1
’ 1
.0.5 L L L L l L L 1 L L L x L
0 LC LC") 2 .J

 

DEPTH(X3/X)

FIG. 45. Normalized ampli‘rude versus depfh for 0;) : O,
Qg“ z: [.0 and 0": 0.0 h 0.5.

LS

NORMALIZED VERTICAL AMPLITUDE
p
U1

177

 

'o

 

 

 

 

0
Z. .1
P m
. I l L l L l l I l 44L L L L L L l L
O 0.5 l.O 1.5 2.0
DEPTH (Xs/X)
HG. 46. Normalized vertical amplifude versus depTh for

0;” = l-io, Q;' = o and <r= 0.250

NORMALIZED VERTICAL AMPLITUDE

I78

 

I 5 i T I r I I T l T r I' r l l T r fir r I 1
. _'_ _
_ 0K - o _
P 0-: 0.35 A
,. -I

 

 

 

 

I. ..I
J l j I J l J I J l l l l l 1 l l l 1
o 0.5 I.O l.5 2 .o

DEPTH (xg/M

FIG. 47. Normalized verfical amplifude versus depfh for
05" = I-IO, 0;! = o and «r: 0.35.

NORMALIZED VERT1CAL AMPLITUDE

 

 

r 7 f r T 1 I T V I ‘7 I ‘ r I r V T
P -I- '4
P- QK-O _‘
- 0'=OAS .

 

 

 

as LO l5
DEPTH ms/X)

FIG. 48. Normalized verfical ampliTude versus depfh for

q*

l—IO, Q,;' = o and <r= 0.45.

NORMALIZED VERTICAL AMPLITUDE

ISO

 

 

 

 

 

[.5 I T V T T T T Y I r r I T r I I I T
F —' - d
__ OK '0 ‘4
» o- = 0.5 -
r— . —¢
>- A -d
r- m -4
.4
LO -~
'— \ _
0.5 ~—
y-
L
r
y.
0 b—
b o—
J L J I l I I l l 4 L j J l I l L J
O 0.5 LO |.5 2.0
DEPTH (X3/M
FIG. 49. Normalized verTica[ amplifude versus depTh for
05" = l-IO, 0;” = o and 0‘ = 0.5.

NORMALIZED VERTICAL AMPLITUDE

I5

I81

 

 

 

 

 

4L J I l I I I I J g I J I J I I I
0.5 LO I. 5 2.0
DEPTH (xs/x)
FIG. 50¢ Normalized verfical amplifude versus depTh for

05" = I-IO, Q;' = 0.: and 0'“ = 0.25..

NORMALIZ ED VERTICAL A MPLITUDE

E82

 

'5 I I T I I I I I 1' I I I I I I 1 T T r
r- —l_ '1
__ QK -O.l
. 0‘=O.35 I
- 4
~ -+

 

 

 

 

’- -‘
L I
>- J
A A A A J l 1 1 L l L 1 1 1 I 1 l A 4
O 0.5 |.O LS 2.0

DEPTH (x3/x)

FIG. 5|. Normalized verfical amplifude versus depTh for
—r _ -I a
OS — l—IO, QK = 0.! and U“: U0359

C

 

L _l 4
__ QK=OI _*
r 0'=O.45 4

'O

NORMALIZED VERTICAL AMPLITUDE
.0
U"

 

 

 

l l A I l I I I J l L L I A l l #1 I l

0 0.5 I.O 1.5 2.0
DEPTH (x3/x)

 

FIG. 52. Normalized verTicaI amplifude versus depfh for
0;“ = l-IO, ox” = O.| and cr= 0.45.

lid/l

 

 

 

 

 

 

|.5 T V 1 V r T T T r I T I T r
b -I '1
_ K 0.: _
~ 0- = 0.5 +
.‘
UJ |.C)" “q
o r-
3
t _ -x
J y\\‘
n. ' \
ii p 4
<1 -1
" \
3’ ~ _
2 _ -
F- F \ "a
II
LL] p «1
>
0.5 -— w
O '1
LL] P
5 v- “
.1
d P '1
2 __ q
a:
C) F .
Z P _
p. a
P- -1
,. .J
b— .
O _.....J
T , ,--—.-
- 4
r— ..
r J
l A I I I l A 1 L I l L l l
O 0.5 LO l.5 2.0

DEPTH (x3/ A)

' go 5fi. Normalized verfical amplifude versus depTh for

Fl
Q; ‘l-IO, Q;’ = O.| and cr: 0.5.

NORMALIZ ED VERTICAL AMPLITUDE

in

| 85

 

 

 

L— Qg' = LO -
. o- = 0.25 -
_ _
L ..

 

 

r q

* ”i

O 0.5 IJO LS 2.
DEPTH€X3/M

FIG. 54. Normalized verfical amplifude versus depfh for

0;! = [—50, 0;] 2 !.O and 0': 0-25-

NORMALIZED VERTICAL AMPLITUDE

L86

 

l'5 I' r r I' l' r Y T r I F T U I I r I 7 fir
P _l_ ‘
— QK ' '.O u
. 0' =0.35 q
r- 4
+ a

 

 

 

 

0 0.5 a. 0 LS 2.0
DEPTH (x3/x)

FiG. 55. 'Normalized verfical amplifude versus depfh for
057' = l-IO, QR" = I.O and 0—: 0.35.

NOR MALIZ ED VERTI CA L AMPLITUDE
,0
U

{87

 

r- ‘1
. og'= L0 _
a-=o.45 .

'O

 

 

 

A I L l J L A 1 l l A A l I I l 1 A l

0 0.5 1.0 l.5 2.0
DEPTH (x3 A)

 

FIG. 56. Normahized ver+ical amplifude versus depfh for
05" = I-IO, Q; = l.0 and 0—: 0.45.

 

 

 

 

 

l.5
m 1.0
O
D
t:
_J
a
2
<1
.1
<1
2
.—
[I
uJ
> o 5
Q .
LIJ
E
.J
<!
E
I
0
Z
O
p— q
A A k l l I A L A l A L l A J l I L l
O 0.5 LO l.5 2.0
DEPTH (x3/x)
FIG. 57. Normalized verTical amplifude versus depfh for

i-IO, Qi' = [.0 and0'= 0.5.

NORMA L [ZED HORIZONTAL AMPLITUDE

 

 

I
1

I
l 4 AL

 

 

 

-O.5 --
— -4
I l I A J A A J 1 l l L I l J J I l L
O 0.5 LO l5 2.0
DEPTH (xa/x)
FIG. 58. Normalized horizonfal amplifude for Q;' = l~!O,

and Q;’ r O, and 0': 0.25.

NORMALIZED HORIZONTAL AMPLITUDE

.0
or

C3

I90

 

 

 

 

 

and Q;' = 0, and 0': 0.35.

-O.5 - _.
L— _.
'- -1
l L I I l 4 A1 I l A l 1 A l l l
O 0.5 LO |.5 2.0
DEPTH (xa/x»
. FIG. 59. Normalized horizonfa! ampli+ude for le l-IO,

NORMALQED HOWZONUH.AMPLWUDE

 

 

 

 

 

 

.L.11....i.4.411...
0.5 LC) L5 2.0
DEPTH(X3/M
FIG. 60. Normalized horizonfal ampliTude for 0“ = 1—10,

5
and Q;' = 0, and or: 0.45.

NO RMAL iZED HORIZONTAL AMPLITUDE

 

I I V T I r T U I l r r I V r r T

Lo~— _, -—
. QK '0 .1

0‘ $0.5 -

i -<

y- 4

,.. _

P 'l
0.5— _.
r- -1

 

CD

 

 

A J l l l l l I L I I A 4 l l J J

 

 

o (15 L0 L5
DEPTH(X3/A)

Normalized horizonfal ampli+ude for og‘
= O, and 0': 0.5.

FIG“ mo
and Q;'

NORMALIZ ED HORIZONTAL AMPLITUDE

 

 

 

 

 

— 4

rm 4
0.5 —- d

I—- —4

In -4

A L l A l L A l l R I L l A l l 1 4L 1
O 0.5 [.0 LS 2.0
DEPTH (Xg/M
FIG. 62. Normalized horizonfal amplifude for Q;' = l-IO,

and 0;” = O.l, and 0': 0.2”.

NORMALIZ ED HORIZONTAL AMPLITUDE

f94

 

 

 

 

 

“0.5- ..
p— —.
i- -<
4 A a A _L L4 1 A L L L j 1 J L l J _L__J

O 0.5 LO l.5 2.0

DEPTH (X3/M‘

Fch 5:50 Normalized horizonfal amplimde for Q;‘ = 5—!0,
and Q;' = O.I, and <r= 0.35.

NORMAL [ZED HORIZONTAL AMPLITUDE

l 9 '2

 

 

 

 

 

 

 

and Q;'

i—' -I
P 1
J l L J 1 4 4 J 1 l l I I l l l J J I
0.5 LO LS
DEPTH (Ks/M
HQ. 64. Normalized horizonfal ampli‘rude for Os" = l-lO,

= O.|, and 0’: 0.45.

2.0

NO RMALIZED HORIZONTAL AMPLITUDE

£3
01

O

'05

 

 

 

 

£3
*1.
I.
0
41414

1L1

l

 

 

.. AJAA.AJLJJLJJ..JI
0 C15 I.C) L5 2.0
DEPTH (X3/k)

HQ. 65:. Normaifized horizom‘fa! ampii‘fude for (9;: = l-IO,

and Q;'

= O.|, and O'= 0.5.

197

 

NORMALIZED HORIZONTAL AMPLITUDE

 

 

 

 

'05 P- -'*
L ‘
_ J
l A L J I A i ‘ R R A I I 4 l l I l l
O 0.5 LO l.5 2.0
DEPTH (Xs/x)
FIG. 66. Normalized horizonfai amplifude for Q;‘ = l-IO,

and Q;’ = n.0, and <r= 0.25“

398

 

.0
UI

NORMALIZED HORIZONTAL AMPLITUDE
O

 

 

 

 

“0.5 -— ._.q
k I l A l i l J L i l _j_ J A l A 4 J l
O 0.5 LO |.5 20
DEPTH (X3/A)
FIG. 67. Normai‘l’zed horizonfai ampli‘fude for 05" = 1-10,

and of = 5.0, and (r = 0.35.

NORMALIZED HORIZONTAL AMPLITUDE

.0
w

I99

 

 

 

 

0,2' = ID d
0' = 0.45 ~
.4

 

 

O
“0.5
4
.4
J I L J J A L l L R i j J A J l l
O 0.5 LO [.5 2.0
DEPTH (X3/X)
FIG. (38. Normalized horizonTal ampli‘rude for 0;, = 1-10,

(7“!
and ‘n

= [.0, and 0”= 0.45.

NORM ALIZEO HORIZONTAL AMPLITUDE

I.O

0.5

200

 

 

 

 

 

 

*- -1
l .1 I l L L. L 1 _4L AL 1 1 A A l l l l L
O 0.5 |.O LS 2.0
DEPTH (x3100
FEG. 69. Normaiized horizon+ai amplifude for Q;' = l-EO,

and Q;’ = ;.o, and 0': 0.5.

20|

This is no longer True, and The depTh aT which The parTicle moTion
reverses is The depTh aT which The lengTh of The minor axis becomes
zero. AnoTher criTerion for The parTicle moTion reversal was given
in secTion 7.5 (reTrograde if cos S:>0, prograde if cos 5 < O ).
The raTio of The semi—minor axis To The semi—major axis mulTiplied
by The algebraic sign of cos 5 (hereafTer called axis raTio)
has been compuTed (figures 70 — 8|).

The following conclusions regarding The axis raTio are
apparenT from figures 70 - 8|:

I - elasTic and low—loss ( 0 é 0;. S qu é O.l)

a) The axis raTio as a funcTion of depTh for elasTic solids
is The same (wiThin Three significanT digiTs) as The
corresponding axis raTio for low—loss solids,

b) The depTh aT which The parTicle moTion changes from
reTrograde To prograde decreases from approximaTely
O.|9 A for O‘ = 0.25 To O.l3 A for U = 0.5,

c) The decay in The axis raTio wiTh depTh is monoTonic in
The depTh inTerval of Two wavelengThs,

II - moderaTe—loss (O.| 4 C§J or O.| A 05')

a) The difference in The curves for moderaTe-loss and The
corresponding curves for low—loss decreases as 0‘
increases,

b) The depTh aT which The parTicle moTion reverses from
reTrograde To prograde varies beTween O.l A and 0.4 A ,

c) for 0;! 5 g?” g | The parTical moTion reversals occur
beTween O.IA and 0.2 A ,

d) The decay in The axis raTio wiTh depTh is noT monoTonic

AXIS RATIO

'05

 

202

 

 

 

 

 

 

 

'0 T I V r T r T T T T I r I I' l’ T U V T
Q§'=O '1
0‘=O.25

q

.1

4

.1

.4

J 4

d

2 U“

4

.. \\ 3 —

r- -1
p “==E=:u3

._ . - -
r l l l L R J I A L l l I l l 1 l L I J

O 0.5 LO l.5 2.0

DEPTH (x3/M

51o. 70. Axis raTio (see Texf) versus depfh for Iowflloss and
moderaTe—Ioss maferiais wiTh Q;' = O and 0’: 0.25.

LO

0.5

AXIS RATIO
CD

203

 

I I T I I I I I I T I I I I I I I r I
0;: = o _
0" = 0.35 -
4
—J
‘M .1
Hi. .4
.\\ J
\x. I
- 1

 

 

l I T T I

 

   

 

 

       

 

 

—o.5- \\
.— \\\\\\ 6 i
_ "'.\‘\\*x.>m ¥i_ V 7‘
. 8
.PL11414114L.,
- I '00 0%5 I.O 1.5 2.0

7-30.

DEPTH (X3/M

7i. Axis rafic (see fexf) versus depfh for low—loss and

moderafe—loss maferials wiTh 0;] = O and 0': 0.35.

204

 

l-O 1' I I fl l I’ I I f l T 1 I I I I I I I I.
' -1 -
r- QK ’ "‘
- o-=a45 4
_ 4
0.5 —'

 

 

AXIS RATIO
(3

 

 

 

 

_LO .- 4 4 A A l J L 4 L l 4 4 ; 4 J 1 L l A
O 0.5 LO |.5 2.0

DEPTH (x3/x)

FIG. 72. Axis rafio (see Texf) versus depfh for low—loss and
moderafe-loss ma+erials wifh Q;' = O and G = 0.45.

205

 

 

 

 

'0 T r T V I r V I T I T 1 r Y I I r V r
F -
r— Q;'=O .4
' a- :05
F' -
r— .—
' 4
7- _.
.J
—1
9 -
P ’III
< d
a:
m -
x ‘
‘4
b ...
b n
‘O.5r- ._
#— ~
r—I —
r- 4
#— a-d
r. ..

