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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1381 
 
 ON THE DETERMINATION OF CERTAIN BASIC TYPES OF 
 
 SUPERSONIC FLOW FIELDS 
 
 By Carlo Ferrari 
 
 Translation of *Sulla determinazione di alcuni tipi di campi di 
 
 corrente ipersonora," Rendiconti dell'Accademia 
 
 Nazionale dei Lincei, Classe di Scienze fisiche, 
 
 matematiche e natural!, serie VIII, vol. VII, 
 
 no. 6; read at the meeting held on 
 
 December 10, 1949. 
 
 Washington 
 November 1954 
 
 u. 32611 -/un us^ 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM I58I 
 
 ON THE DETERMINATION OF CERTAIN BASIC TYPES OF 
 SUPERSONIC FLOW FIELDS* 
 By Cajrlo Ferrari 
 
 SUMMARY 
 
 A quite universal mode of attack on problems which arise In super- 
 sonic flow, whether connected with flow over wings or over bodies of 
 revolution, is explained, first, in great generality, and then in more 
 detail, as specific applications to concrete cases are illustrated. The 
 method depends on the use of Fourier series in the formal definition of 
 the potential governing the flow and in the setting up of the boundary 
 conditions . This new formulation of the many problems met in supersonic 
 flow Is really an extension of the doublet type of "fundamental solution" 
 to higher order types of singularity. The limitations and, in contrast, 
 the wide field of applicability of such a means of handling these prob- 
 lems with complex boundary conditions is discussed in some detail, and 
 a specific example of a wing-body interference problem is cited as proof 
 of the versatility of the method, because the •^esnJ.ts obtained by applying 
 the techniques expounded herein agree well with experimentally determined 
 data, even for the quite complex configurrtior j.srd to exemplify the kind 
 of problem amenable to such treatment. 
 
 1. INTRODUCTION 
 
 For purposes of analytic treatment of the flow problem to be con- 
 sidered here the usual rectangular Cartesian coordinate system is employed 
 with the X-axis taken to lie in the direction of and having the same sense 
 
 as the uniform (undlstiirbed) free-stream velocity, Voq. This free-stream 
 
 velocity, Voo, is taken to be supersonic in the discussion that follows; 
 i.e., Vco > Coo where Cm denotes the velocity of sound in the undis- 
 turbed stream. The flow of the gaseous fluid to be investigated is to 
 _ 
 
 Siilla determinazione di alcuni tipl di campi di corrente Iper- 
 sonora," Rendiconti dell 'Accademia Nazionale del Lincei, Classe di 
 Scienze flsiche, matematlche e naturali, serie VIII, vol. VII, no. 6; 
 read at the meeting held on December 10, 19^9- 
 
NACA TM 1581 
 
 be considered as resulting from the supeirposition upon the free -stream 
 velocity of a nonuniform flow, having velocity components that are des- 
 ignated as V^, Vyj and V^, and lying in the direction of the respec- 
 tive ELxes (x,y,z) of the coordinate system. This nonuniform super- 
 imposed flow is supposed to be small enough, in comparison with the 
 speed of sound, Ceo, that it is permissible to neglect the ratios Vx/Coo, 
 /C^, etc. in the equations governing the flow. 
 
 It is taken for granted that, under the conditions stated above, 
 there exists a velocity potential describing the flow in question, and 
 in practically all cases which are of any interest for actual designs 
 it will really be true that this assumption can be made legitimately. 
 If it is then agreed that the nonuniform superimposed part of the flow 
 is to be denoted by the potential 0, It will be recognized that this 
 potential will have to satisfy the relationship: 
 
 dx'^ dy^ 02^ 
 
 where the free-stream Mach number, M^^, is defined as Hjo = ^mr^oa an<i, 
 of co\irse, the potential will also have to obey the boundary condi- 
 tions which are peculiar to each stated problem. 
 
 A way of handling the determination of the function 0, so that it 
 will satisfy equation (l) and so that it will obey the imposed boundary 
 conditions, will now be explained, and its usefulness illustrated by 
 consideration of problems which can be attacked by this means, both in 
 the case of lifting surfaces (that is, wings) as well as in the case of 
 bodies of revolution. The proposed method is based on the use of Fourier 
 series. Although this technique does not afford complete universality 
 in treatment of all the posed problems, as will be more clearly pointed 
 out in what follows, it can be used to fine advantage in a. goodly number 
 of situations by replacing the procedures which are based on the Fourier 
 or the Laplace transforms (which, for that matter, have just as restricted 
 limits of applicability as the analogous ones which arise in connection 
 with the approach being discussed herein) or by being substituted in 
 place of the techniques which stem from use of the "fiondamental" (source, 
 sink, doublet) solutions to equation (l), or from use of transformations 
 carried out in the complex plane. 
 
