X I B RAHY OF THE U N IVERSITY OF ILLINOIS 6ZI.3S5 UG55te no.2-H cop.3 Digitized by the Internet Archive in 2013 http://archive.org/details/closelyspacedtra14hyne Antenna Laboratory Technical Report No. 14 CLOSELY SPACED TRANSVERSE SLOTS in RECTANGULAR WAVEGUIDE by R.F. Hyneman 10 December 1956. Contract AF33 (616) -3220 Project No. 6,(7-4600) Task 40572 WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois 3 Derivation of the Characteristic Equation for y 16 III. Theoretical and Experimental Results for the Case of Slot Length Equal to Waveguide Width 25 3.1 Characteristic Equation for y 25 3.2 Theoretical and Experimental Results 3 3.3 Aperture Distribution Synthesis 42 3.4' Surface Wave Mode 48 IV. Theoretical and Experimental Results for Small Slot Length 58 4,1 Characteristic Equation for y 58 4'. 2 Theoretical and Experimental Results 60 4'. 3 Surface Wave Solution 66> V. Experimental Verification of the Theory 79 5.1 Details of Construction of the Model 79 5.2 Measurement of y in Cases of Large a 81 5.3 Measurement of y in Cases of Small a 83 VI. Conclusions and Recommendations for Further Investigation 85 References 87 Appendix A - Proof of the Stationary Properties of Equations 2 18 and 2-21 89 in IV CONTENTS (CONTINUED) Page Appendix B - The Evaluation of I n (3/k, L/X) 93 Appendix C - Approximations for the Series S3 and Sc in Equation 3-3 103 Distribution List 113 ILLUSTRATIONS Figure Number Page 1 The Transversely Slotted Waveguide 5 2. The Traveling-Wave Slot Antenna 13 3 F <^ J I i:tr '° rl " } 4W 2 4' Attenuation Constant versus Frequency for Fixed Waveguide and Slot Dimensions 33 5. Velocity Ratio versus Frequency for Fixed Waveguide and Slot Dimensions 35 6. Attenuation Constant versus Frequency 36. 7. Attenuation Constant versus Frequency for Fixed Waveguide and Slot Dimensions 37 8: Attenuation Constant versus Slot Spacing 38 9 Attenuation Constant versus Slot Spacing 39 10. Velocity Ratio versus Slot Spacing 40 11. Velocity Ratio versus Slot Spacing 41 12. Attenuation Function and Average Slot Spacing for Uniform Aperture Distribution 44' 13. Polar Coordinate System for Pattern Measurements 45 14'. E-Plane Radiation Patterns for Uniform Aperture Distribution (Figure 12) 46. 15 Input Standing Wave Ratio versus Slot Spacing 47 16. Velocity Ratio versus Frequency for Surface Wave Mode 5l 17. Axial Phase -Amplitude Distribution with Fundamental and Surface Wave Modes Present, W < X/2 53 V I ILLUSTRATIONS (continued) Figure Number Page 18. Measured Axial Phase -Amplitude Distribution under Conditions of Single Mode Propagation 54' 19. Dielectrically Loaded Feed Section to Improve Surface Wave Launching Efficiency 55 20. Evidence of Asymmetrical Surface Wave Mode Resulting from Small Structural Dissymmetry when W/X Near 1.0 56< 21. Evidence of Third Order Surface Wave Mode for W/X Near 1.5 57 22. Real Part of J [( 3/k), (L/X)] versus L/W 61 23. Imaginary Part of J [( 3/k), (L/X) versus L/W for .02 ^L/W <. 0.25 62 24. Imaginary Part of I Q [(o (3 / k ), (L/X) versus L/W for • 02 < L/W <_ 1 . 63 00 25. s'= Y cos 2 , (mrcL/2W) Km 2 X 2 /4W 2 ) - 11 64 - "^ [l-(m 2 L 2 /W 2 ) 2 ] (m 2 - I)X 00 26,. s' = ^ cos 2 (mnL/2W) f(m 2 X 2 /4W 2 ) - 11 65 m^ [l-(m 2 L 2 /W 2 ) 2 ] (m 2 - 1)^ 27 Attenuation Constant versus Slot Length for Inf initesimally Spaced Slots, W/X = 0.686, 67 28 Attenuation Constant versus Slot Length for Inf initesimally Spaced Slots, W/X ■ 95 68 29 Attenuation Constant versus Slot Length for Infinitesimal Jy Spaced Slots, W/X 1,27 69 30 Velocity Ratio versus Slot Length for Inf initesima lly Spaced SI ol 70 V I I ILLUSTRATIONS (continued) Figure Number Page 31. Measured Axial Amplitude Distributions with Fundamental and Surface Wave Modes Present 73 32. Measured Axial Amplitude Distributions with Fundamental and Surface Wave Modes Present 74' 33. E-Plane Radiation Patterns Showing Relative Excitation of Surface Wave versus L/A. as Frequency is Varied 76< 34. E-Plane Radiation Patterns Showing Increase in Surface Wave Excitation with Decrease in D/A. 78 35. The Experimental Model 80 36>. Schematic of Experimental Setup for Phase -Amplitude Measurements 82 37. Po K/d) 107 38. p 2 (Vd) 108 39. p 4 , U/d) 109 40. q Q (l/d) 110 41. q 2 U/d) 111 42. q 4 , U/d) 112 ABSTRACT The traveling wave modes associated with the infinite, periodic structure consisting of closely spaced transverse slots in rectangular waveguide are considered. An approximate equation for the propagation constants of these modes for the case of infinitesimal thickness of the slotted wall is derived through Fourier Analysis techniques and an approximate application of the Reaction Concept. Theoretical results are given for two special cases, a) discretely spaced slots which extend completely across the broad face of the waveguide, and b) inf initesimally spaced slots of arbitrary length. In the homogeneous case considered, it is found that two dominant modes may exist, an attenuated fundamental mode representing a perturbation of the dominant TEjq mode in closed rectangular waveguide, and an unattenuated surface wave, with phase velocity less than that of free space, which is similar to the wave associated with a corru- gated surface waveguide. Essentially independent control of the attenuation constant and phase velocity of the fundamental mode is possible over a wide range through the appropriate variation of the various physical parameters. Curves are given for the propagation constant in terms of these parameters, and the results of experimental measurements are shown to be in close agreement with the theory. V III ACKNOWLEDGEMENT The author wishes to acknowledge the guidance of Profo V.H. Rumsey, without whose help this report would not have been possible. He is also indebted to Drs , R.H. Duhamel and E.J, Scott for their many helpful suggestions, to Mrs, Allen Blankfield, who performed most of the compu- tations, and to Mr. J.J. Stafford who performed many of the experimental measurements . IX LIST OF SYMBOLS c Free space velocity of propagation c/v Waveguide velocity ratio d Slot width (narrow dimension) D Waveguide depth (narrow dimension) (2), H Q ^'(z) Hankel Function of the second kind I (f,M See Appendix B, page 99 K A j j -/T k Free space propagation constant, k = w yj [i Q e Q K Q (z) Modified Hankel Function of the second kind. 't Slot spacing L Slot length (long dimension) L Total length of aperture. v Waveguide phase velocity, v = ((i)X fi /2rc) W Waveguide width (large dimension) a Attenuation constant a/k Normalized attenuation constant (nepers per radian) 0n 3 n = k2 -j^2 3 3 £ 6 = jk 2 " y 2 n Y n Y n = Y - (2nn/-£) Y Y - Y , waveguide propagation constant, Y = (w/v) - jot Q Y Unperturbed y for closed rectangular waveguide, Y =/k^-(n;^/W^) Y Euler's constant, y = .5772 ... r Reflection coefficient at beginning of aperture e Permittivity constant X I Tii.n ^,n - J Vn =J (^/W) 2 + Yn 2 - k 2 K m,n Vn = ik 2 -Y n 2 -(m 2 rr2/ W 2) = Jk 2 -(m7i/W)^ ■ ■ Y 2 'n K K = K l,0 = J k2 " (7r / W)2 - Y 2 A Free space wavelength \ Guide wavelength V Permeability constant £ m n = for m ■ 1 when n = 0, = 1 otherwise. p p = Jx 2 + (y - y') 2 ^n = J 3 n -J Y n 2 " ^ o n w u) = 2rc frequency I . INTRODUCTION The general subject of traveling-wave antennas has been of considerable interest for a long period of time. Early investigations have dealt with the characteristics of such long wire devices as the Beveridge Wave Antenna and the Rhombic. More recently, as the result of the trend to higher frequencies, attention has been primarily directed to studies of open, radiating waveguide systems. Dielectric and corrugated surface waveguides^ "^ in their various forms belong to this latter class. Their usefulness arises from the existence