 

 

 

-I.O 1 A L 1 J L L L A J L L L n 1 J J 1 1
O 0.5 LO [.5 2.0

DEPTH (x3/x)

FIG. 75. Axis rafio (see Tex?) versus depfh for low—loss and
moderaTe—loss ma‘ferials wh‘h Q;’ ~ = O and er: 0.5. ~

206

 

 

TwT

 

AXIS RATIO

 

 

 

‘05

 

T T T I 1 1 1 l 1 T T ‘l 1 I T T T T 1

 

-|.O A A L ; J l # L‘L I

A I J l L A

 

 

O 0.5 LO

I.5

DEPTH (xs/x)

FIG. 74. Axis rafio (see Texf) versus depfh for Iow—Ioss and

moderafe—Ioss maferials wifh Q;

==O.l and O‘= 0.25.

 

 

 

 

 

 

 

 

'0 r r 1' T I r 1' I v T T T r r I I T l r
-I _ 4
OK ' O.| d
0"-' 0.35
.J
0.5 —~
‘2\ -
\
9 r— "
.—
<I * .
a: o ;\
m \ .
x — mu
4 \
_ \\\ 4
.— A
p— -1
“0.5"— _‘
,. 4
-|.O A J_ A A l A A A A 1 4‘ A A J l J A A A
O 0.5 LO |.5 2.0

DEPTH (xs/x)

FIG. 75. Axis rafio (see TeST) versus depfh for low-loss
and modera+e—loss maferiais wifh Q;’ = 0.l and 0': 0.35.

208

 

 

 

AXIS RATIO

 

l L l L A J L J l I L J J

 

 

A J

 

0.5 LO l5 2.0
DEPTH (Xs/X)
FIG. 76. Axis rafio (see Tex?) versus depfh for low—loss and

modera’re—loss materials wifh 0;" = 0.! and 0': 0.45”

209

 

[.0 T l’ T l’ I I I V Y T V V T T T I T I T
p —
'- o;' = 0.I ~
. .
_ CT' =13.5 _
P n
F. -
F- "l
~ 4
0.5 —
F -4
_ -
q

 

 

AXIS RATIO

 

 

 

 

 

--4

b fi

‘05 —' __

I- '1

I— .1

—'.O- A A A A l A A L A J A 4 A A l A I A A q
0 0.5 I.O L5 2.0

DEPTH (xs/x)

FIG. 77. Axis raTio (see fexf) versus dep+h for lowvloss and
mOderaTe—loss ma+erials with Q; = 0.I and G'= 0.5.

2iO

 

 

 

AXIS RATIO

 

 

 

’ 0.5

 

 

 

 

—IO A 4 A A l L A A A l A A A A l A A A A
O 0.5 [.0 LS 2.0

DEPTH (x3/x)

FIG. 78. Axis rafio (see Tex+) versus depfh for low-loss and
moderaTe—Ioss ma+erials wifh 0;! = [.0 and W = 00253

 

 

 

 

 

 

 

 

 

I O T T T I I I V V T I I T f T I T V f
-| _ __‘
oK - LO
0" = 0.35
0.5 ‘ —
.J
p— —-
O P J
.— F— ‘ —-{
g P K‘A‘J“: -
0 ”5:3“
m \‘x .
‘>‘< ’ ‘\‘\\
<I _ \\ ..
\\ ‘
_. N
* \\ Q§' ' *
, \\ 3 . I
‘05 — x -
‘— V: ’ _‘
—l O 1 L l l 1 A A L J l A 1 L I l L J
O 0.5 LO l.5 2.0
DEPTH (x3/x)
FIG. 79. Axis rafio (see TexT) versus depTh for low—loss and

moderaTe—loss maferials wifh Qfl =

l.O and 0'= 0.35.

 

'0 Y T Y Y T T r if V I' Y r I r I T I I I'
t- ‘1
P- Q- = I'O —
P -4
0' = 0.45
F- -—1
P u
b- —
, ..

 

0.5

 

AXIS RATIO
0

 

 

 

 

 

 

 

”0.5

P u!
I— ~
r- ..
_ —-.

 

 

”IO 1 A 4 1 J 1 A L J l n L L L l n L A I
O 0.5 LO '5 2.0
DEPTH (Xs/X)

FIG. 80. Axis rafio (see Texf) versus depfh for low—loss and
quera‘re-Ioss ma‘rerials wifh 0;] = |.O and O” = 0.45.

 

213

 

 

 

 

 

 

 

 

I0 I l' I I’ T V r I I l’ T T I I I’ I’ 1' I I ~
*' 0.2' = LO ‘
r ‘
_ a" = 0.5 _
)- -1
C155{— “‘
F -1
9 -
+- ._
:g .
o
(n d
3; q
.,
-0.5— ..
_l.O P J L J_ l l A l J A l A L l J J A I A l
o 0.5 1.0 LS 2.0
DEPTH (X3/A)

FIG. 8!. Axis ra+io (see Texf) versus depfh for low-loss and
moderaTe—loss ma+erials with Q;' =

|.O and_0'= 0.5.

e)

2l4

in The depTh inTerval of 2 A , in conTrasT To The
monoTonic decay for low-loss,

for The range of parameTers considered The parTicle
moTion changes from reTrograde To prograde or vice—

versa only once in The depTh inTerval of Two wavelengThs.

The TilT of The parTicle moTion ellipse has been compuTed

using equaTion (7.96) for Q;‘ = o, 0.1 and (25": o, O.l, 0.2,

~-, |.O (figures 82 - 89). The following conclusions concerning

The TilT are apparenT for The range of parameTers considered:

O S

l)

2)

3)

4)

5)

6)

The TilT is non-posiTive aT The surface, rapidly increases
To a maximum negaTive value beTween 0.05‘A and O.lA , Then
decreases and for CT#CL5 and cg” ¢ 0;’ becomes posiTive
beTween 0.7A and IA ,

for low—loss (o .4. 0K“ £— 0520.” The maximum TilT as a
funcTion of depTh varies beTween O and - l.5 degrees,

for larger amounTs of absorpTion, O"’> O.l The values

of TilT beTween 0.05A. and O.l A are sufficienle large

To suggesf The design of experimenTs for iTs measuremenT,
The TilT aT each depTh increases as 0;} increases wiTh
fixed values of 0' and 65",

The TilT decreases aT each depTh as 0' increases for fixed
values of G§4 and Q;',

for Q¥":= Gfi'lThe TilT is zero aT The surface, rapidly

increases To a maximum negaTive value beTween 0.05)A and

O.l A , and does noT assume any posiTive values.

The TilT for larger amounTs of absorpfion (l é cg” 5 IO and

q§"4 l ) are ploTTed aT a reduced scale (figures 90 — lOl).

~

TILT (DEGREES)

 

 

 

r r T T l r7 f r r I Y r T r " fi' T
Sb- QRI=O
- 0' =0.25 J
p -|-
#- QS -
O

 

 

 

L L L l L L

 

 

,l .0
DEPTH (x3/x)

FIG. 82.
depfh for IOW’IOSS and moderaTe—Ioss maferials wiTh Q;‘
and 0': 0.25.

Tilf of +he axis of The parficle mofion ellipse ver5us
= O

TILT (DEGREES)

2!6

 

 

 

 

 

 

 

 

 

 

 

.rrvTrrrrfijrw.r,r1.
, .
5—- Qk'=0 _,
P o-=o.35 1
- _
P
v- fl
_
r 1
PQs-l:
r- ~<
o 0 ,,,
P /.. 1
,
e
4
e
4
q
q
._
‘
q
,
a
4
a
,
-
1
.q
1
_ 4
r 1
_ _
. e
#- --1
. .
VLLLLLL‘LLLLLLL1L4eL
0 0.5 LO LS 2.0

DEPTH (x3/x)

FIG. 83. Ti|+ of The axis of The parficle mofion ellipse versus
dep‘rh for low-loss and modera‘re-loss maferials wifrh Q;' = O
and 0': 0.35.

TILT (DEGREES)

 

-|_
SF- QK‘O
' 0"O.45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A L I L l L g I I L 4 l l L L A l J

I

 

 

0 0.5 1.0 |.5
DEPTH (x3/x)

FIG. 84. Tilf of The axis of The parTicle moTion ellipse versus

depfh for low—loss and moderaTe—Ioss maferials wiTh Q;'
and 0‘: 0.45.

2.0

= O

TILT (DEGREES)

2l8

 

 

 

 

 

 

 

fir I' V I I Y Y fir T I I IV V T l I V 1 V
k I q
+— Q»? = —.
¥ 0‘ = 0.5 ‘
P— -I
hit .1
P 1
F "‘
P —| - 4
p 05" 4
r 4
O
J ‘
.2 j
.3 '4
.4 "
.5 ‘
~ +
v- ‘7 '*
P 9 q
,_ 1
P 1
__ 4
P 1
F- ‘1
r- H
r ‘1
r 4
L ‘1
r '1
*- —1
P '1
Jr 1 L L L L I L I l l 4 1 1 l L L l L
O 0.5 I.O LS 2.0
DEPTH (x3/x)
FIG. 85. TilT of The axis of #he parficle mofion ellipse verSUS
depfh for low-loss and moderaTe—loss maferials wiTh Q;’ = O

and 0‘: 0.5.

TILT (DEGREES)

219

 

 

O 0.5 LO l.5

V V V T T r T fir I l' T I 7 I V V V I
Q": 0.: _+
0‘ = 0.25 .

 

 

 

PLLIJJLlLL;JilJlgLJILLLLJIJ

 

l L L L l l J_ 4 I l l l L L l L k i 1

F"
o

DEPTH cx3/x)

FIG. 86. Tilf of fhe axis of +he parficle mofion ellipse versus
dep‘rh for low-loss and moderafe-loss ma‘l‘erials th Q;' = 0.!
and 0‘= 0.25.

TILT (DEGREES)

220

 

Y Y T I r l I’ I 1 l V T 'Y I' T l' I 1 T
og' = 0.: _q
o- = 0.35 *

 

 

 

 

 

 

fl
-‘
4
—q
J
4
>— -—.
- 4
y— '-I
_L L l l l A A; L l l L L l #L i L L L L
O 0.5 I.O 1.5 2.0
DEPTH (x3/x)
FJG. 87. Til+ of The axis of The pariicle mofion ellipse versus
depfh for low-loss and moderafeeloss maierials with Q;' = 0.!

and 0‘: 0.35. ‘

T'LT (DEGREES)

 

1
fi
w
.4
1
-+
4
.4
j
r -1
_ l 1
~ %
L +
'lO — 4
~ +
r 4
- 4
- —<
_ J
— J
- J
L A; . L l . 1 l l L l .
0 0.5 LO [.5 2.0
DEPTH (xs/x)
FIG. 88. Tilf of fhe axis of The parficle moTion ellipse ver5us
depfh for low—loss and moderaTe—loss ma+erials wifh 0;” = 0.!

22!

 

T I' fit I T‘Y‘r 1 T I' I I [fl ‘ T 1 l
-|-
0K-o.u
o-=o.45

 

 

it

 

 

 

 

 

 

 

 

 

and o— = 0.45.

(DEG REES)

TILT

ZZZ

 

 

 

 

 

 

 

r r r r I r eT r l V r r a I . i r r
F ,
5__ op: 0.: __
r- 0—=O.5 -
L— _
_ -
E -
L
E -
y. .
#— -~
q
q
.q
_ -
>— 4
~ 1
- -1
L .
-|Or—— -+
L- -«
L— —-.
L -«
*- "1
_ q
_ ._
. 1 J L 14 . 1 . 1 . .4 4 J 1 J A .
O 0.5 1.0 I.5 2.0

DEPTH (X3 /)\)

FIG. 89. Til‘r of +he axis of The parflcle mo‘rion ellipse versus
depfh for low—loss and moderaTe—loss ma+erials wH‘h Q;' = 0.!
and O” = 0.5.

7/ 3

 

TILT (DEGREES)

 

 

 

 

P If Y 1’ Y r V T V Y I r Y r I ‘ V V
' 02' = 0 :
y.
20 ’—' 0" = 0.25 ——.J
y- ..
C 1
r- "-4
I q
IO r-—- ——<
P +
, 4
I. J
}- -4
L-
o f— - 4
» ~4
; 4
r- ~1
P —l _ 4
+ 0s ' .
f I ~
“IO _ <
r- 1
C. ' 2
t /‘
F 3
P 9'1
'20" 7 lO '1
4 6 4
5 '1
1
.1
’30 "f‘
4L 1 A l l l L L A I L l l I l l L L .‘
O 0.5 LO l.5 2.0
DEPTH (x3/x)
FIG. 90. TilT of The axis of fhe parficle mofion ellipse versus

depfh for Q" = l, 2, --—, IO, Q" = o and 0' = 0.25.
S K

TILT (DEGREES)

 

 

 

 

P r r V Y T T Y r T 1' Y I Y I T Y I
: Q.. - O
, K '
20—- 0- = 0.35
r
r 1
IC)P‘ 5:
I J
C 4
- '1
)- q
l- -
I 4
C)?—
“ /
: -l -<
L 03‘ J
i I. ' 3
-IO -:
C 2
. A
’ 1
3 '1
h . 1
y- .i
-20.— 5 -i
' 4
6 1
9 +
r— 7 8 ‘1
4
q
/ 1
. // 1

 

\\

ILLLALLLLPALJ’LLLILILI

o 05 ID |.5
DEPTH (x3 A)

P°
(3

FIG. 9|. TilT of The axis of The Rarficle moTion ellipse versus
depfh for Q;‘ = I, 2, -—-, 10, Q; = o and 0‘: 0.35.

TILT (DEGREES)

 

£3

*1.

II

CD
1L1 L1

N

O

9

O

‘c.

on
I

r. .2

5 .
‘LLijjlillej

 

  

 

 

 

JLLLLLJJL‘JJLJlJJLJ‘lJAlLJJ11111L111AIL.

L L L I L l L A J L L A L k L L I

o 0.5 LO L5
DEPTH (xg/x)

b
p

{‘0
0

FIG. 92. Til+ of The axis of +he par+icle mofion ellipse versus
dep+h for 05 = l, 2, --~, IO, QK" = o and cr = 0.45.

TILT (DEGREES)

22‘;

 

20

lllllljljllljlllllilelJ

ITIYY‘IVfirYW1771111T1j‘IYVj7

 

   
   

 

IO

-20

'30

Illl‘lAJA‘AAILLAJJEl‘LJAJ1111llLlLJ

VYV‘IYIYV‘V‘Fr'YrWVTTTYTY

 

 

 

N
O

0.5 I. 0 I5
DEPTH (xs/x)

CD

FIG. 93. Tllf of +he axis of +he parTicle moTion ellispe versu;
dep’rh for 05' = I, 2, ---, [0, Q;’ = o and cr = 0.5.

TILT (DEGREES)

227

 

' fl ‘7‘ Y i t V ‘ Yj V j r V ‘ I ‘ T V
0’ = O.| ;
20 0': 0.25
1
IO —

 
 

JL‘AAVAJJ4JJJIJA

It

 

 

U

y- .

N j 1 J C)
JLJJ JAJJ jALJ LIJ 111

7...AL.L.L1L4.4ILLL

O 0.5 LO L5 0
DEPTH (XS/X)

FIG. 94. TilJr of fhe axis of The par‘ficle mo’rion ellipse versus

depfh for 05" = l, 2, ———, lO, 0;’ = 0.1 and 0’ = 0.25.