 2. PROBLEMS HAVING TO DO WITH FLOW OVER WINGS 
 
 As usual, the wings are imagined to be very slender and so placed 
 that the wing span lies along the y-axis; i.e., the long dimension is 
 
NACA TM 1581 
 
 out the y-axis (see fig. 1). Let the equations which define the ventral 
 and dorsal surfaces of the wing surface be, in fact, given in the form: 
 
 z (x,y) and z = z,(x,y) 
 
 and then the slenderness of the wing is supposed to be slight enough 
 
 that the above -defined values of z will be so small at all locations 
 
 on these siirfaces as to make it possible to accept the fact that the 
 
 5z 
 derivative -— is, to all intents and purposes, equal to the direction ' 
 
 ox 
 cosine, with respect to the free -stream x-axis, of the normal to the 
 surface. It is further assumed that the wing is immersed in a stream 
 of supersonic flow which has a constant value for its component lying 
 in the direction of the x-axis, of magnitude Vm- The component in the 
 direction of the z-axis, meanwhile, is assumed to be known, but of rela- 
 tively small size in comparison with the V^o velocity, and it may take 
 on various values, which will be denoted by V^ ' • Ifj now, the potential 
 
 describing the flow perturbed by the wing is denoted by this poten- 
 tial will have to satisfy equation (l), and it will also have to conform 
 to the conditions which are imposed at the boundaries. These further 
 (boimdary) conditions may be stated as follows: 
 
 (2) Upstream of a certain surface, which may be immediately defined 
 just as soon as the wing-like body is specified which is to invest the 
 impinging stream, the value of is zero; i.e., the basic condition is 
 
 0=0 (2) 
 
 (3) On the wing surface, it must be true that 
 
 (t) = -V»|^-V,' cos (^,t) =H(x,y) (5) 
 
 wherein the value of z to be employed is either the Zy or z^ quanti- 
 ties, depending on whether one is concerned with a point which is lying 
 on the under ventral siirface or on the upper dorsal surface, respectively. 
 
 The notation cos (,n,z] signifies the cosine of the angle between the 
 .z-axis and the \init vector taken in the direction of the exterior normal 
 to the wing surface in question; i.e., this vector is represented by the 
 vector n, and \inder the present hypothesis cos vn,zj = ±1. 
 
NACA TM 1581 
 
 It is convenient to distinguish between two basic types of problem 
 which come under this kind of analysis, and to make the differentiation 
 on the basis of the sort of boundary conditions met with in each type; 
 that is J 
 
 Symmetric Types of Configuration 
 
 In this case, the boundary conditions to be satisfied on the wing 
 may be expressed in the form 
 
 pi =H(x,y) (5') 
 
 and 
 
 if) - -«('''^> 
 
 Asymmetric Types of Configuration 
 
 In this case, the boundary conditions are expressed as 
 
 P =fM\ =a:x,y) (5") 
 
 The first type of problem corresponds to a conf igtiratlon for which 
 the wing has a zero angle of attack with respect to the free -stream 
 
 undisturbed flow, Voo, and which possesses a symmetric profile. The sec- 
 ond type of problem corresponds to a configuration for which the wing 
 is a flat plate, but which has any local angle of attack whatsoever, 
 
 with respect to the free -stream vector, Voo, so long as it is small. 
 
 3. DEVELOPMENT OF THE CASE OF THE SYMMETRIC TYPE OF CONFIGURATION 
 
 In this case it will siiffice to examine the flow solely in the 
 upper half -plane, where z > 0. If 0^^(x,y,z) stands for the flow 
 
 which takes place in this upper region, and if 0(2)(x,y,z) represents 
 the flow in the nether region, then, of course. 
 