of a guided wave mode of propagation for which the phase velocity is less than the free space velocity of light. Thus when employed in their rudimentary form as antenna elements, these structures possess essentially end-fire radiation characteristics. When an arbitrary radiation pattern is required, a surface waveguide may be employed as a transmission line for the excitation of a suitable configuration of discretely spaced radiating or scattering elements. '"' The Channel Guide" and the Traveling Wave Slot Antenna"" 10 have also received considerable attention. Basically, these structures consist of conventional metallic, cylindrical waveguide in which the internal fields are coupled to the exterior, or free space region, by means of a continuous axial slot. In the homogeneous case , the propagation constant is complex and the traveling wave, which has a phase velocity greater than that of free space, is attenuated with distance along the guide. A useful means of radiation pattern control is available since the aperture illumination may be adjusted by suitably varying the transverse 1 geometry of the waveguide and the width of the axial slot. 1 As a result, a wide variety of antenna patterns may be synthesized. In recent papers*^*^ , Elliott has reported on an investigation of the Serrated Waveguide Antenna. This structure is essentially a rectangular waveguide .having a corrugated surface for one of the interior broad faces. Transverse slots placed in this face at the bottom of the corrugations provide coupling to the exterior region. Because of computational difficulties, the analysis was limited to the case of inf initesimally spaced slots and corrugating teeth. For this case, the Serrated Waveguide is essentially a thick walled rectangular waveguide which has been pierced by a series of inf initesimally spaced transverse slots. This report is concerned with the characteristics of finitely spaced transverse slots in a thin walled rectangular waveguide. Thus, from a physical standpoint, the structure is in some respects a complement of the one considered by Elliott. In addition, from an electrical point of view, the transversely slotted waveguide differs from the rectangular case of the Traveling Wave Slot Antenna only in the type of coupling mechanism which is employed The resulting differences between the characteristics of the two structures are in the modes of propagation and in the type of polarization of the radiated fields The radiating aperture of the transversely slotted waveguide consists of a collection of discretely .spaced radiating elements. However, rimentl have shown that when the slot spacing is less than one-half fr«v -.\,.Kf wavelength, the character of the distant radiated field is ■pproximately the same as for a continuous distribution that is the average of the discrete elements. Thus the existing techniques for antenna pattern synthesis which apply to the continuous or axial slot case give good results for the case of transverse slots. In order that these techniques may be employed, a knowledge of the attenuation constant and phase velocity of the traveling wave is necessary. In the treatment of the uniform, infinite, periodically slotted case, Fourier Analysis techniques may be employed to obtain an approximate expression for the propagation constant. By following this procedure, the need for an explicit and difficult consideration of the large mutual coupling between the closely spaced slots is obviated. Approximate theoretical results, which are in close agreement with experimental observations, are given for the attenuation constant and phase velocity as a function of the structure geometry. In addition, it is found that an unattenuated surface wave with velocity less than that of free space may exist under certain conditions. Thus the transversely slotted waveguide combines some of the characteristics of an open corrugated surface waveguide and the Traveling Wave Slot Antenna, II. DISCUSSION OF THE PROBLEM 2,1 Simplifying Assumptions A theoretical analysis of the problem is possible only for the infinite, uniform, periodic case. Although these conditions are not met in a physical situation, previous investigations have shown that the results of an analysis of the idealized case closely apply when the structure is long, and where any departure from uniformity and periodicity is gradual, in terms of wavelengtn. Several simplifying assumptions are made, With reference to Figure 1, it is assumed that the slot length to width ratio L/d is sufficiently great, and d/X is sufficiently small that only the z component of electric field in tne slot is appreciable. The thickness of the slotted wall of the waveguide is taken to be infinitesimal. Finally, consideration is re- stricted to the case where the permittivity and permeability of the internal and external regions are equal to those of free space. However, the somewhat more general case where the two regions are individually taken to be homogeneous represents only a simple extension of the theory. 22 Space Variation of the Fields All components in the internal and external regions can be written in terms of the unknown tangential E, in the plane of the aperture. The imposition of continuity conditions on the various components at the ■perture then Leadi to an integral equation for the unknown aperture ■ i j but i on Undei tin- Initially aeaumed conditiona of periodicity, it can be '' 'ill field component! m;iy be expressed in terms of 4' > o CO 0) (/) l_ > c a CI 6. exponential and periodic functions of position. The space variation of the fields in both the external and internal regions is of the general form (e J ut time convention), F(x,y,z) = G(x,y,z) e"JYz (2-1) where y is independent of x and y, and G_ satisfies the relation G(x,y,z) = G(x,y,z, z-l). Let the origin be placed at the center of a typical slot as in Figure 1. Let the tangential electric field at x = and within the period _/ C/2 <. z <. 1/2 be given by Jkan s L E z (0,y,z) = z g(y,z) fo < |y| < L/2| |0< | z | < d/2 1 = Tl/2 < |y| id/2 <_ |z| < 4/2? (2-2) where g(y,z) is an unknown scalar function of position, and z_ is the unit vector in the z direction. In accordance with Equation 2-1, E z e + JY Z i s periodic, and thus may be expanded in a Fourier series valid for all z. CO E z (0,y,z) - e-JY* V a n (y) e j(2n7tz/^) i n (y) e-JYn^, ( 2 -3) 00 a. n= -oo where .ind where Y n - Y - 2nn/<£ 1/2 ;.„(y) \/l \ E 2 (0,y,z) eJY* e -j (2nrczA0 dz ■£/2 .d/2 l/4\ g(y,z) e JYn z dz 0- |y |< L/2 , " d/2 L/2<|y |.(2-4) The field components may be expressed in terms of the aperture field distribution through the use of the equivalent magnetic current sheets defined by M = E x n, where n^ is the unit normal to the aperture pointing into the appropriate region. Thus in the internal and external regions respectively, Mint = "-I E z (°'y> z) (2-5a) ^ext = I E z (0,y,z) (2-5b) Let £ be the magnetic vector potential defined by E. = -V x £. From Equation 2-5 and in view of the symmetrical boundary considerations, F has only a y component, F =_y f. Thus, E = - V x X £ = y x Vf = x Of/Bz) - zOf/Bx) (2-6) and all fields are TE with respect to the y axis. Then H = - 1/jwn V x £ = - 1/jwn (Vxyf) ■ - 1/jwM- [-xO^/^xBy) + jy_02f/Bx 2 + B 2 f/Bz 2 )+ - z (3 2 f/3yBz)]. (2-7) Since in any source-free region f must satisfy the scalar Helmholtz equation, V 2 f + k 2 f m 0, then H v = 1/jwn (k? + 3 2 /By 2 )f. (2-8) 8 2.2.1 Internal Field Distribution Since E tan vanishes on the conducting surfaces, then from Equation 2-6, f must satisfy the following boundary conditions in the internal region: Bf/Bz =0 y = + W/2 Bf/Bx =0 x = - D Bf/Bx = -E z (0,y, Z ) 00 ■ -V a n (y) e-JY n z \y\&/t n=-oo = L/2<|y x = 0. It