 

 

V ' V Y I T I' 'fi 1 V 1 I V 1 I V I T ‘

Q = 0.! <

20 0': 0.35 _-1
.1

IO .

TILT (DEGREES)

 

 

411L111lLlALllilAlLJJlllngllili

P
l-
p
F"
b
D
+
P
>-
D
’
p-
F—
P
y-
p

0 0.5 LO L5
DEPTH (X3/X)

FIG. 95. TilT of +he axis of The par+icle mofion ellipse versus
dep‘l'h for 05" = I, 2, --—, IO, of = 0.1 and 0" = 0.35.

.N
o

229

 

q
‘
q
q

—
q

6
q
q
1
4
.1
1
L44J¢JLJilJJLlLJL

q

og' = 0.:
20 a- = 0.45

 

 

 

TILT (DE GREES)

 

A‘AAJ‘LJAL11*J_1AIJAALLLLJ+1J4111J¢11LL 41“"

 

L I L I L A L L L L L 1 i L 1

l L
o 0.5 LO LS
DEPTH (xs/x)

r

 

l“
a

FIG. 96. TiH' of fhe axis of ’rhe par-Hole moflon ellipse versus
depfh for o;‘ = I, 2, ---, IO, Q;' = 0.! and cr = 0.45.

TILT (DEGREES)

 

20

IO

JAILJL;AJJJLJ4JAALJIJJJLJ

 

 

J.
C)

'30

L4;ALJLJAAJAJJLJLIILLLAAJJJJLLLALLJ

1Y7T‘IIYT1'YTITTVVVTIVITV

 

 

L A A L l A A A L J. A L A A L L P J L

LO LS
DEPTH (x3/x)

 

C3
.0
on
.N
C)

FIG. 97. Tilf of The axis of fhe parficle mofion ellipse versus
depTh for 05" = I, 2, -—-, IO, o;' = 0.1 and (T = 0.5.

23!

 

V Y T I l V T V V I V V I' V T V 7 f I 1

Q: =I.O 3

20 o- = 0.25 1
u‘

1

1

fi

W

q‘

,

:0 e
,

.4

\
\\

 

    

 

 

 

 

A 4
u] ’4”/ V-
LIJ
a: ,
0 1
DJ 4
9 1
1
p— .
g 1
|- .p ""
. 1
.J
‘V
—-1
u‘
1
1 ‘1
1
-20 —+
P -1
h ’ 1
~ 4
r- -<
r 1
~30 ~— -~
L 1
b J I l A L 4 LL L L l L L L A l L l L L
O 05 ID l.5 2.0

DEPTH (x3/x)

FIG. 98.V Ti£+ of The axis of The parTicIe mofion ellipse versus
depfh for Q; = I, 2, —--, IO, 0““ = 1.0 and cr = 0.25.

TILT (DEGREES)

[‘0
DJ
M

 

V ‘ V U r 1 I Y r 1 I r V l’ V 1 r I
Q: =I.O
20 0‘ .= 0.35
IO .
,.aiFi

J

 

 

J

 

ALJJJLAJJJJJJLJJJAALJJJALJJJJJJJiELJJJ

 

 

 

;.
r-
-30 __
L L A; A L L l L EL L 1 L 4 A l l L A l L
O 0.5 LO l.5 2.0

DEPTH (x3/x)

FIG. 99. TilT of The axis of The parTicle mofion ellipse ver3us
depth for 05" = I, 2, ---, IO, of = [.0 and cr = 0.35.

TILT (DEGREES)

20

IO

 

 

 

.
q
q

-1
4
q
4

—_
‘
d
J
4

—-1
4
4

fl
1

9

II

.0

1)

UI
JLJJJJJJJLJJJJJ

2J1

 

 

 

 

 

 

JQJIJJJJJIJLJIJJJJJIJIJJJJJIJJLJJJ

 

I A l A L A L L L 1 L

,-

 

I.O 1.5
DEPTH (x3/x)

FIG. IOO. TilT of The axis of +he parficle mofion ellipse versus
dep‘fh for Q;‘ = I, 2, —--, IQ, of = 1.0 and cr = 0.45.

l“
C)

.334

 

 

 

 

7 ‘ fl

1

1

-—I-‘

{

.4

.{

q

1

4

1

‘1

.1

A 4
W

“J 1

“J +

0: 4

E3 %

e 1

}- d

:1 1

5— —--4

fi

1

4

.4

.1

1

1

—-—¢

4

4

1

q

‘1

+

1

4

'3C) ‘4

1

1

q

l L A l L A L I I l L A J L L L J L 1 +

(3 ClS LC) |.5 213

DEPTH (X3/A)

FIG. IOI. Tilf of fhe axis of The par+icle moTion elli '
_ pse versus
dep‘l‘h for 05’ = l, 2, ---, IO, of = |.O and <7 = 0.5.

235

The TilT for 0— =0.25, QK"=0, 5" = 4, 5, and 0" =

-l -l
0.35, 0K = O, 5 = 8 , 9 , l0 decreases away from The
surface. For The remainder of The values of The parameTers The
TilT rapidly increases away from The surface. The TilT decreases

aT a given depTh as 6’ increases for fixed values of C§4 and GK—L

8.4 ParTicle MoTion CharacTerisTics for a few RepresenTaTive

MaTerials

An experimenTal range for The parameTers GE" and QQJ was
presenTed in secTion 8.22. ParTicle moTion characTerisTics for
This range of The parameTers were compuTed in secTion 8.3. The
parameTers for a few represenTaTive maferials are TabulaTed (Table
ll). For These maTerials measuremenTs of velociTy and aTTenuaTion
exisT for boTh shear waves and dilaTaTional waves. The firsT six

seTs of parameTers are for maTerials which may be classified as low-

A

loss maTerials (i.e. 0 5- O;"€ O.|; O 5 G['._ O.l). The
remaining four seTs of parameTers are for wafer safuraTed soils
which mighT be classified as moderaTe-loss maTerials ( Q;l> O.l).
Given The velociTies and absorpTion coefficienTs for shear and
dilaTaTional waves and The frequency aT which The measuremenTs were
made, The parameTers Os" , Q;' , 0' , and 0;! may be quickly

compuTed from The following formulas,

95"; (”WU/H — _|_<"-°s"s)z)
, w 4 w

 

 

 

o;’ = (”PM/u—

J_(§_‘EE_V_F’)2)
L0 4. LU

(34)

TABLE II

Paramefers for Represenfafive Maferials

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v xlo’3 v x10“3 Q‘fo105 Q‘foIOS
. 5 p S p 0"x102 Q"x102 Q"x102 0 Ref

Maferlal (m/sec) (m/sec) (sec/c) (sec/c) s p k '
Dural
(mefal) 3.00 6.02 .351 .215 .335 .410 .450 .335 Vikforov
Glass
(mirror glass) 2.60 5.31 1.23 .314 1.02 .530 .300 .342 Vikforov
Polysfyrene
(plasfic) 1.05 2.28 9.29 .178 3.1 1.29 .580 .366 Vikforov
earfh's manTIe
(1500 km) 6.50 11.8 .210 .0613 .435 .230 .0907 .282 Kanamori
earfh's man+1e
(8| km) 4.30 7.75 .914 .507 1.25 1.25 1.25 .278 Kanamori
Pierre McDonal,
shale .802 2.162 40.8 4.51 10.4 3.10 1.47 .421 ef al.
wafer safurafed Hamilfon,
soil (BS-14B) .108 1.458 162. .948 55.5 .441 0 .497 ef al.
wafer saTuraTed Hamilfon,
soil (DS~147) .101 1.457 1950. .717 62.5 .332 0 .498 e1 81.
wafer saTuraTed HamiITon,
5011 (08-151) .088 1.454 2750. .883 76.9 .408 0 .498 ef al.
wafer saTuraTed Hamilfon,
5011 (Tower) .197 1.798 3190. 4.62 200. 2.64 0 .495 ef al.

 

 

QEZ

237

“=ll—‘é‘3 9L5)f_2)/(2(_V_Pfl9§" )HI)

5 siog’) Vs 6(05')
-I_ l—G‘ —l _,2 -l
_ .___(3 ~ Z(._._._._' 0‘) .
Q“ l+o- Qp 1—5" Os )

CompuTaTion of 5%, from Oilor vice versa may be simplified wiTh
The use of Table (l2).

Using The parameTers for shear and dilaTional waves, characTer~
isTics of a Rayleigh-Type surface wave on a half-space of The
corresponding maTerial were compuTed (Table l3).

For The maTerials dural, glass, and polysTyrene, VikTorov
also reporTs measuremenTs of The velociTy and absorpTion coeffici-
enT for a Rayleigh—Type surface wave. Comparison of The TheoreTi-
cal values lisTed in Table I3 wiTh The measured low—loss values
shows reasonable agreemenT.

For The low—loss maTerials TabulaTed, The velociTy of a Rayleigh~
Type surface wave is 0.92 To 0.94 ThaT of a homogeneous shear wave.
The absorpTion coefficienT for a Rayleigh—Type surface wave is l.02
To l.4 ThaT of a homogeneous shear wave. The TilT of The major
axis of The parTicle moTion ellipse is less Than - 0.337 degrees
aT The surface and less Than - |.Ol degrees aT a depTh of beTween
0.05A and O.lA . The only maTerials lisTed for which 0;, is
less Than 0;! is dural. For such maTerials The TilT of The

parTicle moTion ellipse aT The surface is posiTive, i.e.

TABLE «2
0,;70; as a Function of 03/0; and a

 

 

 

 

   

 

  

 

 

 

 

 

 

 

 

 

 

 

°:fias O. .025 . ' .{"f -" .1" .“\ .775 .32: .753 .375 .40C .425 .45f .475 .5CC
O. .657 .656 . 3‘ ."‘ .“‘3 .571 .549 .515 I L .379 .767 .222 .174 .121 .063 .075
.". .€’i at q .Lv- .ETE .‘ 3 .59? .95 .317 .742 .325 .275 .24? .195 .743 .'%7 .'25
. .58" .‘ T . .*1‘ . l! .””3 .37; .5»( .378 .342 .3“? .261 .215 .165 .119 .f5C
. ‘ o 97": .‘ ‘7 .4“: . 3‘) .“ '4 ."T‘ .‘~'1 .7545 .36) .‘22 .2731 .236 .1117 .134 .575
. ' .‘ a 1 . at 1 .>‘K .&14 .‘“4 .fi7? I .472 .411 .377 .34“ .357 .257 .20? .157 .!09
. - . n .: _ . 31 ,-.3 .. . . -w .)?v . 7 . 3n .427 .39w .T:3~1$=¢ .£?7 .2;1 .1“: .125
. A- .f7' .“54 .f‘* .517 .557 .52fi .512 .474 -4L4 .472 .777 .339 .299 .353 .274 .150
.w7‘ ."- o'44 .644 .57” .‘65 .‘4? .315 .43q .46' .429 .595 .353 .!18 .375 .227 .175
.3 ‘ .:”' .t74 .577 .6?” » .574 .DJ6 .431 .5‘5 .477 .446 .4l3 .373 .339 .297 .251 .230
.7." .7 . .f‘”. .’~{"‘1 .5')’ .’."Z .c‘? .99’ .fi'vq .946 .9!) .493 .463 .432 .397 .?00 .319 .274 .225
.7T< .7C .574 .“7q .htZ .344 .62} .6”; .391 .‘b’ .515 .5”? .43} .1‘5’7 .417 .333 .341 .293 .253
- . ~ 7 ‘7 ' .17 . . ..“ rW: .-“’ .57; .f31 .5?¢ 1‘V- .‘fi? .‘3? .L’i I}53 I3E§ .2’5
7' .5”4 .569 .5r“ .531 .511 .59” .567 .542 .515 .437 .455 .422 .73‘ .344 .3"C
"" .544 .52 .574 .392 .536 .531 "5 .475 .342 .4“7 .358 .325
.656 .533 .519 .509 .575 .55“ .523 .494 .463 .429 .391 .355
.571 .5: .634 .613 .591 .557 .542 .514 .464 .451 .415 .375
.58“ .6é7 .549 .679 .5’7 .585 .55C .533 .Efih .47? .433 .490
cj‘v .41; .-», .1 v .0.“ .fi'f ., a . 73 .f?: o. f . -2 .v3:
.71; .604 .57d .65? .645 .b‘° .537 .972 .545 .517 .495 .45:
.723 .7“8 .492 .675 .655 .637 .615 .592 .566 .53€ .538 .475
.737 .7? .777 .69” .673 .654 .633 .611 .9%7 ' .532 .SCG
.75'7 .736 .7?2 .7“6 .49¢ .677 .552 .631 .673 .58 .555 .525
.763 .75“ .736 .72! .776 .fiflfi .670 .65C .673 .655 .579 .55C
.11" .T,T.rfi,v . IJI .rgc .111 .a In .rfi, “.07 .317 07.! .31)
.789 .778 .760 .752 .738 .723 .7'7 .689 .67” .648 .625 .Cnfi
.ar_z .703 .Nv .75»? .7-3< .74.": .125 .7-‘-'»s .500 .mc .549 .1325
.836 .576 .7“5 .783 .771 .7‘5 .743 .725 .711 .b“2 .672 .659
.329 .419 .Q3Q .TQW .7q7 .775 .76? .747 .73? .714 .é96 .575
.342 -3 .“24 .314 .R34 .79: .79W .767 .752 .7?¢ .719 .7FC
.53: .3a: . .' .fi. ..(u .51“ .rio .'Un ..!J .rjfi .'5 .ILD
.65.: .351 .931 .M‘S .nkh .ny7 .817 .‘Hfi .701 film; .76!) ,7S",
7 .193 .1 .“nh .“bl .837 .944 .325 .525 .314 .612 .753 .775
» C; .395 .689 .q§3 .975 .ebq .862 .855 .R44 .335 .924 .“13 .840
1 .fi?‘ .QTI .717 .GEZ .03“ .C3‘ .HQ7 .QQZ .BHS .879 .372 .fi64 .855 .flhb .835 .925
“<7 .S‘h .Vfé -“LU .°25 ."21 .017 .C12 .037 .QL? .896 .370 .983 .E7b .4OE .“47 .95?
~. .vw. .44“ HI' .‘fir .-.h n!) .1”! .VA: .va .Il. .VJW .74! .VfV .WYY . r, .,A;
v>n .957 .155 95‘ ."K1 . 47 .944 .541 .958 .935 .931 .927 .922 .“17 .912 .9“6 .900
C74 .Qé” .WIO .*~4 .943 .96! .0‘4 .056 .934 .951 .94? .945 .942 .099 .934 .977 .°?5
‘VC ."7" .“77 .G7L .97“ .97“ .Q7? .QTX .th .047 .C65 .963 .961 .959 .956 .953 .?50
3‘1 .§79 .QJG Uri .087 .957 .936 .9QS .QQQ .984 .983 .982 .931 .Q79 .975 .977 .975
1.! .“ . I." 1.7 z.” 1.: .4 ..' ..C 1-3 1.U [.3 !.5 1.0