 /2)(x,y,z) =0(l)(x,y,-z) 
 
NACA TM 1381 
 
 The boundary conditions in this case are composed of equations (5')> 
 together with the restriction that 
 
 1)\ 
 
 = (for locations lying beyond the region 
 
 occupied by the wing surface) (2') 
 
 Now let the definition of the function describing the velocity com- 
 ponent at the wing surface, and also the potential function itself, be 
 cast into the convenient forms 
 
 H*(x,y) = V„ ^ H^ (1) cos 2^ = V<„ ^ H^(|) cos | mri 
 
 and 
 
 5;i(l)(x,y,z) = 0(1) = V« b ^ K'^i,^) COS I mri {k) 
 
 m 
 
 wherein | = x/b, r[ = y/b, and ^ = z/b, while b is a suitable length 
 used for purposes of nondimensionalization. The value used for b will 
 be equal to the semispan of the wing in the case where the leading edge 
 of the wing is supersonic everywhere, and provided that the wing tips 
 are cut off in such a way that the wing siirface remains outside of the 
 tip Mach cones emanating from either one of the wing-tip extremities out 
 at the farthest reaches of the wing span. The value used for b will 
 be larger than this semispan Just defined, if, in contrast, these geo- 
 metrical relationships do not hold; the magnitude employed for b in 
 this latter case is illustrated in figure 2. 
 
 Finally, it should be observed that H* is a periodic function 
 of y, which is equal to the values taken on by the function H at the 
 wing's surface and it is zero for points lying out of this region, and 
 this definition is to hold throughout the spanwlse interval for which 
 -b < y < b. 
 
 The fact that it is possible to write H*(x,y) in the form given 
 as equation {k) (i.e., the possibility of expressing the component- 
 velocity field describing the normal velocities to the wing surface by 
 means of a Fourier series Instead of in terms of a Fourier integral) 
 stems, from the property already noted to the effect that the perturba- 
 tions, which are created at any arbitrary point P(x,y) whatsoever, do 
 not make themselves felt anywhere outside of the Mach cone emanating 
 from P. As a result of this situation, therefore, as far as the 
 
MCA TM 1581 
 
 determination of the field of flow about the given wing is concerned, it 
 makes no difference to this flow whether one considers the wing to be 
 operating by itself as an isolated entity within the impinging stream or 
 whether, instead, one imagines it to be accompanied by an infinite nijmber 
 of reflections of this primary wing in the planes y = ±mb. 
 
 If one now inserts the second of the expressions given as equa- 
 tion (k) into equation (l), it will be seen that this differential equa- 
 tion reduces to 
 
 ^ 
 S^2 
 
 _ b2 ^ = k20, 
 
 m 
 
 (5) 
 
 wherein B^ = Moo^ - 1 and where k replaces the constant 2^. 
 
 Meanwhile, it is also evident that, on the basis of the first of 
 the fonnal developments given as equation (k) , the boundary condition 
 reduces to 
 
 
 ^=0 
 
 = Hjn(0 
 
 (6) 
 
 The expression given as equation (5) above is formally analogous to 
 the so-called "telegraph equation, " and its solution, which is suitable 
 for applying the type of boundary condition exemplified in eqixation (6), 
 is 
 
 ^- = 1 
 
 ^^-B^ 
 
 
 
 hm(l') J( 
 
 
 
 I \Jii^)^ - b2^2 
 
 dV 
 
 (7) 
 
 where Jq is the cylindrical Bessel function of zeroth order. 
 Consequently, the vertical derivative turns out to be 
 
 
 |^:|.)2 _b2^2 
 
 B 
 
 d^' 
 
 j\/^rr^7^ 
 
NACA TM 1581 
 
 and, because of the boundary condition (6), it follows that 
 
 hm = -^ Bin 
 
 so that the sought potential must have the form 
 
 1 / ^-^^ 
 
 B.^ 
 
 k. DEVELOPMENT OF THE CASE OF THE ASYMMETRIC TYPE OF CONFIGURATION 
 
 The possibility of being able to find solutions to such asymmetric 
 problems by means of the method being propounded here is restricted in 
 this case to those configurations for which the leading edge as well as 
 the trailing edge of the wing are supersonic, and where the wing tips 
 are cut off in such a way that the wing surface lies outside of the Mach 
 cone emanating from the very tip of the leading edge where the maximum 
 span occurs . 
 