follows from the wave equation that f is of the form CO 00 f int. = /^ Za Cm,n COS n,7ly/W "° S Vn (x+D) e - JYn z m=1 *""" Vn sln Vn» (2-9) where and where . /« K m)n ^ - (mrt/W)2 - Yn 2 J c m,n = 2 /W V a n \ d,' \ dz'{ »=2 )> (2-11) J-L/2 J» I R J where R =J X 2 + (y-y') 2 + U-Z') 2 . For complex y> Equation 2-11 does not converge, and hence is not a valid representation. However, the vector potential function for an infinite line source y My dy'e-JY n z of arbitrary propagation constant y n can be determined. "' For such a source in free space located at x, y, the potential function is given by F - y My dy'(-j/4') H < 2 > (0 n p) e-JY n z, where ^n ^ 2 -Y n 2 > p = /(x-x') 2 + (y-y') 2 . The radiation condition at infinity is satisfied when the root of (* - Y n ) having a positive real part is taken for 3 n < 10 Upon substitution for M v from Equations 2-3,4', 5 X/2 f ext = -J/2^\ d 4Z a " (y0 H ° 2 (6 " P) e " JYnZ ) ^L/2 (n=-* J L/2 r- d/2 ( -j/24^ d/JV \ g(yU') e -JY„U-z). ^-L/2 |n : *ti - d/2 >t •H n 2 (6 n p) dz' 2.3 Integral Equation for the Aperture Field From Equations 2-7,8 the condition f ext (°.y> z ) = f int (o.y-z) for y,z in a slot region satisfies the requirement of continuity of tangential jl. For this condition to be met Equations 2-9,10,12 yield (2-12) p L/2 ^d 2 /®i \ dy' «-'-L/2 00 L 'C , ' f / / 'n cos K m n ( x+D ') / /,„\ \ dz Jg(y,z) m - Lj5 cos(mTly/W; J-d/2 \ K m,n sin K m,n D cosfmriy^e-JYn^- 2 '^ d/2 F ^ g(y! Z ')H (2) (0nP)a-JYB< M W| n^oo J- d/2 =0. x=0 (2-13) Because of the singularity of H Q ' (fip) at the origin, some care must be exccrised in interchanging the operations of summation and integration in Equation 2-13. However, it can be verified for a suitably well behaved g(y'z) that both term* are uniformly convergent for any x/0, p-yx^ + (y-y)^>0. , although not explicitly stated, a limiting process is assumed in the 11 following whereby x-0 after the interchange of operations has taken place. Hence, Equation 2-13 may be written J-/2 _d/2 '-L/2 J-d/2 0=\ dy'\ dz'JgCyU') 00 «> V 2/4W V COt Km ' nD cos (mTty/W)cos(mrcy^W). _n = -<» m=l x m,n #e jY n (z-z') + (cj/2-e) H o 2 (0 n |y-y|)e-jYn( Z -/)|. (2-14') Expression 2-14 is an equation of the form JV2 d/2 V dy'C dz'fj^z') K (y,z|y,'z')) J.T/9 J-d/9 (2-15) ■L/2 ^-d/2 There is an infinite number of solutions, each solution g p (y,z) associated with an eigennumber y p and related constants Y p = Y p - (2n7t/<£). However, the equation is not of standard form and no technique for exact solution exists. Thus a variational method will be explored for obtaining an approximate value of y based on an assumed distribution function, g. 2.1 Variational Solution for y From Equation 2-14, we have that the kernel given by oo r— oo f K (y,z|y',z') = Y (2/^)^] ^ *""'" - cos (mny/W) cos (mrcy'/W)- n=-oo m=l K m, n . e - JYn (z-z') + j/2^H (2) (3 n |y-y'|) e'JYn <»^ (2-16.) 12 where Y n = Y - (2n7t/ *° for the general case where y is complex. However, approximate formulas for Y may be obtained which do have stationary properties for the restricted conditions that y he real, or that the slot spacing be infinitesimal For example, assume y real and y>k. Then 0n ' "J /Y n 2 - ^ Vn = -J^ (WW) 2 + Y n 2 - k2 are pure imaginary for all m,n. Since j H Q (-jx) - - (2/n) K (x), cot(-jx) _ . c oth x -jx X are then pure real, it is readily verified that K is Hermit ian , i.e., K (y,z|y!z) - K*(y,z\y,z). (2-17) Under thia condition, a .stationary formula can be established directly f rosi Bquati on 2-14. ' I' ' h(y,z) be an approximation to the unknown g(y,z). By making use oi the property in Equal ion 2-17, ii ia shown in Appendix A that 13 the value of y which satisfies h(y,z) h* (y,z) K (y,z|y,z) dy dy dz dz = (2-18) is stationary for small variations in h about the correct function g. Thus a first order error in the assumed aperture distribution leads to a second order error in the value of y derived from Equation 2-18. When y is not real, an alternative formula having stationary properties for the case of infinitesimal slot spacing can be obtained from a consideration of the Reaction Concept. 2.5 Approximate Application of the Reaction Concept Through the use of the Reaction Concept, Rumsey has treated extensively the case of tne continuous axial slot or the Traveling Wave Slot Antenna." A sketch of this structure appears below. Figure 2 The Traveling-Wave Slot Antenna 14 In the continuous slot case, all fields vary with z as e " J Y z . Let E^ (y) be an assumed tangential electric field distribution that is postulated to exist in the slot. Assuming that E is incorrect, the ~ a tangential components of the internal and external magnetic fields, H„ . (y) and H~ (y), that fit this distribution are discontinuous — a int — a ext in the slot. Hence, an equivalent electric surface current distribution J^Cy) eJ Yz must be established in the slot to support the initially assumed E a . The required current distribution is obtained from J a Cy) - n x [H. (y) - H a . (y)] , (2-19) - a - - a ext - a int with n the unit outward normal. The self reaction^ 1 -' of this surface current source distribution is given by L/2 =\ [J a E a -J a E a ]dy, (2-20) i y y z z J -L/2 y y where L is the width of the axial slot. Since J a (y) includes y as a parameter, enforcement of the condition ■ (2-21) leads to an expression for y which is a function of the assumed field t.nbution Rumsey has shown that this is a variational formula for i vhich is stationary for small variations of the assumed aperture tribution about the correct distribution of an allowable mode of propagation The proof of this r ** -■ ; u 1 1. is it prated j ii Appendix A. It should be not.e(| that this result holds when y is complex. In cases re cmly an E, component oi tangential aperture field is present, 15 Equations 2-20,21 give , n \ E a [H v - H v . ]dy. (2-22) = A . ##% a z "ext >int Expression 2-22 does not apply to a periodic aperture distribution, since, in this case, E a is a collection of traveling-wave fields of differing propagation constants y n , n=l,2,3, ... A third aspect of the approximate method is brought out by considering the case of an aperture of finite extent, S . In this case the conventional form of the reciprocity tneorem applies, ° = \ \ (J 1 E 2 " J 2 E V dS > (2_23) S and thus the self reaction of a source J^ a is given by =\\ J a E a dS. (2-24) It is apparent; that the application of Equation 2-24' to the transversely slotted waveguide is not strictly justifiable because of the infinite extent of the aperture Thus, although the relation 0=\\ E a [H v -H v . US', (2-25) J J , a z "ext "int where S is the infinite aperture, gives an approximate expression for Y in terms of the assumed field distribution, the expression is not necessarily stationary. However, in the limit as 't/X-O, Equation 2-25 reduces to Equation 2-22 since the aperture field then becomes essentially continuous and the z dependence can be eliminated. Thus y derived from 16, Equation 2-25 is approximately stationary for small Z/\- This contrasts with the characteristics of Equation 2-18, whicn is stationary for finite -£ but only in the event y is real. The lack of exact stationary properties in tne general case is probably not of great consequence since the degree of disparity between the actual field and any assumed field is unknown. Thus, experimental verification of the theory is necessary anyway. There is some question as to whether Equation 2-18 or 2-25 provides the better approximation for y It would appear that Equation 2-25 more nearly fits the actual circumstances, since, in general, y is complex. In addition, the limiting case of infinitesimal slot spacing is one of special interest. 