 

 

 

8E2

Characferlsflcs of a Rayleigh Type Surface Wave

TABLE l3

for Represenfaflve Materials

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- 5
valo aR/fxlo age: Max. Tm R21}: at HIO $237.14) 4P¢z.xl) 4%.“) dam”) 4‘1'AW.’ Apoz"¢z’ Ang'sz)
Mafer'a' (“/5”) (see/m) (deg.) Meg.) surface ”30 4 (deg.) Meg.) Meg.) ((195).) (deg.) (deg.)
Dural
(mefal) 2.80 .383 +.009 —.025 .638 .638 .I7 —.080 .,o39 .IO9 .267 89.97 89.77
Glass '
(mirror gJass) 2.43 l.28 -.052 -.|l5 .634 .634 .I7 «.282 «.078 .3l8‘ .79I 89.96 89.29
Polysfyerene
(plasfic) .983 9.60 «.I57 -.346 .622 .622 .I7 —.877 «.23I .954 2.47 89.92 87.76
earth's
manfle 6.02 216 -.O33 —.057 .665 .665 .I8 —.IIB -.03| .l38 .3ll 89.98 89.72
(1500 km)
earth's
manf|e 3.97 .990 -.000 -.l|5 .668 .668 .I9 —.308 *.l37 .4I7 .936 89.89 89.20
(8| km)
Pierre
Shale .757 42.4 -.337 -|.Ol .59I .591 .13 -2.99 -.803 3.ll 8.84 89.88 8|.96
wafer safurafed
soil (DS—I48) .l03 I69. —.083 -3.84 .545 .545 .l3 °l4.5 —4.35 I4.5 4|.l 90.00 53.29
Mafer safurafed
soil (DS-l47) .0964 2040. -.062 -4.2l .545 .545 .I3 -|6.0 —4.83 I6.0 44.0 90.00 50.80
wafer safurafed
soil (OS—l5!) .0840 2880. -.077 -4.96 .545 .545 .I3 -I8.8 -5.72 l8.8 48.9 90.00 46.83
wa?er saturated
soil (fower) .l88 3340. -.498 -8.72 .547 .547 .l2 —|0.2 3|.9 64.0 89.98 36.I3

 

 

~3|.9

693

240

 

However, The TilT quickly changes To negaTive values wiTh depTh.
For The small TilTs predicTed, The raTio of The minor To major axis
of The ellipse and The raTio of The horizonTal To verTical
ampliTude are The same To wiThin Three significanT figures. The

angles which The propagaTion vecTors 5;
2

and fi; Torm wiTh The
2

surface are less Than one degree. The angles which The aTTenua—

Tion vecTors form wiTh respecT To The verTical are less Than 2.5

degrees. The angles for Pierre Shale are lllusTraTed schemaTically,

—30° 5' -
’3 F","

    

For moderaTe-loss maTerials, few measuremenTs of maTerial
properTies exisT in The liTeraTure. AccuraTe measuremenTs for
such maTerials are needed for experimenTal verificaTion of The
disTincTions beTween a Rayleigh-Type surface wave on anelasTic
maTerials and one on elasTic maTerials. The measuremenTs for waTer
saTuraTed soils reporTed by HamilTon, eT. al. ([970) are based on
The assumpTion ThaT Q;‘ = O. AddiTional measuremenTs are needed

To verify This assumpTion. Using The measuremenTs reporTed by

24|

HamilTon, The velociTy raTio, ”h /\g , TheoreTically predicTed is
.95 for each of his four sTaTions. The absorpTion raTio an’las
is l.O45 — l.046. The predicTed TilT of The major axis of The
ellipse is less Than O.l° aT all sTaTions excepT The "Tower"
sTaTion for which The TilT is O.5° (see Table l3). However, The
predicTed maximum TilT aT a depTh of 0.|A for each sTaTion is
considerably larger ( - 3.8° To - 8.7°). A carefully designed
field experimenT should be able To measure The maximum TilTs. The
angles which The propagaTion and aTTenuaTion vecTors make wiTh
respecT To The surface and The verfical, respecTively,are much
larger for The moderaTe—loss maTerials Than for The low—loss

maTerials. The angles for The ”DS—l47” sTaTion are illusTraTed.

 

 

The paTTern of The parTicle displacemenTs in a Rayleigh-Type
surface wave aT a fixed Time are illusTraTed (figures l02 - 106).
The inTersecTions of The grid lines may be inTerpreTed as maTerial
parTicles. In The absence of a Rayleigh-Type surface wave The grid
lines form perfecT squares. The grid begins aT The sTarT of a
wavelengTh. The parTicle displacemenTs normalized by The maximum
verTical ampliTude aT The surface were mulTiplied by a scale facTor
for illusTraTion purposes.

The paTTern of The parTicle displacemenTs for an elasTic

 

FIG. 102. Parficle displacemenf paTTern a+ a fixed Time for an elasfic haif-space
(scale facfor = O.|, see fexf).

ZVZ

 

FIG. l03. Parficle displacemen+ pafTern a+ a fixed Time for a half—space wifh
+he paramefers of Pierre Shale (scaie facfor = 0.1, see Texf).

 

FIG. [04. Parficle displacemenf pa++ern aT a fixed Time for a half-space wifh
The paramefers of wafer safura+ed sedimenTs (sfafion DS—l48; scale facfor = 0.2,
see fexf).

VVZ

FIG.
parame’rers 0;' = 05“ = I and (r = 0.25 (scale fam‘or = 0.3, see +ex+).

I05.

 

Parficle displacemen+ paffern af a fixed fime for a half-space wifh

QVZ

 

FtG. l06. Parficle disp!acemen+ paffern af a fixed Time for a half-space

wifh paramefers Q;" = 0.l, Q;'
fexf).

IO and 0': 0.499 (scale facfor

0.3, see

9V2

247

half-space wiTh 0' = 0.25 is shown in figure l02. No horizonTal
decay in ampliTude occurs for an elasTic half—space. A slighT
horizonTal decay in ampliTude is apparenT for The parTicle
displacemenTs on a maTerial wiTh The parameTers of Pierre Shale
(figure IO3). The parTicle displacemenT for a moTerial wiTh
parameTers of waTer saTuraTed soil (sTaTion "DS-l48") aTTenuaTe
rapidly over a horlzonTal disTance of one wavelengTh (figure IO4).
The increased curvaTure of The verTicaI grid lines for a moderaTe-
loss maTerial (waTer saTuraTed soils (figure l04) shows The superim-
posed sinusoidal dependence on depTh of The parTicle displacemenTs).
Two exTreme examples of parTicle displacemenTs are illusTraTed
(figures l05 and l06). The parameTers for These solids are noT
very realisTic; however, The superimposed sinusoidal dependence on

depTh of The parTicle displacemenTs is readily apparenT.

248

8.5 References

HamilTon, E. L., H. P. Bucker, D. L. Keir, and J. A. WhiTney,
VelociTies of compressional and shear waves in marine sedi-

menTs deTermined in siTu and from a research submersible,

J. Geophys. Res., 12, pp. 4039-49, I970.

Jackson, D. D., and D. L. Anderson, Physical mechanisms of seismic-
wave aTTenuaTion, Rev. of Geophys. and Space Phys., §J pp. l—

60, I970.
Kanamori, H., SpecTurm of P and PcP in relaTion To The manTIe-core
boundary and aTTenuaTion in The manTle, J. Geophxs. Res., 1g,

pp. 559-7l, l967a.

Kanamori, H., SpecTrum of shorT-period core phases in relaTion To
The aTTenuaTion in The manTle, J. Geophxs. Res., 12, pp. 2|8l—

86, l967b.

Knopoff, L., Q, Rev. of Geophxsics., 2, pp. 625-659, I964.

Kolsky, H., STress Waves in Solids, Clarendon Press, l958.

 

Love, A. E. H., A TreaTise on The MaThemaTical Theorv of ElasTlciTv,

Dover, I927.

McDonaI, F. J., F. A. Angona, R. L. Mills, R. L. Sengush,

R. G. van NosTrand, and J. E. WhiTe, ATTenuaTion of shear and

249

compressional waves in Pierre Shale, Geoghxsics, 22, pp. 421-

439, I958.

Press, F. and J. H. Healy, Absorp+ion of Rayleigh waves in low-Ioss

media, J. A991. Phxs., gg, pp. 1323-25, 1957.

VikTorov, I. A. (frans. by W. P. Mason), Rayieigh and Lamb Waves,

 

Plenium Press, I967.

250

9. CONCLUSIONS

The basic maThemaTical framework for problems of plane mono—
chromaTic wave propagaTion in a generalized isoTropic linear
viscoelasTic maTerial has been presenTed. Classical elasTiciTy
Theory for plane wave propagaTion is a special case of The general-
ized formulaTion.

The energy in a sTeady-sTaTe viscoelasTic radiaTion field is
given by equaTion (4.7). The fracTional loss in energy densiTy per
cycle of forced oscillaTion Tor homogeneous bulk, shear, and dilaTa-
Tional waves is given by expression (4.l5), (4.29), and (4.34),
respecTively. These expressions are independenT of The low—loss
assumpTion.

AnalyTic soluTions for The reflecTion of a general (i.e. eiTher'
homogeneous or inhomogeneous) incidenT SV wave and The problem of a
Rayleigh—Type surface wave have been derived. For each of These
problems many of The TheoreTical resulTs for anelasTic media can
22: be predicTed from The corresponding resulTs Tor elasTic media.
The basic reason is The Type of inhomogeneous wave which can pro—
pagaTe in The Two Types of maTerials. For elasTic media, The only
Type of inhomogeneous wave which can propagaTe is one for which
planes of consTanT ampliTude are perpendicular To planes of consTanT
phase. For anelasTic media This is The only Type of inhomogeneous
wave which can 29: propagaTe.

A general incidenT SV wave upon inTeracTing wiTh The Tree sur—
face of a HlLV half—space generaTes a general SV wave and a general
P wave. An exTended version of Snell's law is valid for The

reflecTed waves, in which The veolcoTies are Those of general

25l

waves (equaTion 6.26). The componenT of aTTenuaTion along The
surface for boTh reflecTed waves is equal To ThaT of The incidenT
wave. For an incidenT homogeneous SV wave, resulTs of The reflec-
Tion problem are:

l) The reflecTed SV wave is homogeneous,

2) The reflecTed P wave is homogeneous if and only if

XI
sin 263. S R

.. I __ ‘
, K + 4:2k BHd QS = QP I, (ExperimenTal
R 3 R

evidence shows ThaT for mosT anelasTic maTerials
05—! # QP-'-),

3) no criTical angles of reflecTion exisTs for anelasTic
solids such ThaT QP_| < QS-l, (ExperimenTal measuremenTs
in The laboraTory show ThaT QP-|<( QS—' for a majoriTy
of maTerials. All in—siTu measuremenTs To daTe for The
earTh show Qp-| S 03".),

4) The compleTe conversion of a non-normally incidenT

homogeneous SV wave To a P wave can occur only for

viscoelasTic solids such ThaT QP— = Q8. '
For an incidenT inhomogeneous SV wave, in an anelasTic half-space
resulTs of The reflecTion problem are:

I) The reflecTed SV wave is inhomogeneous,

2) The reflecTed P wave is, in general, inhomogeneous,

3) for every angle of incidence 951, There exisTs an
associaTed Yfi such ThaT 6Si is a criTical angle for
reflecTion,

4) for normal incidence The ampliTude of The reflecTed

dilaTaTional wave is noT zero.

In addiTion, a general incidenT SV wave upon inTeracTing wiTh The

252

free surface does noT generaTe a Rayleigh-Type surface wave.

TheoreTical resulTs for a Rayleigh—Type surface wave on a HILV

half—space are:

|)

2)

3)

4)

5)

The general anaIyTic soluTlon of The complex Rayleigh
equaTion (7.7) is The analyTic expression (7.29), where
The parTicular rooT ég'of (7.29) which saTisfies l§§1 < l
yields The desired soluTion for a Rayleigh—Type surface
wave,

The propagaTion vecTors B and associaTed wiTh The

‘$

4’2 11’2
dilaTaTional and shear displacemenT poTenTials respecTively,
are inclined wiTh respecT To The surface (The only Type of
viscoelasfic solid in which boTh are parallel To The
surface is elasTic),
similarlyJThe aTTenuaTlon vecTors7§¢2 and-K1p2 are inclined
wiTh respecT To The verTical (The only Type of viscoelasTic
solid, in which boTh are perpendicular To The surface is
elasTic),
The normalized horizonfal and verTical ampliTude disTri—
buTions (equaTion 7.90) for anelasTic solids exhibiT a
superimposed sinusoidal dependence on depTh, which does
noT exisT for elasTic solids,
characTerisTics of The ellipTical orbiT described by a
parTicle as a funcTion of Time are:
a) The moTion is reTrograde if cos S >() and prograde if

cos 8 <fO (S is defined by equaTion (7.95),

b) The TiIT of The principal axes of The ellipse is given

by equaTion (7.96), which shows for anelasTic solids

The TiIT depends on depTh,

6)

7)

253

c) The ellipTiciTy is given by equaTion (7.97) and
depends on depTh,
For The special class of viscoelasTic solids in which

05-! : QP—I

= QK-', addiTional resulTs are:

a) The velociTy raTio vR/vs depends only on o and is
independenT of The magniTude of 05" = QP-| = QK—I,

b) The absorpTion coefficienT raTic>aR/as for anelasTic
solids depends only on o and is independenT of The
magniTude of QS-l = QP-I = QK-I # 0.

c) The TilT of The parTicle moTion ellipse is zero aT The
surface, buT is nonzero aT depTh Tor anelasTic solids,

d) The raTio of The maximum horizonTal To maximum verTical
parTicle displacemenT aT The surface (H.0/H50) is The
same as The axis raTio of The parTicle moTion ellipse,

e) boTh The raTio H'o/H30 and The axis raTio aT The

surface depend only on o and are independenT of The

magniTude of GS. = QF,‘| = QK-l'

For The special class of viscoelasTic solids in which

0 = 0.25 andQ ‘ =Q ' =0", (i.e.}\=/U) resulTs in
s P K

addiTion To Those lisTed in "6") are:

a) The velociTy raTio VR/VS = 0.9l94,

b) The absorpTion coefficienT raTio aR/QS = l.0877,
c) The angles e and eS which The propagaTion vecTors

P

5¢ and fl; , respecTively are inclined wiTh respecT To
2 2

The surface are resTricTed by relaTions (7.57) and

(7.58) and for anelasTic solids lesl < lepl,

254

d) The angles 6 and es which The aTTenuaTion vecTors

P

-l
I; and Aw , respecTively are inclined wiTh respecf To
2 2
The verTical are resTricTed by relaTions (7.63) and

(7.64), and for anelasTic solids 6 4 95.