 Under these circumstances the boundary conditions are constituted 
 from the restrictions given as equations (5")^ an<3. of equation (2') 
 once again. If one then follows the same procedure as was utilized in 
 section 3; i"t follows that the expression for the sought potential is 
 formally given as (ref . 1) 
 
 = tthjn 
 
 '0 
 
 I -Blcj]+«k^ r " ' h^(|')Ji|/(| - |')2 -b2^2 
 
 d^' 
 
 1/(1-1 •)2-b2^2 
 (8) 
 
 where h^ is, a priori , an undetermined fionction, and where it should 
 
 be recognized that the + sign is to be employed for the lower half -plane 
 where ^ < 0, ajid where the - sign is to be employed for the upper half- 
 plane where ^ > 0. It is evident, therefore, that the derivative of 
 
 with respect to ^ will be continuous along the plane ^ = 0, but the 
 
8 MCA TM 1381 
 
 derivative of <f)^ with respect to | will be discontinuous, and the 
 "jump" will be of such size that 
 
 holds true . 
 
 It is cleEirly permissible here again to concentrate attention solely 
 upon the disturbed flow in the upper half -plane where ^ > 0, therefore, 
 because the observation just made above will tell one how to compute what 
 the flow will be in the other lower half -plane, once the former is 
 obtained. 
 
 The boundary conditions in this instance may now be recast into the 
 form 
 
 ^m^U) + n^J h^(i') Jo /k i-^j dr - 
 
 Ttk J h^U^) Ji ^k ^-^^] d| • = G^(0 
 
 provided, as in the previous section, one sets up the convenient con- 
 vention that &{x,y) is to represent a periodic function in y that 
 is to be equal to the values taken on by the function G(x,y) at the 
 wing's surface, and it is to be zero for points lying out of this region. 
 This definition is to hold throughout the spanwise interval for which 
 -b < y < b. In addition, the form of G*(x,y) is to be assumed, specif- 
 ically, to have the appearance 
 
 G*(x,y) = V„^ 0^(0 cos |mTi 
 m 
 
 while it has also been assumed that the derivative of a function by the 
 sole parameter upon which it depends is to be denoted by a dot over the 
 function, that is. 
 
NACA TM 1381 
 
 The integro-differential eqiiation defining h^ may also be Immedi- 
 ately simplified to the compressed expression 
 
 «Bhjn(| ) + n |£ r hin(r ) (jo + J2) til ' = Gm(0 
 
 (9) 
 
 Now apply a Laplace transformation to this integro-differential 
 
 equation (i.e., multiply through by the factor e~P5 and integrate 
 from to 00 j . Thus , one obtains 
 
 2 
 
 v2 - 
 
 . (v^-^ 
 
 2 k2 k2 / 2 ^ k2 
 
 - Gm 
 
 where a bar over a symbol serves to indicate that this quantity stands 
 for the Laplace transform of the function so designated. 
 
 Standard tables of Laplace transforms could be consulted to check 
 these results, which may now be simplified by noting that 
 
 X , {J^--T i^p^^g^p^-^/ 
 
 p^^n 
 
 ^^^p SA^^S 
 
 ^ ,/p2 + ^ 
 
 -^f-ii- 
 
 Thus the Laplace transforms of equation (9) simplifies to 
 
 jtBpl^ + itB ,p^ + 
 
 .2 ^ kf 
 
 B" 
 
 pi Ito = Gm 
 
 or the explicit expression for the Laplace transform of the unknown func- 
 tion hju is given in the form 
 
 =^ = 1 
 
 p2.4 
 
10 
 
 NACA TM 1581 
 
 so that finally one may inverr the transformation to obtain 
 
 ^ = ^X' ^^^'^'^ S^^ -^'3 ^^' 
 
 (10) 
 
 Once having obtained the value of hj^, it is easy to write down the 
 expression for the component of velocity lying in the x-direction and 
 located at the wing-surface, because one has simply that this component 
 is given by the partial derivative of the potential 0, taken with respect 
 to I, and evaluated at the plane of the wing; i.e., one finds that 
 
 Furthermore, the formula giving the lift on the wing is just 
 
 mT] 
 
 m+-l , , 
 m 
 
 (11) 
 
 for m = 1,3,5,- 
 
 
 where the symbol, Z, is used to denote the distance along the x-axis 
 measiired back from the leading edge of the root chord to the projection 
 into the plane of symmetry of the trailing edge of the wing-tip profile. 
 
 In regard to the moment taken about the y-axis, it is apparent that 
 it may be computed from the relation: 
 
 2>.5 
 
 My = 2«p„V„-b 
 
 m 
 
 >z/b 
 
 cos \^ ^j dTi j 5hjn(|)d| 
 
 m+l 
 
 --»^vz(^)(-)^ ^"^(a-X ^''''' 
 
 -l/h 
 
 (12) 
 
 for m = 1,3,5,' • •, etc, 
 
NACA TM 1381 11 
 
 5. PROBLEMS HAVING TO DO WITH FLOW PAST BODIES OF REVOLUTION 
 
 The procedure discussed in the preceding sections can be extended at 
 once to apply also to the solution of problems which are concerned with 
 the flow over bodies of revolution. 
 