2.6 Derivation of the Characteristic Equation for y The expression 2-25 is taken as the basis for an approximate formula for y. The integration is to be performed over the infinite aperture surface, but since all fields vary between adjacent slots as e "J ^ , 2-25 is equivalent to _» ^L/2 _d/2 - prt J-i/2 J-d/2 ^ \ /M , dz N* (H W "y.nt'} , (2 " 26J i^oo J-I./9 J-d/9 where the integration is performed over any typical slot, e.g., the slot centered at z=0 A sufficient condition for Equation 2 -26. is, then, X/2 d/2 ■L/2 ^-d/2 ipproximation to E a is taken as dyC dz [K a (Hy - H y . )} . (2-27) J-IV9 1,1/9 l l VeXt Yint ' g a (y,z) cos (ny/lj )\z\ £ L/2 z\ <.d/2 (2-28) |y| > 172 d/2 £ |z| < 1/2. 17 The substitution of Equation 2-28 into 2-8 and 2-10 gives for the y component of internal magnetic field at the aperture, 03 00 H =(l/j^)[k 2 4^2/3y2)]y Y 2/4* cos (mrty/W) - ° S V» (x + D) e 'JYn z . lnt fel n^-« Vn sin V n D JL/2 d/2 , dy'\ dz' (cos (rty'/L) cos (mny^W) eJY n z J, '-L/2 J-d/2 (2-29) Similarly, from Equations 2-8 and 2-12, .L/2 r . d/2 i dy ' uJ\ '-L/2 (ji^o J- d/2 H yext (I/j«n) fk 2 ^ 2 /3y 2 )J -j/2-e ^ dy'J Y C cos (*y}t). In — ® >J- • eW H < 2 > (3 n p) dz', (2-30) where p =)x 2 + (y-y) 2 . Upon substitution of Equations 2-29 30 into 2-27, with the results L/2 \ cos (umy/W) cos (rry/L) dy = (21/ n) QosimnL/Ml- , (2-31) J- L/2 l-(m 2 L 2 /W 2 ) r d / 2 r d/2 \ e'JV dz\ e JYn z dz = d 2 sin 2 f(n7td/l) - ( Y d/2)J , (2 „ 32 ) J-d/2 J-d/2 f(nrtd/^)-M/2)] 2 18 and upon passing to the limit x-*0, we obtain 00 f 00 0= V sin 2 r(nnd/^)-(Yd/2il (16L 2 /TI 2 W) V [" k 2,/ m 2 n 2 /vv 2 )] c os 2 (mnL Z2Wl_ . [(nnd/^)-( Y d/2)] \^ £=! [l (m 2 ! 2 ^ 2 )! 2 n=-<=° X/2 JL/2 5 0t >'.n D + jC V cos (7ry/L) cos (ny / /L) , K m,h J-L/2^-L/2 '[k 2 + (a2/3 y 2)] Ho (2)(3 n | y . y »| ) dydz /. (2-33) The limiting process employed in Equation 2-33 is as previously stated in Section 2.3. Since the integrals in Equation 2-33 converge for x=0, the limit as x-0 exists. It is convenient to consider the normalized propagation constant, y/k ■ c/v - j a/k, where c/v is the waveguide velocity ratio and a/k is the normalized attenuation constant in nepers per radian. Substitution for y n » where Y n - Y (2nnA0 k (c/v - j a/k - n\/<£), yields for Y and 3, K mn k J I - (nX/-t - c/v - ja/k) 2 - (m\/2W) 2 , B n ■ k y/l •• U\/l -c/v - j a/k) 2 . Principal interest is attached to cases where •i < A/2 and W < 1.5X. Thus, foi reasonably sroi 1 1 a/k, both K m _ and P r are approximately pure imaginary 19 whenever m/1 or n/0. Under these conditions, it is convenient to let Vn = "J C m,n - ~jk/ (nX/ so that a recomputation of these terms for succeeding iterations is usually not necessary. Unfortunately, the use of either Equation 2-39 or Equation 2-42 is quite tedious because of the slow convergence of the infinite series for cases of small L/W and d/"5. Under some conditions, alternative series representations which have more favorable convergence properties may be obtained. An example is the series derived by means of the Euler-Maclauren transformation, whose rate of convergence varies inversely with L/W and d/'t. The first term of this series is essentially the integralapproximation of the infinite som. Unfortunately^ the ^italuation of this integral presents considerable difficulty in the application of the transformation to Equations 2-39,42. 24' The double sum in Equations 2-39, 42 reduces to a single sum for either of two cases of special interest; 1) for L=W, and 2) for the limiting case as 'd/X-O. These special cases are considered in further detail in the following sections. III. THEORETICAL AND EXPERIMENTAL RESULTS FOR THE CASE OF SLOT LENGTH EQUAL TO WAVEGUIDE WIDTH. 3.1 Characteristic Equation for y In Equation 2-39, let L - W. Since m jt L = n/4 cos 1 - W 2 Equation 2-39 becomes = 1. f 1, cot kD X l, 25 then assuming Re ( ^) > ® k aX _ x > « I k 26. (3-2) for all n(n = excluded), except for possibly n = + 1 Without excessive error in the final results, it can be assumed that Equation 3-2 holds for n f Then in the first two terms in the right hand side of Equation 3-1 _k_ a ±n X nA t I V *) 1 CnX 1 ni 1 1 nX -r I L-i k V' 1 1 + -T o nX + x, L_ Z U k — I and ^ .1 Jl. C +n <°£ T $ /iu/2 X2 - 4W I — -t k - I k' where in each case n is now restricted to positive values only Since the remaining terms under the summation sign in Equation 3-1 are demonstrably small compared with the first two, they may be treated even more approximately For these terms, the approximations sre n (QA-;- i) v v I k As h further consequence of Equation 3-2, it can be verified that 2.5 nD V I • ft, assuming reasonably snail a, ^ is approximately a positive re; ,1 27 for all |n| _> 1, then for Di^, coth ^D -L Finally, since small slot widths have been assumed. d < 1 X 8 xd 2 x! d < TL k | X 8 and si&_X CL 1. Id 2 With, these approximations, 3-1 gives pot xD ., X /•£ kD 2rcD X ■ sin 2 |o|d^ldJ sin 2 'nM+xd] n 3 i-lii J n 3 i + H_l°J n X k 1 n X k 1 I £ r - 2 i la f--f l , »" 2 tei n=l L_n<* 1 4 i_H_ X 21 n X k n 4 | 1+ l£x] 4 J \ n X k| 4 ^2/ X 1 - ., X' 4W' n_1 l n X k) n\k 1 1 4W 7_ _4ti 3 " T7 :1 /-M 3 — iWi dD 2 /asLxd) sin 2 /n^d+xd)^, 00 ►+ : VL 1 r g £ TtD 1-- *- |k X 4W 2 (3-3a) 28 kD 71 D -fe: s 4 +x IP"! iT^W 4 \2 -1 i 2d s . 1 1 U 4W< 4W 4n J W J \2 1 o 4W 2 » (3-3b) where S3, S4, Sc, Sg are defined by this equation. For W < „7X, it is readily verified that each term of S4 and Sg is negligible compared with the corresponding terms of S3 and S5. Computations have also shown that for smaller W/X, the final term of Equation 3-3 becomes dominant, and S4 and Sg are then negligible with respect to that term. Thus in the following, only S3 and S5 are considered. These are expanded and approximately summed in Appendix C. Polynomial representations are given, which are of the form Sq-2 S5-4 i 1 P +p 2 Uk \2 + P4 \ I 1 X k \ 1 _j red *♦* tif + «»fjf where the p j , qj are functions of t/d that are defined in Equations C- 4 and C-6 and graphed in Figures 37-42. Substituting S3 and S5 into Equation 3-3 gives 2 1-0 \ / V 1 1 e w k ' X A K (3 4) 29 whe re 1 - - X - - k 2W k '£ and I Q U W ° k X \ is given in Appendix B. The corresponding perturbation formula, obtained from Equation 3-4 using Equation 2-41, is I: 6 X = 6 £ - j \ k \ r k a r~> \ I 2D . 2 i. 8«/^¥ D l 3X X 2 i=0 Tp2i r* + V 2 'i r\ 2i x k ♦ J ■ — ^ r [fi. I x 1- _J^ ik X (3-5) Because a assumes relatively large values for the case L = W, the practical value of Equation 3-5 is somewhat questionable. However, Equation 3-5 is of interest in that it shows the general dependence of y on the various physical parameters Both a and 6(c/v) increase wi£h a decrease of D/X or of 't/X For small £/X the term in Equation (3 -■ 5)within the brackets [ ] is given approx- imately by From Equation C-4 S -2 X" Po p = ln 2^2M + LA d\ Z ■ nd °" z H 2n< Thus large values of £/d enter Equation 3-5 only as the logarithm It follows that for -t/X held constant, a is relatively insensitive to changes in d/X Further, from Equation 3-5, a is independent of i/6 in 30 9 the limit as f i* . The last term of Equation 3-5, F ( P .„ -j*-i 7 I H. ■ is plotted against W/A in Figure 3, with (3 equal to its unperturbed value 3 - k 1 - ol = * ,k | f ' The real part of F( Q (3W) is relatively small compared with the first three terms of Equation 3-5, the sum of which is approximately real. The imaginary part of F is relatively constant with W/\. Hence, for small perturbation of y» so that that quantity in Equation 3-5 contained in the braces {} is relatively slowly varying with changes in W/X„ Under these conditions, a is approximately proportional to 1 - — n- if all other dimensions are held constant, \ 4W 2 / Since the variation of this latter factor is slight except for W in the vicinity of A/2, a is relatively insensitive to changes of W/\. Thus a measure of independent control over a and c/v exists, since, to the first order, c/v depends on W/X only 3.2 Theoretical and Experimental Results. Expression 3 4 was solved for the lowest order root giving the prop- Bgation constant for the fundamental mode Because of the transcendental nature of the expression the University of Illinois automatic digital cooputei (|I.I.IA(;) wu '•mplrjyrii in i.hf solution. 