P

The characTerisTics of a Rayleigh Type surface wave on a HlLV

half-space have been compuTed numerically as a funcTion of The Three
parameTers QS-l, QK-|' and 6'. The numerical compuTaTions are
independenT of a parTicular viscoelasTic model. A general summary
of The resulTs from The numerical compuTaTions is:

l) for low-loss solids (O < Q_lé O.l) The error percenTages
for The velociTy and absorpTion coefficienT of boTh body
and Rayleigh—Type surface waves are less Than I percenT,

2) for moderaTe—loss maTerials (0") O.l) The expressions
for The velociTy and absorpTion coefficienT of a Rayleigh—
Type surface wave derived wiTh The low—loss approximaTion
are in general noT adequaTe (e.g. for The wafer saTuraTed
soils of HamilTon errors up To 43% in Vh and |84% in QR
were compuTed),

3) The velociTy and absorpTion coefficienT for a Rayleigh»
Type surface wave on low-loss and moderaTe—loss maTerials
were compuTed (figures l7-23),

4) The TilT, axis raTio, and raTio of The maximum horizonTal
To The maximum verTical parTicle moTion aT The surface of
low—loss and moderaTe-loss maTerials were compuTed (figures
25—35),

5) The normalized verTical and horizonTal parTicle moTion

ampllTude, The axis raTio and The TiIT were compuTed as a

6)

7)

8)

255

funcTion depTh for low—loss and moderaTe—loss maTerials

(figures 36-l0l),

The majoriTy of The parTicle moTion associaTed wiTh a

Rayleigh-Type surface wave is concenTraTed wiThin one

wavelengTh of The surface, and The parTicle moTion becomes

more concenTraTed near The surface as The amounT of

absorpTion increases,

The TilT increases rapidly To a maximum beTween 0.05) and

O.lA below The surface, also The TilT increases aT each

depTh wiTh an increase in The raTio QS-I/ QK_I,

for low-loss (Q_'é 0.!) anelasTic maTerials (e.g. dural

(meTal), polysTyrene (plasTic), glass,crusTal and manTle

maTerials of The earTh, and Pierre Shale), The general

numerical compuTaTions show:

a) The magniTude of The TilT aT The surface for 0.25 élr
is less Than l.2°,

b) The maximum magniTude of The TilT aT a depTh of
approximaTely O.IX is less Than l.6°,

c) The axis raTio aT each depTh is To wiThin Three
significanT figures The same as for elasTic maTerials,

d) The normalized parTicIe moTion ampliTudes aT each
depTh are To wiThin Three significanT figures The
same as The corresponding elasTic ones,

e) The magniTude of The angles which The propagaTion
vecTors are inclined wiTh respecT To The surface are

less Than approximaTely 3°,

256

f) The magniTude of The angles which The aTTenuaTion
vecTors are inclined wiTh respecT To The verTical are

less Than approximaTely |O°.

9) for moderaTe—loss (Q'—I

> O.|) anelasTic maTerials (e.g.

waTer saTuraTed soils), The numerical compuTaTions show:

a) ThaT The magniTudes of The TilTs and The magniTudes of
The angles associaTed wiTh The propagaTion and
aTTenuaTion vecTors can be subsTanTially larger Than
The magniTudes lisTed in "8"), wiTh The upper limiT of
The magniTudes depending on The upper limiT of The
absorpTion defined for moderaTe—loss maTerials,

b) for The measuremenTs on waTer saTuraTed soils reporTed
by HamilTon, The predicTed magniTude of The TiIT aT The
surface is less Than O.|°, The predicTed maximum
magniTude for The TiIT aT depTh is beTween 3.8° and
8.7°, The angles associaTed wiTh The propagaTion

vecTors E¥h and 3W; , range beTween —l4.5° To 3|.9°

and 4|.l° To 64.0°, respecTively.

The preceding resulTs for a generalized viscoelasTic half-
space suggesT consideraTion of several addiTional problems con—
cerning plane wave propagaTion in generalized viscoelasTic maTerials.
The Theory and The numerical resulTs are presenle being exTended To
mulTiple layers of viscoelasTic maTerial. ExperimenTal resulTs
(laboraTory and in-siTu) arelneeded for confirmaTion of The Theory.

The resulTs of This disserTaTion suggesT a general Theory of
plane wave propagaTion in viscoelasTic media is especially relevanT

To earThquake engineering and geophysics. in earThquake engineering

257

such a Theory is useful for describing sol! response and The
associafed earthuake hazards. ln geophysics fhe Theory is

useful for obfaining refined esfimafes of fhe infernal sfrucfure

and composifion of fhe earfh.

257a

IO. ACKNOWLEDGMENTS

The U.S. Geological Survey's NaTional CenTer for EarThquake
Research generously provided financial assisTance during The course
of This research.

A discussion concerning The energy in an elasTic radiaTion
field wiTh Dr. D. L. Judd (AssociaTe DirecTor of Lawrence Berkeley
LaboraTory, Berkeley, California) was very helpful.

LockeTT's paper (I962) sTimulaTed consideraTion of The problem
of incidenT inhomogeneous plane waves. I am very graTeful To
Professor J. L. Sackman, who called my aTTenTion To LockeTT's paper.

Professors P. W. Rodgers, T. V. McEvilIy, J. L. Sackman and
L. R. Johnson kindly provided Their Time and energy for discussions
and reviewing of The disserTaTion.

I especially appreciaTe numerous discussions wiTh fellow

sTudenT and good friend, J. C. Coggon.

258

APPENDIX I

Some properfies of The Riemann-Sfielfjes convoluTion are
sTaTed below. Their proof is given by Gurfin and Sfernberg (1962,
p. 298-9).

Suppose f is confinuous on [0,00] and HH =0 for f<0 and

suppose The firsT derivafives of g and h exisf on [O’Do] and

gtTJ=hLH=0 For +<0, fhen

(I) f :e dg = 9 w dh (Commufafivify),
(2) f *dtg*h)=(€nd9)«dh = F*dgudh (AssmflafivHv),
(3) F*d(g+h) =Ffl'd9 + Fur dh (Disfribufivify),

(4) de9=0implies i=0 0" 3:0,

“
9(0):”) + J; mum dflc‘f‘) d“? {or Os-réoo.
w

(5) {gas

259

APPENDIX 2

The purpose of This secTion is To prove Theorem I sTaTed in
secTion 6.2.2. The proof given here is similar To ThaT given by
LockeTT (I962). To prove The Theorem, iT is convenienT To firsT

esTablish a lemma.
LEMMA l- Aka/Ila is a real number if and onl '1‘ Q" - Q"— “
' p 5 Y' S _ P _ QK '

a 1
Using equaTions (3.30), (3.3!), (4.29) and (4.34), fip / I;

may be expressed in Terms of The parameTers of The solid as

11“ l + i Q;'
“wig-“n niog‘ ‘"

 

Lemma I follows from relaTion I and equaTion (4.37).

To esTablish The necessary parT of The Theorem, suppose The
incidenT SV wave and The reflecTed P wave are homogeneous, (i.e.,
a; = a; = 0 ). Then, since 5;: and ix}: are assumed parallel,

There exisTs a non—negaTive consTanT c such ThaT

P, = ax, . (2)

For anelasTic media, definiTions (6.3) and (6.4) TogeTher wiTh (2)

imply

260

it = 3:: ('_C_t_L1 = ‘3. <3)
1% 1i: c-mi) it,

Therefore, ¢‘/.k is a real number and The definiTion of d,‘z implies

2 2
d“ 2, <4)
’3. .. .2 — l 2.0 -
5

Now, since The incidenT wave is homogeneous, equaTion (6.l3) shows

relaTion (4) for 95. 1: 0 may be wriTTen as
I

 

Z Z
d k
1-2"; z —|zO. (5)
R “s sin 65.
I

The desired conclusions follow from relaTion (5). FirsT, relaTion
(5) shows ibPz/fi: is a real number and hence by lemma 1, qg' :Q;I=Q;'.

Also, relaTion (5) shows

p
P

‘°~
h

 

(6)

”l
“N
x
x
4.
all:
:3
a:

EquaTions (4.|5) and (4.34) show Tan 85 = Tan 8K . The preceding

argumenT esTablishes The necessary parT of The Theorem.

26l

To esTablish The sufficienT parT of The Theorem, suppose

I) The incidenT SV wave is homogeneous ,

2) Q: = p" y

and

3) sin 19 5 11,

By lemma I, The assumpTion Q§'

= 03' impHes -&:/ k: is a
real number and

(7)

The Third assumpTion TogeTher wiTh equaTion (7) implies for 95.¢0
ThaT

___..—._. . (8)
ha sin tég,
I

Now, since The incidenT wave is homogeneous,

z z . 2
R - -&5 sun asi

(equaTion (6.l3)), hence, inequaliTy (8) may be rewriTTen as

Z. 2. 2
lip-1! = f; 20, (9)

from which iT follows ThaT d‘)/3 is a real number, say c.

dq =: c R and d = c 2 . These expressions show ThaT The
R R “z I

Hence,

262

propagafion and affenuafion veclors for fhe reflecfed P wave may be

wriffen as

-* A A

P = R ( x ‘- C X

“’2 R ' 5) (10)
and

A¢z—lkxl(§,—c>’25l. (H)
Equafions (IO) and (l I) show Ei,2 is parallel To i?;a , which

esfablishes The sufficienl par? of The Theorem.

263

APPENDIX 3

The purpose of This secfion is To prove Theorem 2 sfafed in
6.2.2. For The exisfence of a crifical angle, 13¢: musl be
parallel To The boundary, Thai is, d“a musT be zero (definifion

6.3)). Hence, fhe definifion of d“a implies

mm}; = id.“ d‘! = 2 up 1i, + law“, i sin 95_
z ' I

sin(95i-rs;))=0. (ll

Definifions (6.3) and (6.4) imply

Pw'Aw='* Jis , (2)

which for anelasfic media may be wriffen as

" A ._ ”s *3
la]! IAw" - "' R t o (3)
£05 6;;

 

Subsfiiufion of equaTion (3) info equafion (l) shows

. 3 9?
8m 95. sin (65‘. - X53) = cos 3'5; __'_&_..‘.'.I_
I

ll,“ *5; <4)

Simplifying equafion (4) gives fhe desired relafion

,2 2?,
3m 9,; —- 7%
fan [5;7' 5: 5: 95,-?‘037’73' (5)

sin 9% cos 9:;

ReIaTion (5) esfablishes The validify of Theorem 2.

264

265

APPENDIX 4

The purpose of fhis secfion is To prove +heorem 3 sfafed in
secfion 6.2.2.

If will be sufficienf To esfabllsh Thai if The incidenf SV wave
is homogeneous and 637 is a crifical angle fhen 01’s 0;.-

For an incidenf homogeneous SV wave (5;: O and fheorem 2
(secfion 6.2.2) implies Togelher wifh relafion (6.28) +ha+ for

anelasiic media

fi
sinz(95fl = '3 && . (H
1‘ fi‘

Z Z . l
a = *s 5'" 9s; - (2)

For 6,), To be a criTical angle 5.,2 is parallel lo fhe sur—

face, hence, definifion (6.3) shows
dd : O. (3)

The defini‘rion of d: ; jgpz—fiz and (3) Together wiTh (1) show

266

z z . 2
-d“: = (1”,,I - 1b,: ) — smzesltk‘e - R; ). (4)

Since —da1

2

is a negative real number, equafion (4) Togefher wifh

(I) and relafions (3.32) and 3.34) show

-2 a“ a,‘ é - a a,“ 2),: (5)
Z 7. 2 l
ks - *3: “p. — fi’x

The lefT—hand side of inequalify (5) is 0;” (equafion 4.34) and

The righT-hand sideis di (equafion 4.29). Hence, inequaliiy (5)

is The desired inequalify,

(pg' s 09".

267

APPENDIX 5

The purpose of This secTion is To show ThaT The planes of
consTanT phase associaTed wiTh The Two poTenTials ¢ sand W for a
Rayleigh-Type surface wave are noT boTh perpendicular To The surface
unless The solid is elasTic. ln oTher words, if The solid is ane—
Iasfic, The propagaTion vecTors f5}: and 75“,?- are noT boTh
parallel To The surface.

To prove This resuiT , suppose The conTrary, ThaT is, suppose
fi?éa and 5}: are parallel To The surface. If This is The

case definiTions (7.7) show

bqufi=0. (I)

RelaTion (6.2!) for The principal values TogeTher wiTh equaTion (I)

imply

2 2
lb? I - Re L b5 J

H
0

(2)

and

lbji- Retbfl

n
O

(3)
2 2 . .
EquaTion (2) TogeTher wiTh The definiTion of bk = h - h: Implles

268
and
z a
a—A‘>Ic,a-&‘. (5)

a z
‘1

Similarly, equafion (3) Togefher wlfh fhe definifion of b: = fiz— Q2

P
implies
14,,be = 2," ll" (6)
and
2 z 2 2
hR—n,>fi,‘—lkplao, (7)

For a Rayleigh-Type surface wave 3 musT safisfy equafion (7.4),

Thai is

4lizbdbs = (&2+ bazlz. (8)

Using equaiions (l) The real par? of equafion (8) is

(b2

8“ + (a: -&f))’ - <an lg)" = 4bdnbaatlbf-kal (9)

and fhe imaginary parl of equaTion (8) for anelasfic media is

2. 2
a-fi, . (l0)

b’ a

269

SubsTiluling equalion (l0) inlo equaTion (0) and simplifying yields

The following expression in Terms of “a and ‘kL ,

2_ z 2. z 2.
(2R QIHQS‘!‘ “5) -Jé_(ksR—asi) +zafafzofl'”

Solving equaTion (ll) TogeTher wiTh equaTlon (4) for anelasTic

solids gives The following pairs of rooTs for Q: and 9:: ,
z _ z _ 2
an ~ .L( 1‘s“ *s,’ (12a)
2 _ 2 z z z
81 ‘ Z 35R “5‘/( “Sa ‘ fis‘) (I2b)
and
z _ _ 2 z 2 z
“a Z Jga.fisx /l 959- RE: ) (l3a)
2_ _ I 2 z
“r * “a" *5“ ~ “51) - (l3b)
. 2 1- .
For anelasTic solids Rs, - R3! > 0 (relaTIon 3.32) and also
R: is a non-negaTive real number. Hence, The second seT of

rooTs (13a) and (l3b) is noT physically possible. This leaves The
firsT seT as The only possible seT of rooTs for .2: and .2:

BuT relaTions (IZa) and (l2b) imply

270

z z 2 2
9R: _ k: 3 '32 _ “32 _ ( Z *5 k, + a, - I}; )
" 1 2 2 ' (l4)
‘3, " 5x a

The quanTiTy Inside The parenThesis of equaTion (I4) is posiTive

. . 3 z . .
for anelasTIc SOlldS. Hence, ha - k: > 0 Implies

& - a 4 5 — 9‘ . (l5)

BUT relaTion (l5) is conTrary To relaTion (5), which esTablishes The
desired resulT. ThaT is, if The solid half-space is anelasTic Then
boTh of The propagaTion vecTors associaTed wiTh The displacemenT
poTenTials for a Rayleigh—Type surface wave are 22: parallel To The

surface.