 For this purpose let a cylindrical coordinate system (x,Y,9) be 
 set up, and then the equation which governs the potential, 0, being 
 so\ight will have the form 
 
 Now let R = R(x) be the equation of the meridian line of the body, 
 and let it be assumed that R is sufficiently small at all locations 
 along the body so that the direction cosine of the normal to this merid- 
 ian line, measured from the Y-axis, may be taken to be equal to unity. 
 
 Furthermore, let Vjif' stand for the component, taken in a direc- 
 tion perpendicular to the circular cross -section of the body, which 
 arises from the impinging flow which invests the body. Then the boundary 
 condition which must be satisfied at the surface of the body may be 
 expressed mathematically by the relation 
 
 = -Vnf' 
 'Y=R 
 
 For sake of simplicity, it is also now assumed that the treatment 
 to be developed is to be restricted to the case where symmetry with 
 
 respect to the semlplanes 9 =±— exists in the incident flow. Under 
 
 this hypothesis it is convenient to write the normal velocity components 
 and the potential being soLight in the following explicit formulations: 
 
 V.^' = V^^ Fjjj (1) sinm 9 
 
 r\ 
 
 m 
 
 = V„ > 0in(x,Y)Y™ sin m 9 
 
 •00 y^ rin\ 
 m 
 
 (14) 
 
12 NACA TM 1381 
 
 If one now inserts the second of the expressions given as equa- 
 tion (l4) into the differential equation governing the flow (15), it 
 will be found that the defining equation for the potential will have 
 the form 
 
 
 and the boundary condition turns out to be 
 
 ^(^"Vm) 
 
 By 
 
 = -Fin (15') 
 
 Y=R 
 
 A suitable solution to equation (l5)j which can be made to satisfy 
 the boundary condition being imposed as equation (l5')j will be found 
 to be 
 
 0„ - (^ D" ^0 (16) 
 
 where 
 
 ^0 
 
 0Q = / fQ(x-BY cosh u)du 
 
 ^arc cosh ^ 
 
 BY 
 
MCA TM 1581 
 
 15 
 
 Thus, the successive individual potentials are given by the expressions-'- 
 
 nO 
 Y01 = -B / fi(x-BY cosh u) cosh u du 
 
 o 
 
 'arc cosh :^ 
 hi 
 
 Y02 = ^^ / f2(x-BY cosh u) cosh^ u du 
 
 "^arc cosh ^ 
 BY 
 
 etc . 
 
 > (17) 
 
 Upon imposition of the requirement that the boundary condition (15') 
 is to be satisfied, one obtains a set of integral equations which serve 
 
 -"Translator's note: It was pointed out on page 65O of an article 
 by R. H. Cramer in the Journal of the Aeronautical Sciences, vol. I8, 
 no. 9> September 1951j entitled "Interference Between Wing and Body at 
 Supersonic Speeds - Theoretical and Experimental Determination of 
 Pressures on the Body," that the result given here for <^, for m > 1, 
 is incorrect; the correct formula is, for m = 2, 
 
 Y% 
 
 = b2 / 
 
 "^arc cosh ^ 
 
 fgCx-BY cosh u) (2 cosh^ u - l) du 
 
 while, in general, the use of hyperbolic functions of multiples of the 
 argument u gives a more compact form, which is easy to work with; i.e., 
 in general it is true that 
 
 Y™0m = b"(-1)°^ / fm(^-^"^ ^°2^ ^) {^°^^ ^ ^) ^^ 
 
 'arc cosh —- 
 
 BY 
 
Ik NACA TM 1581 
 
 to determine the arbitrary functions f^, which are a priori unknown. 
 Thus, applying these conditions, one finds thaf^ 
 
 Bin+l (_i)m+l / fjji(x-BR cosh u) (cosh u^+l du = -F^ (18) 
 
 "^arc cosh ^ 
 BR 
 
 The determination of the values of the fm's appearing in formula (I8) 
 
 may be carried out by using a step-by-step procediire which is entirely 
 analogous to the one employed by Von Karman in his work on determining 
 the flow about a body of revolution at zero angle of attack. 
 