31 << TZ C*S * c o ■6 o a. a Z c o € c o z ok .02 .01 Solid curve is calculated ond points are measured for -rrr=.OI66.-4r=5. Broken curve is calculated for ^-=0,-4^ = 1. W d 1.3 1.5 ■Jf- = Waveguide Width (Wavelength) Figure ¥„ Attenuation Constant versus Frequency for Fixed Wavegusd* and Slot Dimensions 34' opposite, so that the direction of decreasing phase is opposite to the direction of decreasing amplitude. A comparison of the perturbed and unperturbed c/v ratios is given in Figure 5. Several curves in Figure 7 are of particular interest in that they show the relative insensitivity of a to changes in %/d* The broken curve in that figure, which is calculated for zero -t/X and the maximum possible ■i/d ('t/d - 1) gives the largest attainable CL for the given waveguide dimensions. Curves 3 and 4, which are for impracti - cally small values of d, are only slightly below this curve of maximum a. A moderate degree of control over a is available through the vari- ation of 'C/X,, Typical results are given in Figure 8. The maximum range of a for variation of -C/X alone is seen to be approximately 10°. 1. However, this range can be increased by allowing a simultaneous vari- ation of D Curves for different W/X and D/X are given in Figure 9. Because of the relative insensitivity of a to W and d, fairly accurate results can be interpolated for most cases from Figures 8 and 9. An indication of the corresponding perturbations of the c/v ratio is given in Figures 10 and 11. The waveguide depth is limited by the need to suppress higher order modes. The unperturbed cutoff value of D for the next higher mode is given by X Ic.o 1 2/l ->- \ \2W, nroent.;il r r .,,,,,,,,,,, 1 , have shown this value may be exceeded by 35 W -r- - Woveguide Width ( Wavelengths) Figure 5. Velocity Ratio versus Frequency for Fixed Waveguide and Slot Dimensions 36. 0.7 0.9 l.l 1.3 W/X = Waveguide Width (Wavelengths) Figure 6 Attenuation Constant versus Frequency 37 0.7 0.9 I.I W/\ = Waveguide Width (Wavelengths) Figure 7. Attenuation Constant versus Frequency for Fixed Wavecuide and Slot Dimensions 38 C O •6 <§ i_ & 0) Q. o C I O U« 0.20 0.10 ^ 0.05 o "55 c o o 0.02 0.01 0.005fr o f=4400mc X=6.8cm -^- = 0.7 -|-=0234 W f = 6000mc X = 5cm -y-=0.95 -y^O: A f=8000mc X=3.75cm -^=1.27 -j- .0425 C-Bond (3.95- 5.85 kmc) Waveguide W = 4.76 cm D= 2.22 cm d= 0.159 cm a. = Slot Spacing (Wovelengths) Figure 8 Attenuation Constant versus Slot Spacing 39 20 ©W/X=0 95 D/X =0150 d/X =0317 ©W/X-. 7 D/X =0 326 d/>=0234 ©W/X=0 95 D/> = 326 d/X- 0317 @W/X-0.95 D/X 592 d/X- 0317 ©W/X= I 27 D/X =0 592 d/X-- 0425 0.1 02 0.3 0.4 JL/)* =S'oi Spacing (Wavelengths) 0.5 Figure 9. Attenuation Constant versus Slot Spacing 40 i.o 0.9 0.8 >N > V 5 o o c Q. CO :>. ■*- o o > II ■* X O 0.7 0.6 f=8000mc X= 3.76 cm W/X=l.27 dA=.0425 -A- f=6000mc X= 5.0cm W/X=0.95 d/X=P3!7 a f = 4400mc X=6.8cm W/X=0.7 d/x=.0234 C-Band (3.95-5.85 Kmc) Woveguide W= 4.76 cm D= 2.22cm d = 0.159 L/W=I.O Curves are calculated, points are experimental Broken curves are for equivalent unslotted woveguide 0.1 0.2 0.3 0.4 i/\- Slot Spacing (Wavelengths) 05 Finurc 10 Velocity Ratio versus Slot Spacing 41 io 0.9 ae o o 0) > 8 o a. ! 4) ■o o> at > o 0.7 u o a> > o > 0.6 0.5„ W/X-- Q95 d/X= .0317 L/W= 1.0 Unslotted guide D/X=.592 D/X= .443 ; D/X=.326 All curves are calculated 01 0.2 0.3 JL/\ = Slot Spacing ( Wavelengths) 0.4 0.5 Figure II. Velocity Ratio versus Slot Spacing 42 roughly 25% before the excitation of this mode by an incident TEi q field reaches an objectionable level. This limitation on D/X, together with the requirement that tf-k/2, results in a minimum a of about .005. This in turn restricts the use of structure to applications where the aperture length is of the order of eight wavelengths or less. An increased dynamic range of a can be L obtained by varying \, as considered in the following section. 3.3 Aperture Distribution Synthesis An approximate formula relating the aperture illumination to the attenuation factor as a function of distance along the guide has been given^ as A 2 (z) 2a < z > = : n i o • — *n . (3-6) l/fl-f)f A 2 (z) - fV(z)dz O "O where z is the distance along the aperture, A(z) is the desired aperture amplitude function, L is the total aperture length, f is the fraction of incident power remaining in the waveguide at the end of the aperture. Expression 3-6 may be written in normalized form, A^(z) a (z) . JL k 4rr l/a-f)^ L A 2 (z')dz'- t Z A 2 (z')dz', (3-7) I. whr;rr: ;i I I distances are now in terms of wavelengths Within certain Liaitationa, ;■ function A( i) can be obtained for a desired radiation pattern uaina conventional pattern synthesis techniques. ' 43 For the case of discrete slot spacing, (Equations 3-6,7) apply on an average basis, The uniform aperture distribution is used as a demon- stration of this application. Standard C-band waveguide is used, with an excitation frequency of 6,000 mc. From Figure 9, this gives a c/v ratio of approximately .81, which corresponds to a beam maximum approximately 36 degrees away from end-fire. The proportion of the incident power which is to be dissi- pated in the load is chosen as 1/7. Substitution into Equation 3-6 for a total aperture length of seven wavelengths and with A(z) = 1 gives f(z) = f t-- 1 — = K -P (3-8) k 4tc £ 7 -z 15.6 - z for \ = 1. 97 inches. The required variation of a is plotted in Figure 12a The average slot spacing as a function of distance along the aperture is determined from Figure 8 and plotted in Figure 12b. The actual slot spacing is then taken as a stepwise approximation to this curve. The polar coordinate system used in the measurement of the radiation pattern is defined in Figure 13. The principal plane measured and predicted patterns (qp = 0, Eq polarization) are given in Figure 14. In view of the neglect of the variation of c/v which accompanies the variation of a, the agreement as to both beam shape and side lobe level is very good. Because of the approximate basis of the procedure, this result can- not be taken as further corroboration of the theory, but only as a demon- stration of its application to cases of discrete slot spacing. A crude estimation of the mismatch caused by the discontinuity at the beginning of the aperture can be obtained by considering this point 44 .07 e 8 .06 0> Z c .05 o (A c o .04 c 0) 5 .03 e w O .02 .01 W/X=.95 D/W = .467 d/W=.033 2 4 6 8 10 12 Z = Axial Distance from Start of Aperture (Inches) 14 c A a> > c a. to o (/) 0 guide field From Equation 2-6, H.. - - l - (k 2 ♦ A) f yo JWU 3y -1- (k 2 -4) fo jwn w 2 ° 46 o < n UJ A" X _ "o \ ^,_°^ / V S ^«< CM X ^^. X o "c N — ^V ^^ / •? e %»w Xx X -C X •^ v\ yX H UJ u. 133 ^ i 1 IS c o CD 1 3 X) k. -M V) O CD k. 3 ■M k. a> a. °o E II i_ -o- o c > i C c > 3 // — \ X p \ UJ ro k. O 4- X O M X "■= E c o I / o ® flj ~^ss^ \> x !vA\ / 0) Q. ■o >*^ ^VV&V ^ / -C x <0 4r--^^®f ae As. ^^^^^*w. N? v9 llfir /* c 1 ^»w ^^^ ^ \>> '/j*9> 0) O 1 ^^"^-^. -"Ox js'^y? i lo e CO Q_ 1 UJ =f CD k. u_ o O ii -©- C r- n 47 Thus, h z 2 n Y 2 where the subscripts 1 and 2 refer respectively to the unslotted and slotted regions of the waveguide. Then T , the reflection c ef ^i _: : e: :; at the junction, is given approximately by Z 2 - Z l Y 2 /Yi - 1 r = 5-S Z 2 + Z l Y2/Y1 + 1' where y^ : k /l-(X/2W)^ and Yo are tne propagation constants m regi 1 and 2. 