27l

APPENDIX 6

The purpose of This secfion ls +0 show Thaf each of The Three

roofs z; (n = ',Z ,5) of +he reduced equafion (7.13) is equal

To one of The roofs z;' (n = I, Z, 3)-

Definifions (7.l8) and (7.l9) imply The relafion

5 (l)
WiThou+ loss of generalily suppose p in equaflon (I) is such +haf

(v )"5 - — D
.V. .. " ——-———_.._. -
P 3 p.v.LV+)'/5 (2)

For such a p , if will be shown fhal

z; := Zn+ (n = 1,2, 3 ).

 

(3)
For example, consider n = 2 . Definllion (7.22) implies
Z’ = ~v‘ u — P
z o ,
3‘6.“ _(4)

where

(5)

272

and

u E e:(p L i 2 U'/ 3 J .

Subsfifufion of definlflons (5) info (4) and simplifying wifh

equafion (2) ylelds The desired resulf

”3 ovi/s P
Za=p.v.(v+) e ———_. =2,". (6)
us 4U1/3
3p.v.(V+) 6
Similarly, one may show zn" = 25* for h = I and n = 3.

273

APPENDIX 7

The purpose of This secTion is To show ThaT if cz/flZ is a rooT
of The complex Rayleigh equaTion (7.7) such ThaT o < lcai/’|Bzi 4 l,
Then cz/pzsaTisfies equaTion (7.5) and in Turn saTisfies The orig—
inal equaTion (7.4). Such a resTricTion on The Three possible rooTs.
is useful for selecTing The one which corresponds To a Rayleigh-Type
surface wave. For convenience, equaTion (7.5) is rewriTTen here in

Terms of The complex velociTies

2 2 Z
-c "a _.c "a- _C.. 2
4(l #2) (I :1) .l2 52) , m

where each of The square rooTs shall be inTerpreTed as principal
values. DenoTing The lefT-hand side of equaTion (i) by f and The
righT—hand side by g, equaTion (I) may be expressed by

F=9 . (2)

The operaTion of squaring each side of equaTion (7.5) To obTain

equaTion (7.6) expressed in Terms of f and g is

f = 9 , (3)

which may be rewriTTen as

l?-9)(F+3)=O. (4)

274

Hence, a rooT of equaTion (4) implies eiTher (ii-9) = O or (F+g)=o,

Hence, To show ThaT a rooT saTisfies equaTion (2) iT is sufficienT
To show ThaT The algebraic signs of The real and imaginary parTs of
¥ and g are The same. WriTing equaTion (4) in Terms of real and

imaginary parTs gives

2 2.
$9-9n=fi-9; (5)

and

F1 FR : 9! 93 ' (6)

EquaTion (6) shows ThaT if ell/fig is a rooT of equaTion (4) such
ThaT The algebraic sign of The real parTs of F and 9 are The same
Then The algebraic signs of The imaginary parTs are also The same.
Hence, iT is sufficienT To show ThaT if Cz/Bz is a rooT of equaTion
(2) (i.e., equaTion (7.7)) and 0<Ll§_‘|_:'<: Then The algebraic signs
of The real parTs of F and 9 are The same.

Define

275

and (7)

2
e E;:a.v.(l~-.C..)”2
(3 #2

Wiih definiTions (7) The real pari of € inay be wriffen as
‘FR = 4 ( ed 9,“ - e“ e”) (8)
and fhe real parf of g is given by

- 2 2 Z _ Z 2.
gR—|+2Re[ep] + Reiepl Imtepl - (9)

1
By assumpfion, cz/fiz is a roof such fhai' 0 < %%F% < l , hence

Z
2 _ C.
RetepJ‘RBLI‘jj Z0 (IO)
5
and
22 2 a
Mile 1 =(-Im{_§_]) <l. (H)
fi 52

Subsfiiufion of resulis (IO) and (II) info equafion (9) shows The

algebraic sign of 93 is posifive.

276

Definifions (3.|6) and (3.17) for o(and 3 show

 

Z
'3 I < 31
.___ ' ((2)
la 2I 4
Hence, if follows Thaf
z ’- ’-
ReledJ=Re[l—£__€.J>.L> o
- (l3)
«2 fig 4
In addiTion, relafion (6.2!) for principal values shows
2 z
e“ : lea. + Re[ed1
“ 2
Z 2.
:Uiea‘ -Re[e“J
(l4)

leqll ' 2
e :l’lefl + ReLefiZJ
fig _2

left - PeLszl

Z

 

18px] =

Combining relafions (IO) and (IS) wifh relafions (I4) shows

277

and (l5)

>

Combining relaTions (l5) wlfh equallon (8) gives The desired

conclusion fhaf lhe algebraic sign of f“ is posifive and equal fo

Thaf of QR

278

APPENDIX 8

The purpose of This secflon is +0 show

b := 0. 9 l8 R
q 2 5 (I)
and
% : 0.4278 %

(2)

for The case .A :‘11 , where b5 and bq are defined as
principal values by equafions (7.2). As an example, consider b“:

RelaTion (6.2l) implies

2 2
b : fign [Imlhjll Bil—:BEEELl—y
CI 2 (.3)

where

2 Z Z
bq=fi‘“s- (4)

For +he case A :11 , fig : “350 a: , hence

2 2
bq:0.l830 as . (5)

Relafion (5) implies

279

Sign [Im bf] = si9n[0.l830(2 asnafin , (6)

and The inequaiifies (3.33) and (3.34) Togefher wifh equaiion (6.2!)

show
-I if a

signLImij={ . (7)
+] if Q5 3 0

Referring back To equafion (3) and using relaTion (5) gives

 

bi] ~RcLsz o.i830 9‘ 40,1350 3
a d 51 —- 5: ’

= - \
Z (8;

since inequaiify (3.34) is .Q s 0 . Hence, comparing equafion (8)
5:

wifh (7) gives fhe desired resulf

b“: = 0.92.|8 9181 .
Similarly, one may show
bk“2 = O. 92i8 R5“ ,

“O”
H

0.4275 “5,, ,

280
and

b8: = 0.42 75 k5,.

These relafions combine To give equafions (l) and (2).

28!

APPENDIX 9

The purpose of This secTion is To show The algebra necessary
To wriTe equaTion (7.87 in The form of (7.89).
EquaTion (7.87) is

Retun1= D { F" sinlin + an) + dm)

+Gnsin((wl-+ an) +d5n)} (n:l,3).(H

The Terms u)? and ar‘are common To The argumenTs of boTh The

sinusoidal Terms in (I). Regarding Their sum as a single angle and

using The TrigonomeTric idenTiTy for The sum of Two angles, equaTion
(I) may be wriTTen as
ReELln] = D Sin(wf + an) {Fn cosdm + Gncos dan}

+D cosiw’r + an){Fn sindm + 6" sin dan}'(2)

Now for noTaTional convenience leT
¥n 5 F5 cos dm + Gn cos d3”

and ( 3)

3n; Fn sin dm + 6" sin d3“ (n:l,b).

Using (3) equaTion (2) becomes

282

ReEun1= D {fn 5m tun + an) 1- 9" cos tw+ +an)},(4)
which may be wriffen as

2
ReEun] = D +3, +9: sm(w+ + an + an), (5)

where

an E mug—n ) . (5)
9n

Using definifions (3), equafion (5) can be wriffen in The form of

equafion (7.89), i.e.

ReEun] = D Hh sinLuH + an + 9n) , <7)

where

H a (Fnz + G: + 2Fn G" cos(dm-d3n))'/z (a)

and

Fn sin d,n + 8“ Sin d5”

 

a" : fan—’[ ] (n: l,3)_ (9)

F3: cos dm + 6,. cosdm

283

APPENDiX l0

The purpose of This secTion is To show ThaT The parameTric
equaTions (7.94) describe an ellipse, and To derive The properTies
of The ellipse.

The equaTlons (7.94) are rewriTTen here for convenience

Reta.) = D HI sin(.f(T)) aru. (I)

ll

ReEusl D H3 cos(d’(1‘)+ 5) 21¢ , (2)

where

3(7); W1- + 0.! + 6|

5 a as-a, + 93-9, —1r/a ,

and an , 9n (rizi,3)are defined by (7.88). To show equaTions (I)
and (2) describe an ellipse, The parameTer 5 shall be eliminaTed and
a second degree equaTion in 14 and m} derived. Using The Trigono—
meTric idenTiTy for The cosine of The sum of Two angles, equaTion

(2) can be wriTTen as

w = DHSK cos in) (103(5) - sin Jm Sinis> ) . (3)

SubsTiTuTion of (I) inTo (3) and simplifying yields

284

 

I w 'u .
._ _— (S .
COS(5)(DH3 + DH, 5'" H (4)

COS 4N?) :

Subsfifufion of equafions (I) and (4) info The frigonomefric

idenfify

sinzflr) + cog: in) = i,

yields The desired second degree equafion

Auz+ Buw + sz-i =0, 6)

where

:9
IN

LDchosIS) )‘3

B E SinziS) (‘6)
D ”3”: cos (5)

c a (0H5 cos isn—a

and for a given parficle The coefficienfs A, B, and C are consfanf

wiTh respecf To fime. The discriminanf of equafion (5) is

a"- —4AC = 4 (DszH, coszcsn‘z (sinzm-H. (7)

285

Hence, for s¢ TT/ 2 The discriminanT is negaTive and equaTion (5)
describes an ellipse whose axis do noT coincide wiTh The axis of The
coordinaTe sysTem (Purcell, i958, p. i30).

To deTermine The direcTion in which a given parTicle describes

an ellipse wiTh Time, consider The polar angle defined by

e a Tan"Lw‘/ui

(8)
or picToriaily
9 x'
l11,url
‘3
If GCT) is monoTonicaiiy increasing wiTh Time (i.e., ‘3': > 0 ),

Then The ellipse is described in a clockwise or prograde fashion.
if 9 (T) is monoTonically decreasing wiTh Time (i.e., Egg 4 0 )
Then The ellipse is described in a counTerclockwise or reTrograde

fashion. The derivaTive of 9 wiTh respecT To Time is given by

éfi. : 15:. L cscz flT) cos (S) . (9)
37 HI 1Azl-1mz

Hence, equaTion (9) implies The ellipse is prograde if COS (5 )(l?

286

and reTrograde if c0$l5)>0. If COS£S) =0Then 6 is noT
changing wiTh respecT To Time and The parTicle is moving along a
sTraighT line which is a degeneraTe case of an ellipse.

AddiTional properTies of The ellipse can be derived by defin-
ing a sulTable roTaTlon of coordinaTes, such ThaT equaTion (5)

reduces To sTandard form. Such a roTaTion is given by
u: u’ cos a - w’ sin 5’ (10)

and

w=u’ sinIr + 10" war, (ll)

where The angle of roTaTion is given by

Tan 2X = (PurceH,lQS8,qu)-
'FO" A=C (‘2)

IT equaTlons (l0) and (ll) are subsTiTuTed inTo equaTion (5) The

desired sTandard form resulTs

12 z
3!. + 21’ =1, (l3)
AI-‘ Ca...
where
I 2 . . 2
A=ACOS T+stn¥COSF+Csm X
and

"Asin‘r — Bsmr cosr +Ccoszll'.

h
I

287

Equafion (13) is Thai of an ellipse whose ellipficify is given by

f.
C’

The angle of rofafion r' defines The +l|+ of The ellipse and

equalion (l2) may be rewriffen using definiflons (6) as

. (i4)

288

References (appendicles)

GurTin and STernberg, M. E., and E. Sfernberg, On fhe linear
Theory of viscoelasflclfy, Archlve of Raf. Mech. Analysis, ii,

pp. 29l-356, I962.
LockeTT, F. J., The reflecfion and refraclion of waves al an lnler—
face beTween viscoelasfic media, J. Mech. Phys. Solids, l9,

pp. 53-64, I962.

Purcell, E. J., Analeic Geomefrx, Applefon—Cenfury-Croffs, I958.

289

l2. COMPUTER PROGRAM LISTINGS

The compufer programs and subroufines used for The numerical

calculaTions in chapter 8 are listed.

290

PROGRAM DEPT (INPUTOOUTPUTOTADE 99I

DIMENSION A‘3IIRI3IOCROOTISIOACROTI3IoNUMBI3I0R03I3OIIR05I3OIO

lHAFTlSq)IVAFTI59)0HA159)IVAI59I9TILTDIS9IOSPECSISO)IBUFXISOOII

28UFVISOOIvDLPTHIS9)oPRISlIIARI59InYISII

COMPLEX CSTRMoCBTRMoCVELRcAIR oSCROTpCROOTICCBR
c............'.....CIQC........I...............‘........................

PUPPOSE

NROB I NUMBER OF

ROBIJ) 8 ARRAY FOR

NROS 3 NUMBER OF

ROSIJ) = ARRAY FOR

NPR I NUMBER OF

PRIJI ' ARRAY FOR
REMARKS

AND RCHAR

nnnnnnnnnnnnnnnnnnnnnnnnnnn

PROGRAM DEPT (R.o. BORCHERDT)

COMPUTE AND PLOT CHARACTERISTICS OF A RAYLEIGH TYPE SURFACE
HAVE AS A FUNCTION OF DEPTH

DESCRIPTION OF INPUT PARAMETERS

VALUES OF RECIPROCAL O IN BULK
RECIPPOCAL O IN BULK

VALUES OF RECIPROCAL O IN SHEAR
RECIPROCAL O IN SHEAR

POISSON RATIOS

POISSON RATIO

SEE SECTIONS 7.5 AND 8.3.1 FOR RELATED FORMULAS
FOR COMPUTATION PROGRAM CALLS SUBROUTINES CUBICoSELCT-

FOR PLOTTING PROGRAM CALLS SUBROUTINES AXLILI.SAXLIR.$AXLIT.
PFLIthNXTFRMoAND GDSEND

PROGRAM I5 SET UP TO COMPUTE PROPERTIES AT SURFACE-WITH MINOR
MODIFICATIONS OF INITIAL PARAMEIERS PROGRAM MAY BE USED TO
COMPUTE CHARACTERISTICS AS A FUNCTION OF DEPTH

cI0.0IOIOOICICOGDIIi...G.IICC.OOCOOOIOOOOOOOIO‘IIOII...RUDD...llif.IDill

C
C INITIALIZE PARAMETERS OF DEPTH ARRAY AND POISSON RATIO ARRAY

TEST-IE-IO
AIII"OO
DPR'0001
DD'OoOS
DOI'noOZ
SPRIO.O
50-0."
NPR-SI
NDI‘I
NO'I
REID IOOIINROBI
100 FORMATIIIOI

READ IOIOIRORIKIoK-IvNROR)

I01 FORMATIBFIO.3I
READ lOOoINROS)

READ lOlolROSIKIvK'IoNROSI

TPRISPR-DPR

DO 9 KiIcNPR

TPR'TPRODPR
9 PRIKInYDR

DO 8 XII-30

20

SPFCKlli-o.0
c GENERATE DEPYHS son COMPUTATION
10-50-001
00 20 K:1.N0
100:00
IFtK .LE. N01) 100:001
TD-TDoYDO
ocpru(x)-10
spec5111 - 2.0
spccstzy - 1.0
595(513) - 0.5
sPscs¢u1 - 0.0
595(515) - 1.0
SDECS(6) - ~00.
SPFCSt7) - 5.0
SPECS(B) - e.
595(519) . so.
SPFCSIIO) - «1.
svrcst11) - 1.
Sprcst121 - 99.
specstxan - up:
sprcs1141 - 1.0
svrcst151 - 1.0
sper<tzsy - -0.1s
sorc5127» . -0.15