 It is important to point out that if one only has in mind to calcu- 
 late the force distribution along the axis of the body and the corre- 
 sponding moment, and if one is not interested in knowing the local veloci- 
 ties or pressures around the body, then it is merely necessary to 
 calculate 0^ and 02- 
 
 6. APPLICATIONS 
 
 The procedure that has been propoiuaded above has been applied (ref . 2) 
 to the situation arising in the study of the question of wing-body inter- 
 ference. The wing -body configuration considered in this particular appli- 
 cation of the method is depicted in the appended figure 5- 
 
 The wing used in this configuration is a flat plate, whose semispan 
 is equal to ^Rq' ^^ere Rq is the radius of the circular cross-section 
 
 taken through the body at the location where the body is widest. The 
 leading edge of this wing is located 5^0 downstream from the tip of the 
 nose of the body. The free -stream flow is impinging on the body at a 
 speed which is twice the speed of soiaid in the undisturbed stream. 
 
 ^Translator ' s note: In view of the correction pointed out in Note 1 
 above, it will be seen that this formula for determining the f^ functions 
 is also incorrect, except for m = 1; for higher integral values of m, 
 the correct formula is : 
 
 oO 
 gm+1 (_2_)nH-l / fin(x-BR cosh u) [cosh m u cosh ul du = -F, 
 
 arc cosh — ^ 
 BR 
 
 m 
 
NACA TM 1581 15 
 
 The curves shown in figure 5 give the value of the pressure coeffi- 
 cient, 1^ , out along the span of the wing, in the mid-chord loca- 
 Ip V 
 
 tion (i.e., along the wing axis), for points on the upper (dorsal) side 
 of the wing. These coefficients have been calculated by the method out- 
 lined in section k, and there are shown results for various angles of 
 attack, which apply to such points on the upper side of the wing pro- 
 files at their mid-chord positions. 
 
 In addition, some experimental test points obtained by R. H. Cramer 
 (see ref. 2) are also plotted on these curves. These results were 
 obtained from experiments carried out in the supersonic tunnel of the 
 Daingerfield Aeronautical Laboratory. 
 
 The agreement between the computed and experimentally determined 
 results is very good from a qualitative viewpoint. In regard to the 
 more precise details of the quantitative comparison between the results 
 it is worthy of note that the experimental results exhibit a certain 
 amount of dissymmetry as one passes from positive angles of attack to 
 negative angles of attack. Such a dissymmetry cannot be predicted, or 
 should not be expected, from the type of theoretical treatment being 
 considered here. 
 
 In order to bring about a more valid comparison of these results, 
 it would appear logical, in face of such evident dissymmetry, to take 
 for the representative experimental value, at a given value of the angle 
 of attack, p, the one which is obtained from averaging the result 
 obtained for an angle of attack equal to +p with the result obtained 
 at -p. Such average values have been computed and are designated in 
 the plots of figure 5 ^Y means of solid circles. These adjusted values 
 lie much closer to the theoretically derived curves at almost all 
 locations . 
 
 Translated by R. H. Cramer 
 
 Cornell Aeronautical Laboratory, Inc. 
 
 Buffalo, New York 
 
 REFERENCES 
 
 Ferrari, C: Interference Between Wing and Body at Supersonic Speeds • 
 Theory and Nimierical Application. Jour. Aero. Sci., vol. I5, no. 6, 
 June 19^8, pp. 517-356. 
 
 Ferrari, C: Interference Between Wing and Body at Supersonic Speeds • 
 Note on Wind Tunnel Results and Addendum to Calculations. Jour. 
 Aero. Sci., vol. 16, no. 9, Sept. 19^9, pp. 542-5^6. 
 
16 
 
 NACA TM 1581 
 
 Figure 1.- Orientation of coordinate axes and location of typical wing 
 
 plan form therein. 
 
 figure 2,- Definition of the interval of periodicity required for application 
 of the Fourier series technique when leading edges are subsonic. 
 
NACA TM 1381 
 
 17 
 
 Cp = 
 
 P-P, 
 
 00 
 
 1/2/500 Voo 
 Upper 
 surfoce 
 
 .10 .2 .30 .40 .50 .60 .70 .80 .90 1-00 
 7) :y/b 
 
 Figure 3.- Pressure distribution along the wing axis: Comparison of 
 experimental results with predictions based on the method expounded 
 in section 4, 
 
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