4.0 JO T3 f .07 -^.466 £=1.0 -^•=■0166 Curve is Calculated Points are experimental -f- s Slot Spacing (Wavelengths) A Figure 15. Input Standing Wave Ratio Versus Slot Spacing 48 The calculated results for a typical case with variable -t/X, are given in Figure 15. The measured results show reasonable agreement with the theory except in the range of small -£/X, where the perturbation of the fields by the slots is relatively great. The failure of Equation 3-9 in this region is not surprising since the presence of higher order modes and the differences in the transverse distributions of the fields in the perturbed and unperturbed regiqns are not taken into account. In the majority of antenna applications -fc/X is large at the start of the aperture, and thus a fairly good match is obtained. 3 H Surface Wave Mode. In view of the physical similarities between a corrugated surface and the transversely slotted rectangular waveguide, it is instructive to examine Equation 3-4 for a possible unattenuated surface wave solution. Toward this end, assume y/k * c/v > 1. In this case, both (3 and v. are pure imaginary and may be written P ■ -jkj (c/v) 2 -l 4 =■ -JO K - ~jkj (c/v) 2 +(X/2W> 2 -l ~- -jC (3-10) where both o and £ are positive real With these transformations, h'piation 3-4 becomes -ft" -g^iKfp'M^i-^wt*^ * iw 2 From Equation B 7 for L W, i <-j a *> k X J 2 - \ K (*) (I -^)(1 =.) 4W 2 7x cos r since Hq (2) C-ja) - J * K (a). 71 u For any fixed o define F(~) a A. 49 l n X 2 n -( i ) sin r 4W 2 dr, (3-12) F(W) = ,1 1 r / 'O Wa 'V J r7fi TZ J U: r) i TV 71 k X' 1+ \2 (1- £) cos r - 1 ■ 4W 2 " sin r F-Af 4W 2 dr. 3-13) aWr> The function Kq(-^ e )(1: £) is a positive, real monotone decreasing function for Ox. r < n._ Hence in Equation 3-13, n; jK (fi|^)(l ^)cos r dr > 0, o and for W X/2, F(W/X) '> 0. Thus for W > X/2, both terms on the right hand side of Equation 3-11 are negative and no solution giving a real value of £ is possible For values of W less than but sufficiently close to X/2, the second term in the integrand of Equation 3-13 is dominant and negative Thus for any given o, F(W/X) can be made positive and as large as desired by the proper choice of W/X Alternatively., it can be shown that for sufficiently small w, 50 \ K Q (oWr/rc) [1 - (r/rc)]cos r dr > 1/tc \ K (aWr/7i) sin r dr, ^o o since K Q (aWr/rc) - In (n/oWr) as WA - 0. Hence, F(W/A) again becomes negative when W/A is taken sufficiently close to zero. Thus, for given y > k (assuming always that y*i < k) , and therefore given a and C, there is a value of W < A/2 which will satisfy Equation 3-11 and a surface wave solution is possible. The variation of c/v versus frequency obtained from Equation 3-11 is given in Figure 16' for a typical waveguide and slot configuration. For the limiting case of ■i/X - 0, Equation 3-11 reduces to (coth C Dym = j (W/rcD)[l/l - (\ 2 /4W 2 )] J U-jo/k), (W/A)]. (3-14) For this case, there is no lower frequency limit for progagation of the surface wave mode, i.e., c/v approaches 1 as to - 0. An expression which is asymptotic to Equation 3-14 is easily developed. Since c/v - 1 as V\/A - 0, then, from Equation 3-10, C/k - A/2W, (coth WO)— [coth (tiD/W) /(nD/W)] o/k - /(c/v) 2 - 1 - J2(c/v - 1). (3-15) I <>i small o, only the first term of I Q is appreciable. From Appendix H, Equation B19, and Table 1 , 51 5.0 3.0 2.0 >.s o o I 4) O O Q- <7r o> ■8 1.50 •E 1.20 >» u o 9 > 1. 10 ii o|> 1.05 1.02 025 17.5 X = Wavelength ( centimeters ) 15,0 12,5 11.25 C-Bond (3.95-5.85 Kmc) Waveguide w = 4.76 cm D = 2.22 cm L/W=I.O d = .l59cm i/d = 2.5 d=0 m- 1.0 Curves are calculated Points ore experimental for d = .!5Scm Jl/d = 2.5 0.3 Q35 0.4 W/X = Waveguide Width (wavelengths) 0.45 0.5 Figure 16. Velocity Ratio versus Frequency for Surface Wave Mode UNIVERSITY OF IU.IN0W LIBRARY 52 J H?) ?)--J 2 ^ (1S273) k A Hn2l k A 807 + 1.215 --^!> Substituting into (3-14) gives coth 2E 2 W 1 W_^ 1 '(1.273) 4W 2 4\\f2 Y + ^ln - %in 2 + Y 2 ln (c/v -1) A 2 -807 + 1.215 -Xj, 4W 2 ' which simplifies to c/v = 1 + exp J. 808 - 1.91(A 2 /4W 2 )-(n 2 /4HX 2 /4W 2 - 1) coth tiD/W + In A 2 /4W 2 L ^ (3 17)^ Both the rapidly damped fundamental and the surface-wave modes are excited by an incident TE]^ q field when W < A/2. For this condition, typical variations of measured phase and amplitude with axial distance along the waveguide are given in Figure 17, The computed curves in these figures were obtained by taking a solution of Equation 3-4 for y f° r tne fundamental mode, y/k = - .190 - j.438, and the measured value of y/k =1.77 for the surface wave. Relative phase and amplitude of these fields were adjusted to give the best fit with the measured results. The poor agree- n.r-ri t ;it the beginning of the aperture is possibly due to the presence of bighei ordei evanescent modes. For comparison, typical measured results Undei condition* ol single mode propagation (W > A/2) are given in I i gure L8 53 T3 -10 0) ■o a. < at il -30 ^ = .465 -? X W .=.467 -L=_ = i.o W d - = .0165 -=£- = 2.5 ' d i / v — ^s» «'— »»g j^^- Broken line is theoretical. Solid line is experimental 1.0 2.0 Z s Axial Distance from Start of Aperture (Wavelengths) a) Axjal Amp] itude Distribution 3.0 c o -5 o 2 -ijo L* -2.0 (0 o .c 0- -3.0 -4.0 -5.0 W.:.465 -£- = .467 ^.= 1.0 £,.0,65 f ,2.5 Curve is theoretical. Points are experimental. 1.0 2.0 Z = Axial Distance from Start of Aperture (Wavelengths) b) Axial Phase Distribution 3.0 Figure 17. Axial Phase-Amplitude Distribution with Fundamental and Surface Wave Modes Present, W<\/2 54 .o ■o 0) 5 -10 E < .2? £ -20 ■o g> w 3 (A o 0) -30 W/X=0.97 L/W=I.O D/W=.467 d/W=.OI65 i/d = 5 1.0 2.0 3.0 4.0 5.0 6.0 Z=Axiol Distance from Start of Aperture (Wavelengths) c o |-2.0 t= a> 3 s 5 -4.0 -6 W/X=0.99 L/W= 1.0 d/W=.OI65 D/W=467 i/d =5 1.0 2.0 30 4j0 5.0 6.0 Z = Axial Distance from Start of Aperture (Wavelengths) Figure 18. Measured Axial Phase-Amplitude Distribution Under Conditions of Single Mode Propagation 55 As inferred from Figure 17, the surface wave launching efficiency is relatively low when the slotted guide is excited internally by an evanescent TE^ q field. Dielectrically loading the feed section as indicated in Figure 19 gives some improvement in the launching efficiency. However, it appears that an external surface wave launcher, such as a waveguide horn, is necessary for relatively high excitation levels for this mode. Cut Away Side View. Figure 19. Dielectrically Loaded Feed Section to Improve Surface Wave Launching Efficiency There are also a number of higher order surface wave modes. For example, Figure 20 shows evidence of the lowest order asymmetrical mode for W/X near unity. Initial excitation is the result of a slight dissymmetry in the waveguide structure. Similarly, a higher order symmetrical mode is excited when W is slightly less than 1.5 X (See Figure 21). The excitation level of this latter mode by an incident TE^ o field appears to be of an objectionable level forl.4'< W/X < 1.5. Thus, unless effective measures for its suppression are taken, the presence of this mode restricts use of the structure to cases where W < 1.4 X. 56 ■o V T3 5 -20 "5. E < 5 w o 2 -40 -60 W/ko.97 D/W=0.353 L/W=10 2345678 9 10 Z = Axioi Disionce from Stort of Aperture (Wovelenglhs) 12 1.0 Q. £ 2 0> .76 .50 g .25 p ,— -G ,/"• / 6 / Cos \ A. * -*2- me curve o Measured Amp. T X w '2 Transverse Amp Dist. at Z : 0.5\ W 2 Transverse Phase Dist. atZ=Q5\ n Transverse Amp Dist. at Z -7.0X I ° I Tronsverse Phase Dist. otZ=70\ Figure 20 Fvidence of Asymmetrical Surface Wave Mode Resulting from Small Structural Dissymmetry When W/A Near 1.0 57 3 d. E < o> i£ 0) l_ (A O 0) W = 1 42 D - 467 L - 1 n ^-.467 w -..v, _o A =.0165 4- =2.5 -4 \ *~ -8 -10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Z = Axial Distance from Start of Aperture (Wavelengths) 141 1 ■0 A =.0165 4=2.5 Q. -'0 E < ■a -20 a> iZ (A ^° O a> -50 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 I.I Z = Axial Distance from Start of Aperture (Wavelengths) a. E < CD b_ US O a> 1.0 0.8 0.6 0.4 0.2 w irjr -y-=l.l9 / 6 if ¥ V V V wow 2 2 Transverse Amp. Dist. at Z = ll\ to a> a> o» a> ■o a> o -C a. •o a> i_ 3 CO O a> _w _w 2 2 Transverse Phase Dist. at Z = 1 1 X Figure 21 Evidence of Third Order Surface Wave Mode for W/X Near 1.5 IV. THEORETICAL AND EXPERIMENTAL RESULTS FOR SMALL SLOT LENGTH 4 I Characteristic Equation for y. The results of Section III show that y is relatively insensitive to the values of t and d when t $ X/10. Although these results strictly apply only for L = W, approximately the same condition should exist for arbitrary L provided that hf-i is large. Thus the simplified form of the characteristic equation (2-39) for the limiting case of /£ /A = is of special interest. In Equation 2-39, let Z/\ - 0, with i/d remaining finite. Since lim k/a n = lim 1 ■ Q, R ? fi> f*0 f*0 J(n\/l - y/k) 2 - 1 A A. lim Jc/^s lim • - L ' = 0, n / 0, i-0 trO /(n\/> (4 " 1} re k - k/T~- (A/2W) 2 -• (y/k) 2 , (3 - kJT~- (Y/k) 2 5," kj( f/k) 2 'r (mX/2W) 2 - 1 58 59 and where Iq ((3/k, L/X) is given in Appendix B. The corresponding perturbation formula is obtained using Equa^ tion 2=41. Also, with Y/k = ( 0Y /k) +.6 ( Y /k), where 6(y/k) is small, C„^ k J( oY /k) 2 + 2( 0Y /k)6( Y /k) + (mX/2W) 2 - 1 (4-2) 2*X/2W >/m 2 - 1 for m - 3. Thus 6(yA) 6(c/v) - j at J] -(X 2 /4W% / D )(X/W)| - k/3(1 - X2/4W2) D/W + U - lW) 2 . 8k { cos 2 (rcL/2w) -' cos 2 aSL r-^ (m 2 X 2 /4f 2 - i) 2E w th nn/ff , j m l^L + + j s| (i :> 2 / w2 N ) 2 1 (s,l) ' l (4-3) 16 cos 2 (nL/2w) \k X The first order approximation to y is obtained by using qyA =/l - (X/2W) 2 in the right hand side of Equation 4-3. Thus (Y/k)! = Y/k + 6 ( Y /k) - yA + F ( P/k), (4-4) where 3 « k J 1 - ( oY /k) 2 ■ k / 1 - 1 + (X/2W) 2 = k X 2W" Better approximations to the exact solution of Equation 2-39 are 60 obtained from successive iterations of the form (YA) i+ i - oY /k + SjCy/k) ■ yA + F( i 3A) where (i*l)P = k J l * ^Y/k) i + 1 ■ ky 1 -[( Y/k)+ 6T (yAJJ * k(\/2w)/ 1 - 8(1 - X 2 /4W 2 )^ (W/X) 2 6 £ ( Y /k). For use in the first iteration, curves of In (nfi/k, W/X) with W/X as parameter are given in Figures 22=24. The first N terms of the infinite series of Equations 4-1,3 were W L course of these computations, the coefficients summed for N = 5 ^ . The results are given in Figures 25-26. In the ( m 2 X 2 /4W 2 ) -J. (m 2 - \)K were treated correctly for the first three terms of the series. For the remaining terms, the approximation (m X 2 ]/4W 2 was employed. In addition, the approximation coth (tiD/W) /m 2 - 1 - 1.0 was used for all terms. The results given axe estimated to be within ■bout 5% of the exact value of the infinite sum for cases where D/W - 0.2. For smaller D, the hyperbolic cotangent approximation is invalid and recti Oil terms are necessary. 1 2 Theoretical and Experimental Results Two iterations using Equation 4 3 were employed for the results in 61 CO CD > f GO. o o jC o c 0) 0) H _J ■o w- 3 o ^ o» o CD -M > o 5 to Q_ it _ (0 - * CD CM CM O L. m ._ «N u_ C"7'S)J >° |JOd |09 ^ 62 -0.2 -0.5 •1.0 -2.0 -3.0 ->U % -10 -20 £ o -50 H -.100 ■200 ■500 '/ W/X = l.27/ W/\=/95 W/\= .686 .02 025 Figure 23 .0375 .05 .075 L Slot Length .10 .15 .20 .25 W Waveguide Width Imaginary Part of / (oh/k. LA) versus L/W for •02 ' L/W 25 1.0 63 0.5 0.2 0.1 •0.1 -J* -0.2 . O Z -0.5 o O. d» d e -i.o H -2.0 -5.0 -10 / w/x=u \7 /W/X: = .95 /W/X = .686 0.2 0.4 0.6 _L__ Slot Length W " Waveguide Width o.e 1.0 Figure 24» Imaginary Part of I (()f3/k, L/X) versus L/W for 0.2 - L/W - 1.0 64' c/> 500 200 100 50 \xX 20 W N. 10 5.0 2.0 1.0 0.5 .02 .025 .0375 .05 .075 .10 L Slot Length "W Woveguide Width Figure 25 S 1 (I cos 2 DHL (mV —jfct :¥ .a mtLr.}' ( m 2 V/2 I) I)* ^T5 .20 .25 65 V) 50 20 10 5 2.0 1.0 f\ 9 \W \w/x-686S 'X=.95\ U.c W/X = I.27N 0.1 05 .02 01 0.2 0.4 0.6 L Slot Length W "Woveguide Width oo cos 2 !ML Figure 26 s'= £ ( 0.8 2W > l ,J J m 2 X 2 _| 1W 2 (m 2 - 1)2 i.o 66. Figures 27-30. Convergence of the iterative process was sufficiently rapid that the results for6(y) are estimated to be within about 10% of the values obtained by an exact solution of Equation 4-1. Since Equation 4-1 is based on an assumed aperture distribution, further iteration does not appear warranted. Of some interest is the fact that y i s practically insensitive to variations of L above about A/2. Effective control over a thus is obtained only with relatively small values of L. In this connection, it should be noted that for L < A/2, a is quite sensitive to finite values of wall thickness, because of the evanescence of the fields in the slot region. Thus for cases of moderate wall thickness, significant discrepancies may exist between the actual values of a and those pre- dicted by the approximate analysis. The experimental data given in Figures 27-30 wereobtained under an effective condition of vanishingly small wall thickness (see Section 5.1) The technique of aperture illumination control is as outlined in Section 3.3, Since in the case of infinitesimal slot spacing, there is no essential difference between the results obtained with the trans- versely slotted and the axially slotted guides, detailed examples of this application are not necessary. 1.3 Surface Wave Solution Equation 4-1 admits an unattenuated surface wave solution for L I'-, than a critical value near A/2. As in Section 3.5, assume y/k : c/v ' 1 . 67 L /X= Slot Length (Wavelengths) 0.2 0.3 0.4 0.5 g o cr a. a> Q. a> Z c o w c o c o o c r .002 .005 .0005 0.4 0.6 L Slot Length W Waveguide Width Figure 27 Attenuation Constant versus Slot Length for Infinitesimal 1y Spaced Slots, W/X = 0.686 68 L/XrSlot Length (Wavelengths) c o o a. a> a. a> c o c o o o 3 C Q. a> a. a> Z c o M c o o c .o o 3 C E w O z ii a .0005 L _ Slot Length W S Waveguide Width Figure 29 Attenuation Constant versus Slot Length for Infinitesimal ly Spaced SlotSo W/X = 1.27 70 0.80 - 0.686 0.70 0.60 Curves are calculated for -°- =0,^- = l Points are measured for 4- =.Q22G,~r = 2-5 © D/W=0.l © D/W = 0.2 © D/W = 0.446 0.2 0.4 0.6 0.8 1.0 >> u +— o o a> o > 9i > 4) (A •> O r U o 0- a. CO « •o a> i- 3 u. « > D 0.9 0.8 © D/W = 0.l © D/W = 0.2 © D/W = 0.446 d JZ Curves are calculated for -y =0, -j- =1 Points are measured for — =.0316, 4^=2.5 X d 0.2 0.4 0.6 0.8 1.0 o > ©// '© — — ^— — — — © D/W = 0.l © D/W = 0.2 © D/W = 0.4 i 46 0.90 0.80 w Curves are copulated for *■( I ' d , J_ ) 0.2 0.4 0.6 0.8 L . Slot Length W ' Waveguide Width Figure 30 Velocity Ratio versus Slot Length for inf in itesimal ly Spaced Slots. i.o 71 Then - - j o - - j kvWv) 2 - 1, j C - - j k/(c/v) 2 + (X/2W) 2 - 1 K = With these substitutions, Equation 4-1 becomes ■I c os 2 (mTiL/2W) coth£n - M-I 2 /w 2 ) 2 1 _ CD cos 2 (ti:L/2W) 1-(X 2 /4W 2 ) 2nD-^Uw 2 | (1 - m 2 L 2 /W 2 ) 2 ^ V /m 2 X 2 £-^ ,«,2 • coth 2LD + j Jt fl-(L 2 /W 2 ) 2 1 16D cos 2 (rtL/2W) [1-(X 2 /4W 2 )] I (-j |, L), (4-4') whe re ^ = kj (c/v) 2 + (mrt/2W) 2 - 1. Since c/v > 1, the perturbation of c/v from that of the TE^ q closed waveguide mode is in most cases relatively large. Hence the iterative method of solution using Equation 4-3 is of little value, while the direct solution of Equation 4-4 is quite tedious. However, an asymptotic expression valid for small L/X which is a slight extension of Equation 3-17 is easily developed. From Equations 3-14, 15, Section 3.4, we have as c/v-1 , coth CD */ coth(rcD/W ) CD tcD/W J o (- J ?.?) * " J ^(1-273) k A K Y + In 2 L k X .807 + 1-215 ^ 4W 2 I ' a = k J(c/v) 2 - 1 ~ k J2(c/v - 1) 72 Substitution into Equation 4 4 gives re [1 -(l 2 /w 2 )J 2 -. I - JOn -*| - fcln 2 + -.807 + 1.215^}, Y + Jttn(§ - 1)+ \2 where S has been written for the infinite summation over m„ Simplifying, c/v ■ 1 + exp J .808 -(2X/W) fS - L9l(\ 2 /4L 2 ) + In (\ 2 /4L 2 ) + 4 [l -(x 2 /4W 2 )]- . cps 2 (rtL/2w) coth nD/ A [l -(L 2 /W 2 J 2 J (4 . 