“On

nn

DO FOR RECIPROCAL O BULK

DO 1 lROB-loNROB
ROEF:ROR(IR09)
CBTRM-CMPLX(lovR08F)
CALL AXLILlispECS)

CALL SAXL!R($PEC§l

CALL SAXLIT(SPEC§)
SPECS(25)ISPECS(25)-Oo!5
SPECS(27)ISPFCSIZ7)-O.15

00 FOR RECIPROCAL O SHEAR

DO 2 IEOSIlvNROS

ROSF-DOS(IRO%)
CSYRM=CMPLX(109R05F)
ASTRM-CABStCSTRM)
OSTRM-SORT((ASTRM-l.)/(ASTRH+1.)1

00 FOR ELASTIC POISSON RATIO

DO 3 IPRIIoNPR
YpRtpRthRl

COMPUTE RATIO OF COMPLEX 5 AND P VELOCITIES

TKUR“10§TDR)/(Io‘(Zu‘YPR,’
TKUR=(2-/3o).TKU9
CVELR:(TKUR'CBTRM)+(6¢'C57RM/3o’
CVELR-CSTRM/CVELR

COMPUTE RECIPROCAL 0 FOR P HAVES

ROP-(R08F+((he/(TKUR'3.)I'R0$F!)lll.¢Ao/(3o'YKURll

.“H

292

C FIND CORRECT ROOT OF RAYLEIGH EQUATION HITH COEFFICIENTS A(lI-A(2)uA¢3)

100

A'ZI-2“0-‘160.CVELRI

A‘3I'“!60‘I6.'CVELR

CALL CUBIC‘AORQTESTI

CALL SELCTTADROCROOTITESTDNROOTQ‘CROTQNUMBI
$CROT‘CSORT‘10/CROOT‘III
VELR'REAL(SCROTI¢(QSTRM.AIMAG‘SCRO'II
VFLP'In/VFLR

ABSR'OQO

IFT 057R" .NEU 0.0 I ABSRIREALISCROTI"(AIMAGISCROTII/QSTRMI
(CHRICROOT‘II

PRINT 300'(RQSFIRQBFITPRIROPIVELRIABSRI
FORMlTl6FI907I

C
C COMPUTE CHARACTERISTICS OF RAYLEIGH WAVES ON LINEAR ANELASTIC SOLID

3
2
1

CALL RCHARIDEPTHoNDIOSTRMOCVELRICCBR'VELRIABSRoHAFToVAFToHAoVAI
lTILTDoAR)

Y‘IPRDITILTDTII

CONTINUE

CALL pFLILI(PRoY IBUFXIBUFVISPECSI
CONTINUE

CALL NXTFRNTSPFCG)

CONTINUr

CALL GDSENOISPEC‘I

STOP

END

293

PROGRAM LLOSSIOUTPUT)

DIMENSION AI3IDRI3)ICROOTI3IONUMBI3IOEVR$RIIOOIO

ISVERISOIoSAERl50)

COMPLEX CSTRM oCBTRM o CVELR v A IR I SCROT OCROOI vCCBR
c'II}.‘§*"§..lIQ...‘5"I’.fiI"..CG.§§..Q".§‘§'...“..‘I‘...l*..'..*"li..

C
C
C PROGRAM LLOSS (Rob. BORCHEROTI
C
C PURPOSE
C CALCULATE LOW LOSS ERROR PERCENTAGES FOR THE VELOCITY AND
C ABSORPTION COEFFICIENT OF A RAVLEIGH TYPE SURFACE HAVE
C
C DESCRIPTION OF INPUT PARAMETERS
C NONE
C
C REMARKS
C SEE SECTION 8.2.2 FOR RELATED FORMULAS
C PROGRAM CALLS SUBROUTINES CUBIC AND SELCT
C
EIQII.§5§.§¢«onloioioiognnii.uQQOQQI.00¢lIconqueloI§GIQO§iluIII§IaG5§IDG
C
C INITIALIZE PARAMETERS OF LINEAR ANELASTIC SOLID
TEST=IE~6
A(I)'-8-
DROSIO.005
DROHIOoOOS
DDR=0005
SROS=OoOOS
§R09=0.0
SPRtO.l
NRGS:IO
NROQIII
NPR=7
C
C DO FOR RECIPPOCAL O BULK
TROBtsRoq-OROB
DO I IROQ-l-NROH
TRORcTRGB§DROB
CBTRM:CMPLXII.ITROBI
C
C DO FOR RECIPROCAL 0 SHEAR
TROStSROS-DROS
DO 2 IROS- loNRQS
TROSITROSODROS
CSTRM=CMPLXI109TROSI
XX- 1. 0 (TROS'i2)
XXZISORTIXX)
OSTRMISORTIIXXZ-IoI/(XXZ #1.!)
C
C COHPUTF LOW LOSS ERROR RATIOS

SVER(IROSItSORTI(XX201.I/(2..XX)I
SAERIIROS I=TROS*$ORT( XX IIZo‘IXXZ-IoIII
PRINT SOOoITRQBoTRQS) .
500 FORMATIIXo/lv' 1/0 BULK t .0Fl6070' 1/0 SHEAR - '9F16c7i

on

00 FOR ELASTIC POISSON RATIO
TPRISPR-DPR
DO 3 IPR=1oNDR
TpfisTPRQDDR

COMPUTE RATIO OF RAYLEIGH VELOCITY TO SHEAR VELOCITY FOR LOW LOSS
IFIIROB .NE. 1 .OR. IROS .NE. 1) GO TO 12
ETKURIII. + TPRIIIIo-(2n*TPR)I
ETKUR-(Z./1.)'ETKUR
EVELR=1./(ETKUR+(h./3.))
A(2)12h.-(I6.‘EVELRI
AIEII—16.+II6.GEVELRI
CALL CUBIC(A9R.TESTI
CALL SELCTIApR-CROOToTESTpNROOTpACROT-NUMBI

900 FORMATI' CHECK.)

SCROT=C$ORT(I./CROOT(1))
EVRSRIIPRI=REALISCROT|+(QSTRM'AIMAGISCROTII
EVRSR(IPR)'l./EVRSR(IP9)

COMPUTE RATIO OF COMPLEX 5 AND P VELOCITIES
12 TKURIK1o+TPRI/(l.-(2.'TPR))
TKUR=(2./3o)’TKUR
CVELR:ITKUR‘CBTRM)+(0.’CSTRH/3aI
CVELR=CSTRMICVELR

COMPUTE RECIPROCAL 0 FOR P WAVES
IFITKUR .E0. 0.) GO TO 15
ROP:ITRQB+((Q¢I(TKUR'3.)I‘TROSI)l(lo+ho/I3-'TKUR)I
15 IFITKUR oEOo 0-! ROD-IF¢20

FIND CORRECT ROOT 0F RAYLEIGH EQUATION WITH COEFFICIENTS A(1)uAI2IuA(3I
AI2)=2h.-(16o'CVELR)
A(3)=-16-*16o‘CVELR
CALL (URICIAvaTEST)
CALL SELCTIAoRoCROOToTESTyNROOToACROToNUMB)
SCROTSCSORTIIo/CROOTIIII
VELRxREALISCROT)+(QSTRM‘AIMAGISCROTII
VELR=1./VELR
ABSR-REALISCROTI-I(AIMAG(SCR0TI)IOSTRM)
ACBRICABSICROOTIIII

COMOUTE LOW LOSS ERROR PERCENTAGES FOR RAYLEIGH TYPE WAVES
RVEP=(1.—(EVRSRIIPR)*(lo/VELRI‘SVERIIROSIII'IOOo
VRSR2=EVRSRIIPQ)"2
VRPR2=IloiITKUR+6./3.))‘VRSRZ
XMNSVRPRZ‘I2.‘VRSRZ)‘I1.-VRSRZI
XMD:VRPR2*(2.'VRSR2)*(Iu-VRSR2)
XMD=XMD-(VRSR2*(1.-VRPR2I'I2.-3o‘VRSR2)I
XMtXMN/XMD
ARSRL=I(XM'I(ROD/TROS)-Ia)I*lo)/EVRSRIIPR)
RAEP:I1.-(ARSRL'SAER(IRQSI/ABSQ))‘IOO.

PRINT 300.1TPR.ROP.VELRvABSR.RVEPgRAEPoACBRoARSRL)

300 FORMATIIX98F16.7)

3 CONTINUE

2 CONTINUE

l CONTINUE
TROSISROS-DROS

50
305

51
310

00 so K-loNROS
rnos-tkos+naos
SVEP-(l.-SVER(K))'IOO.
SAEP-(l.~§AER(K))'IOO.
PRINT 305.(7Ros.svsp.5AEPI
F0RMA1t1X.3F25.7»
rpnsqu-oon

DO 51 KIloNPR
YPR-TPR4DDR

PRINT 310.1TPR.EVRSR(K))
FonuArllx.2F33.7)

SYOD

END

296

SUBROUTINE CUBIC (AQROTESTI

DIMENSION AII). RIII

COMPLEX AvRoQIPIDES “”3022 ICACUtWI OZ oCTcCF
c...‘.'..‘§..§"...*§i'f'.CO.*II..'....’.G.’.I..I'.‘...'*.‘....‘..‘fi‘...’

SUBROUTINE CUBIC (R. O. BORCMERDTI

PURPOSE
FIND COMPLEX ROOTS RIK): K8102c3 T0 NORMALIZED COMPLEX
CUBIC POLYNOMIALo XI'3 + AIII X"Z * AIZI X + AI3) '0.

DESCRIPTION OF INPUT PARAMETERS
AIJ) I ARRAY WITH COMPLEX COEFFICIENTS OF RAYLEIGH EQUATION
R(J) = ROOTS OF COMPLEX RAYLFIGH EQUATION
TEST 8 ACCURACY TO WHICH ROOT MUST SATISFV EQUATION 7.7

REMARKS
SEE SECTION 7.3 FOR RELATED FORMULAS

.I'....III..-lOIOUOOUUII..ICIGIOI...§I...I.....O....II.OOOI...GOIIIOOI..
PIIA.‘ATAN(I.I
ACU=2.'PI/3o
X 13:] 0 /3I

nnnnnnnnnnnnnnnnnn

C

C COMPUTE Vl”3
03(2./27c)'(AIII'*3)
030 — IA(II‘A(2)/3¢)+AI3)
Pt-IA(I)"2)/3.0AI2)
CT-I(Ail)"3I/27.I'A(3)
(TucT-(I(A(I)“2)/IOB.)'AI2I"2I
CT:CT—IA(I)’A(2I'A(3I/6o)
(T-(T +(IAI3)"2)/b.)
CT-CTO (IAI2)"3)/27.I

105 FORMAT!IXo19E26clblIOXoIPEZUulbi

OES=CSORTICTI
H13=-O/2--OES

fifl

FIND COMPONFNTS OF HI FOR POLAR FORM
ABN13=CABSIN13I
IF(A%H13 .E0. 0.) GO TO 2
ABVI3A8W135'XI3
Zl:V13/ABHI3
THETA=AIMAGI CLOGIZZ)!
THETA t THETA/3o
TARGSTHETA - ACU

fin

COMPUTE ROOTS RIK).K'19203
DO 1 K=Io3
TARGsTARGoACU
CACU=I0o01oI
CACU=CACU'TARG
WI=ABNI*CEXP(CACUI
1 RIK)=WI-(Pl(3-‘Hl)I-(AII)/3o)
GO TO 3

297

C COMPUTE ROOTS IF PERFECT CUBIC
2 DO 5 K3103
5 RTK)--A(l)/3.
C
C CHECK ACCURACY OF ROOTS
3 DO 1 IROOTIIIB
CF'(R(IROOT)..3)OIA(1)‘(R(IRO0T)'.2))
CF-CF+(A(2)*R(IROOT))+A(3)
ACFtCABS(CF)
IF(ACF 0LT. TEST! GO TO 7
PPINT lOZ-(CF0R(IROOT))

102 FORMATI/l/lo‘ ACCURACY OF ROOT LESS THAN TEST'o/|* REAL ACCURACY I
1'c1p528.16910Xo'lMAG ACCURACY D‘IIDEZBoIQI/o‘ REAL ROOT -&.1P228.1
ZkoIOXo“ [MAG ROOT .n.10528.14.//)

7 CONTINUE
RETURN
END

298

SUBROUTINE POLAR (ZvAZvVMETA)

COMPLEX loZAZ
{IllIIIGGIIQIG00.9.0000...DO.IOGOIOQOIOOOIIOOODOOOOOOIOIIQOIIOIIOIGOIOII

C
C ,
C SUBROUTINE POLAR (R. D. BORCHERDT)
C
C PURPOSE
C CONVERY A COMPLEX NUMBER Y0 POLAR FORM
C
C DESCRIPTION OF INPUY PARAMEYERS
C 2 3 COMPLEX NUMBER TO BE CONVERTED
C
C REMARKS
C NONE
C
C
COOIIDlil‘lilGIIIOOIOIOOO.GOIOIQOOOOIIOI.IIOOOOOIOIIIQOIIGOIOODIOQOOIIIO
AZxCABSIZ)
IF(AZ oEO. 00’ GO TO 2
ZAZIZ/AZ

THEYA-AlMAG‘CLOG(ZAZI’
2 'F‘AZ 05°. 00‘ THETA'O.

RETURN

EN”

299

SUBROUTINE SELCTIAIR‘CROOTOTESTDNROOTcACROToNUHOI

DIMENSION RI I I OAI I. I DCROOII I I ‘ACROTI 1 I ‘NUMBI I)

COMDLEX CerROOToA‘R
ci.“.I*.Q§.O.‘IQ'.’...I.I.OII.‘O‘I...........'..............O...I..§'...

SUBROUTINE SELCT (R. D. BORCHERDTI

PURPOSE
SELECT APPROPRIATE ROOT OF COMPLEX RAYLEIGH EQUATION 7.7

DESCRIPTION OF INPUT PARAMETERS
A(J) I ARRAY WITH COMPLEX COEFFICIENTS OF RAYLEIGH EQUATION
RIJI I ROOTS 0F COMPLEX RAYLEIGN EQUATION
TEST : ACCURACY TO WHICH ROOT MUST SATISFY EQUATION 7.5

REMARKS
SEE SECTION 7.1 FOR RELATED FORMULAS

nnnnnnnnnnnnnnnnn

IIIIII.§I§OQGII9n{Io...-Q0.0.560!00.0...oooolulnibclliiooiilioillillcoo
TVI'CABSIao‘AIZI-AI3II
IFITVI .GT. TEST) PRINT 102
102 FORMATI' COEFFICIENTS ARE NOT THOSE OF RAYLEIGH EQUATION .I
NROOTIO
DO 6 IROOT=193
(ROOTIIROOTI8IOoolE-ZOI
CF-CSORTtlo-RIIROOTI)
CF:4.#CF'CSORT(1.~IRIIROOT)'(26o-AI2I)/16-II
CF-CF-(I2.-R(IROOTII"2)
ACF‘CABSICF)
IFIACF .GT. TEST) GO TO 6
ARICABSIRIIROOTII
IFIAR .GT. 1) GO TO 6
NRO0T=NROOT*I
NUVBINROOTI'IROOT
CROOTINROOT)-R(IROOT)
ACROTINROOTI-ACF
6 CONTINUE
IFINROOT .NE. 1) PRINT 1030INROOTI
I03 FORMATIIXoIbo’ ROOTS FOUND ')
RETURN
END

300

SUBROUTINE RCNARIDEPTHONDOOSTRMOCVELRCCCBROVELROABSROHAFTIVAFTIHAO

IVAITILTDOAR)

DIMENSION FOI3I060I3).DIOI3IcD3OI3)vHOISIoTHOI3IOGISIIDIIB)u
lD3(3)oH(3).TH(3IvAMP(3)oAMPFT(3)-F(3) oHAFTIIIoVAFT(II-HA(I).
IVAIIIoTILTD(I)oDEPTH(1)oAR(I)

COMPLEX CVELRoCCBRvCKKROCKKRZOCBBKR'CBAKRoCTI9CT20CT3ICBBK2

clfiiliiiilllGDIOIIl§OGII!IOilIQGIGIIGIGOOIIQIIOIIIOGGIIOOIOOIGI.GIOIOII.

PURPOSE

DESCRIPTION OF
DEPTHIJ) =
ND '
OSTRM I
CCBR I
CVELR I
VELR I

ABSR -

REMARKS

SUBROUTINE

nnnnnnnnnnnnnnnnnnnnnnnnn

PI-h.‘ATANIch
RIP-2o‘PI
OEGC-360o/PI2

C
c COMPUTE PARAMETERS

SUBROUTINE RCHAR IR.O. BORCHEROTI

COMPUTE CHARACTERISTICS OF RAYLEIGH TYRE SURFACE HAVE
AS A FUNCTION OF DEPTH

INPUT PARAMETERS

DEPTH ARRAY

NUMBER OF DEPTMS

0 SHEAR TERM

ROOT OF RAYLEIGH EQUATION 7.7

RATIO OF COMPLEX 5 AND P VELOCITIES

RATIO OF VELOCITY OF QAYLEIGH TYPE SURFACE RAVE

TO VELOCITY OF HONOGENEOUS SHEAR HAVE

RATIO OF ABSORPTION COEFFICIENTS FOR RAYLEIGH TYPE
SURFACE HAVE AND HOMOGENEOUS SHEAR HAVE

SEE SECTIONS 7.59 8.2. AND 8.3 FOR RELATED FORMULAS

CALLS SUBROUTINE POLAR

IIIOIIOIIIIGI’OIG IDIGODQOOOOI..CCOOIIID.IIOGDOOOIOIOQOOOIIGOIOOIIOIIIIOI'

INVOLVING COMPLEX VELOCITY RATIOS

RKIKRI‘ABSRIVELROOSTRM
CKKR-CMPinlooRKIKRI

CKVR?=CKKP'OZ

C83KR=CSORT(CKKR2‘Ilo-CCBRII
CBAKR=CSORTICKKR2'Ilo-(CVELR‘CCBR)II

CBBK2¢CRRKRO'Z

CT18(CKKR2+CBBK2)/ICBBKR'CKKR)
CT2=ICKKR?#CBBKZIICKKR2

CT3=CBAKRICKKR

CALL POLARICBAKRoABAKRuTBAKR)
CALL POLAR(CT1'AT19TT1)
CALL POLARICTZoATZoTTZI
CALL POLARICT39AT30TT3I

C

C COMPUTE PARAMETERS
FOIIIII.
FOI3)'-AT3
GOIIII-ATZIZo
50(3): ATI/Zo
010(1I'0n

AT SURFACE

30!

010(3)'TT3
D30(1"TT2
030(3)'771
DO 1 N=1.3.2
HO(N)‘SORT(FO(N)*.Z 0 GO‘N)'§Z + (Zo‘FO‘N).GO(N)'COS(010(N)‘030(N)
1)),
XNUM=F0(N)‘$XN(OIO(NI)+ GO‘N)‘SIN(D30(N))
XDEN=F0(N).COS(DIO(N)’4 GO‘N,'C°S(O30(N’)
1 THO‘N)‘AYAN2(XNUM.XDEN)

C

C COMPUTE ANGLES FOR PROPAGATION AND ATYENUATION VECYORS
RRBKR=REAL(CBHKRI
X89KR=AIMAG(C8RKR)
PBAKR=REALtCBAKQ)
X8AKR=AIMAG1CBAKP)
EP:ATAN2(XRAKR.I.I‘DEGC
ES-ATAN2(XBBKR91.)*DEGC
XN=-RKIKR
THP=ATAN2(XN9RBAKR)'DEGC
THStATAN2(XNoRHRKR)'DEGC
GAMP=90.- THD-EP
GAMS=90.- YHS'ES

C

C 00 FOR DEPTH
DO )2 ID‘loND
TDIDEPTH‘XD)

C

C cOMPUTE PARAMETERS AT DEFTH
DEAnspnflmanno
opnwspyqunaKaoro

DRAI=DI7'X°AKR.TD
DBhleplzuxRRKR§To
F(1)=EXP(-08AR)
F(3)--AT3¢EXP(~DBAR)
G(l)I-AT2‘EXP(-DBBQ)/2o
G(3)'ATI'EXp(-DBBR)/Zo
Dl(l)t-DBAI
01(3)I-DBAI+T73
03(1)='DBRI#T12
D3(3)=-DBBI¢TY1

C

C COMPUTE AMPLITUDE DISTRIBUTXONS
DO 13 N=1G392
H(N)=SQR7(F(N)*'2+GIN)I'2 #(2o¢F(N)’GGN)'C0$(01(N)-D3(N’)i)
XNUM:F(N)'SIN(DI(N))+G(N)'S!N103(N))
XDEN=F(N)'CD$(DX(N))¢G(N)‘COS(D3(N)!
TH(N)=AYAN2(XNUM|XDEN’
AMP(N)=H(N)/HO(3)

l3 AMPFTtN):H(N)*COS(TH(N)-THO(N))lHO‘3)

C
C COMPUTE PARAMETERS FOR PARYICLE MOIION ELLIPSE
S:TH(3)—TH(1)
IF( H(l) .EO. H(3)) TlLT'PI/bo
IF( Hll) .50. H(3)l GO TO 15
TILT-.S'ATANZ(2.'SiN(S)/(H(3)'H(1’) I (lo/H(l)“‘2)-(l-IH(3)"2)I
15 TILTD(ID)*TILTGDEGC
ROSS-(05(5)

302

A=I.II(H(1)¢RDES)-oz)
C-l./((H(3)'RDES)"2)
g.(2.ust(5)y/(ut1)uut1)-(Roes-¢2))
AP=(A*(CO$(TILY)'.ZI3+(8¢51N(T1L1)0COS(TILY))+(C'ISIN(TILT)'IZ)i
CP:(A*(SIN(TILT)'.Z)i-(B'SINKTILT)'COS(TiLT))+(CI(COS(TILTI**Z))
AR(ID)-SQ?T(CP/AP)
Ant10)=SIGN(1.0.noEsy-Aattox
PRINT zoo.(ro.AMp(1).AMPF!(1).AMP(3).AMPFY!3).TILTDIIO:.Antlo).
1RDES)

zoo FORMAT! 8F15.7.F10.6)
HAFT(!D)-AMPFT(1)
VArT(ID)-Anprrta»
HAtlotxAMotI)
VA(xo)-Aupt3)

12 CONTINUE

RETURN
END

303

IE. NOTATION SUMMARY

The noTaTion used in The disserTaTion is lisTed according To
chapTer where firsT inTroduced. The equaTion number where noTaTion

TirsT appears is given.

I. General noTaTion convenTions noT included in The following
lisTs are:

A. Subscrist:

id! | -- forTh order Tensor

i J —- 2nd order Tensor

5,J -— EinsTein convenTion for differenTiaTion

l or J -- ifh or jfh componenT of 3 dimensional
vecTor

I —— imaginary parT of complex number

R —- real parT of complex number

“ -- bulk modulus

P —— P wave

5 —- shear wave or shear modulus

0 —— surface of half-space

LL -— low loss

B. STeady STaTe:
For sTeady sTaTe consideraTions The following

noTaTional convenTion is employed. For example, consider
a one dimensional sTress disTribuTion denoTed by p.
For sTeady sTaTe p is considered To be complex and P,
which is complex and independenT of Time,is defined
implicile by p = P eiwt. The preceding convenTion for
sTeady sTaTe is used for The following variables which

are defined in The succeeding lisTs: P, e, le’eiJ’ u

a
| ’

uni

u) 6, ¢’p'

III.

Chapfer 2

No

RLiw)

C(iw)
YiiuJ)
JWiIJ)
8'

[SE/kycle

Ep
0‘!

Chapler 3:

7‘}:>'

304

sfress (I).

sfrain (I).

relaxaTion funcTion (I).

creep funcfion (5).
deviaforic sfress Tensor (7).
devialoric sfrain fensor (7).
(IO-II).

Heaviside funcTion

circular frequency (II-I2).

FT .
Fourier Transform of rtr) (I2).
Fourier Transform of CI?) (l2).
complex modulus (I3).

complex compliance (I3).

phase angle by which sfrain lags sfress (l4).

energy dissipaled per cycle of forced
oscillafion (20—2I).

peak energy sfored during cycle (20-2I).

(A E/cycle)/(ZTTEP} (2l—22).

componenf of body force per unif volume (I).
mass densify (I).

componenf of displacemenf vector (l).
complex Lame consTanT (7).

complex shear modulus (8)

complex bulk modulus (9).

9 ..
0( -—
fl --
IJ' J n5 __
XJ' ‘-
V (VP. V.) --
0. (up! as) '-
k (“p 3 fl.) "'
fil(‘; )iZ) “
Z (A‘,,A‘,) --

I I I
V( V? ,vg) --

{(FP’IS) "'

w --
3, , It, , 4“ --
r, , q, —-
Anin=h2r*

Bn (n:I,2)--

305

V-u’ (l3).

complex dilafalional velociTy (l6).
complex shear velocify (l7).
direcfion cosines (l8, I9).

recfangular coordinale

"z.
XI

*5

phase velocily of corresponding homogeneous
wave (22, 23).

absorplion coefficienf for corresponding
homogeneous wave (24), (25).

complex wave number for corresponding homo—
geneous wave (26), (27).

propagafion vecfor for corresponding general
wave (36), (54), (55).

affenuafion vecfor for corresponding general
wave (39), (56), (57).

phase velocify for corresponding general
wave (43).

angle befween propagafion vecfor and aTTenua-
Tion vecfor for corresponding general
wave (43-44).

scalar pofenfial associaled wifh dila+a+iona|
wave (58).

veclor polenfial associafed wifh shear wave
(59).

complex wave number associaled wifh ¢, ¢3=V,
and u: , respecfively (70—7l), (75),(76).

roofs of auxillary equalions associaTed wiTh
¢ and It, respecfively (73), (75), (76).

complex amplifude for firsf and second Terms
of general solufion for ¢ (74).

complex amplifude for firs? and second Terms
of general soluTion for ¢ (75).

IV.

V.

Vl.

C" (n=l.2)

ChapTer 4:

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ChapTer 5:

ChapTer 6:

306

complex ampliTudes for firsT and second
Terms of general soluTion for u, (76).

loss in energy densiTy per cycle of forced
oscillaTion (8-9).

peak energy sTored during cycle (8-9).

(3 W/Cycle)/(21T Wp)for corresponding homo-
geneous wave (l0), (IS), (29), (34).

absorpTion coefficienT of homogeneous bulk
wave (l7).

phase velociTy of homogeneous bulk wave (l8).
Time average flow of energy densiTy (l9).

mean averaged energy densiTy over a space
inTerval of one wavelengTh (22).

logariThmic decremenT (38).

skin depTh (39).

noTaTion used is inTroduced in previous

chapTers.
'- I.
”if. - 4; j" (2).
l l I
P.v.(§1 ‘ 4p )’z (2).
/
p.v. (5L: - a;)‘ a (2).

propagaTion vecTor associaTed wiTh firsT and
second Terms of The general soluTions
for ¢ and w , respecTively (3).

aTTenuaTion vecTor associaTed wiTh firsT and
second Terms of The general soluTions
for if: and W , reSpecTively (4).

307

P -- posiTion vecTor in x'x3 plane (6—7).
333’ '3' -- angles beTween propagaTion and aTTenuaTion

vecTor for incidenT and reflecTed
wave respecTively (6-7), (l6—l7).

9.] : 93' -- angle of incidence and angle of reflecTion,
respecTively for general SV wave (6-7),
(l6-l7).
Q5 -- angle of reflecTion for F’ wave (22-23).
a}, —— angle beTween propagaTion and aTTenuaTion
vecTor for reflecTed general P wave
(22-23).
“I —- apparenT phase velociTy along surface (25-26).
Vll. ChapTer 7:
z 1
be; -- p.v.(R, -.:2.'_ ) (2).
1
bp -- (will, ~ ’55) (2).
l t
bu -- p-v- 15.. ' g1. ’ (2).
R -- complex wave number for Rayleigh Type surface

wave (3), (29).
C —- W/ & (6-7).

BE, F; -- propagaTion vecTor associaTed wiTh second
Terms of general soluTion for ¢ and
p , respecTively (8).

K it -- aTTenuaTion vecTor associaTed wiTh second
Term of general soluTion for fl and (P ,
respecTively (8).

1
y newz (ll).
Vi -— phase velociTy of Rayleigh Type surface wave
(44).
0R -- absorpTion coefficienT of Rayleigh Type sur-
face wave (45).
5p ’ es -- angle beTween propagaTion vecTor Pg; , p? I

2.
and The surface, respecTiver (52), (53).

308

GP , 9s —- angle befween a++enuafion vecfor K}; , Kw;
and fhe surface, respecfively (59), (60).
5} ’ ’3 -- angle befween propagaron and affenuafion

vecfors associafed wifh fi' and P ,
respecfively (65), (66).

Fl 0" __ I] + Q; ' I
s ) “VT—:_EET + | (84)

D -- (88).

F": 6,. -- (88).

q", d,” -- (88).

H", 9,, -- (89).

fHfi "-WI *Q*%(9”.
5 -- a, - 6, (95).

r -- III? (96).

A', 8' -- (98).

A, B, c -— (99).

VIII. Chaefer 8:

0’ -- elasfic Poisson rafio (2).

 

-z
Glo") __M Z1|+o‘ )
’ W I ‘8’-

A -- apparent wavelengfh of Rayleigh—Type surface
wave along surface (SO-3|).

 

axns rar:o-- (secfion 8.3.2).

 

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