5) From Equation 4-4, S increases indefinitely for fixed L/W as W/X - 0. Thus, since c/v - 1 as co - 0, there is no lower frequency limit for propagation of this mode. Similarly, from Figure 25, S increases without limit as L/W - 0. Thus for fixed W and D, the mode continues to propagate as L -■ 0. However, for L « X/2, the fields are quite loosely bound to the aperture surface with the principal energy content being in the external region. Under this condition, the mode is inappreciably excited by an incident internal TE^ q field. The sequence of axial field distribution diagrams in Figures 31-32 gives an indication of the relative excitation of the surface wave and the fund amen taJ modes as L/A is increased, frequency and all other dinun i mi:-; hf-ing held constant. Presence of the surface wave is indicated by the deviation of the measured db field amplitude from a straight lino f|f pcri'l'-ri' i- on ;ixiaJ distance. For these curves, the field ampli- t u'jf *;i\ rn'-;i\ij r fl ;i t the h.ir;k w;ill of the waveguide. The E component of the iurfac< wavi rari< ipproximately as <■ x as x - - D, while the 73 .o 3 Q. E < o> IZ M O 0) 2 o d/W .95 =.033 0/W=.467 JL/d = 2.5 -5 L/X = .28£ -IO -15 -90 2.0 4.0 6.0 8.0 10 12 Z = Axiol Distance from Start of Aperture (Wavelengths) z) L/X = .285 •o a. 4 E < 2 6 « iZ . 8 t» o 2 10 12 WA = .95 D/W= .467 ■ d/W=033 J-/d = 2.5 \ ^s Lr L/X =.332 *\J* ' j\si /v/v y^ l ~ r S/ V \ \ u 2.0 4.0 6.0 8.0 10 12 Z = Axial Distance rom Start of Aperture (Wavelengths) b) L/X = .333 Figure 31. Measured Axial Amplitude Distributions with Fundamental and Surface Wave Modes Present. (Also see Figure 32. ) 74' SI Q. E < 1 , L . W - Q r- D T =.oo, ■ f-*> "w-^' -5 d 1 r- 033 d =2 - 5 / \ -10 1 -15 / V 1 i ■20 -25 ■30 1 2.0 4.0 6.0 8.0 10 12 Axial Distance from Start of Aperture (Wavelength) c) L/\ - ,381 2 -10 Q. E < — 20 2 iZ -30 o a> -40 -50 -60 1 L . w D -r— - .**c~v —r--.^i} Tr--.tor A A W l-- 033 f =2 - 5 W V 2.0 4.0 6.0 8.0 10 12 Z= Axiol Distance from Start of Aperture (Wavelengths) d) L/X ■ .429 Figure 32 Measured Axial Amplitude Distributions with Fundamental and Surface Wave Modes Present 75 same component of the fundamental mode reaches a maximum at the point of measurement. Thus, since C increases with increasing L/X, the apparent decrease in excitation of the surface wave mode after L exceeds about 0.3 X is somewhat exaggerated. A more reliable indication of the process is given by the radiation patterns (q> = 0, E e polarization) in Figure 33. Frequency is varied with all other dimensions held constant. The presence of the surface wave is indicated by the appearance of a second lobe near 9 = in pattern 33b, and by the interference pattern in 33c. In 33d, for L = .45X, evidence of the surface wave has again disappeared. From this sequence, it appears that for these particular transverse dimensions, excitation of the surface wave reaches a maximum for L near . 35X. A possible reason for the peak in the relative excitation of this mode can be drawn from curve 3 in Figure 30b, for which the transverse waveguide dimensions are approximately the same as for the patterns in 33b, c. The velocity ratio c/v for the fundamental mode reaches a maxi- mum for L = .34X. Since c/v has been increased from its unperturbed value, the fields for the fundamental mode are evanescent with x, i.e., decay approximately exponentially as x — -D. Thus the transverse distribution of the internal fields of the fundamental mode most closely approaches that of the surface wave mode when L — , 34X, and the relative excitation of the latter mode is thus increased in this region. Following the same reasoning, it would appear from curves 1 and 2 in Figure 30b that excitation of the surface wave would be further 76. e=o° f =3500mc fs4500mc e=o ( 9=0° f =5000mc f =6300mc MI.76 ca D 2 22 cm L 2.14 cm d 397 cm <£/d-2.5 L-76 cm All patterns are voltage measurements with 0° Figure 33 E-Piane Radiation Patterns Showing Relative Excitation of Surface Wave versus L/A as Frequency is Varied 77 increased by a reduction in D, since this results in a greater perturbation of c/v for the fundamental mode. The patterns in Figure 34, with (c) and (d) being obtained with decreased D/X, show this to be the case. In addition, patterns (a) and (c) indicate excessive c/v ratio of the sur- face wave for the given aperture length L. Patterns (b! and (d} are for the same conditions as, respectively, (a) and (c), except for a reduction in L/X. Alternatively, when effective suppression of the surface wave mode is required, L must be held to relatively small values. The upper limit is dependent on the transverse dimensions of the waveguide. For D/W = .467, I - 15X and for uniform L/X, approximately 12db suppression of the end- fire lobe is obtained with L - .27Xfor W = ,7X, or with L - . 18X for. W = 1.4X. The result is an effective upper limit on the value of a which may be obtained. For use in antennas of short aperture length where relatively high a is required, control by means of variation of -t/X is to be preferred (Section III). 78 e=o< 9=0° e=o° e=o° W/A - 1.06 L/A 32 d/X 035 l/d - 2.5 All patterns are voltage measurements with 0-0° Fir-ure 31 E-Plane Radiation Patterns Showing Increase in Surface Wave Excitation with Decrease in D/A V. EXPERIMENTAL VERIFICATION OF THE THEORY The validity of the approximate theory was checked by measurements of the propagation constant on a laboratory model. The results of these measurements were reported in previous sections. A brief discussion of the experimental procedure is included below. So I Details of Construction of the Model . . A view of the model including the important dimensions is given in Figure 35. All measurements were performed in the 2-7 kmc frequency range. Sectional construction of the model was employed in order to allow variations in the transverse dimensions of the waveguide. In the course of assembly, the interior seams which resulted were covered with adhesive backed aluminum foil tape. The slots were milled in a separate 1/32 inch thick aperture plate. In order to reduce the number of mechanical operations, the slots were cut in each case to a uniform length equal to the width of the assembled waveguide. The effective slot length was then varied by masking part of the outside surface of the aperture plated with foil tape, While the actual thickness of the aperture plate was about .02X at 7 kmc, it should be noted that for small L/W, the masking process gave an effective wall thickness which was considerably less and roughly corresponded with the 10~3 inch thickness of the foil. The fields were probed at discrete intervals through a series of closely spaced holes located along the center of the back wall of the 79 80 Figure 35 The Experimental Model 81 w aveguide as shown in Figure 34. The use of an array of holes instead of the usual longitudinal slot improved the mechanical stability of the model and eliminated the possible excitation of an asymmetrical slot mode result- ing from a dissymmetrical location of the slot. At several points along the guide axis, transverse arrays of holes were provided to enable observation of the transverse distribution of the fields. Since the thickness of this back wall was somewhat greater than the hole, diameter , radiation losses through the holes were negligible. 5=2 Measurement of y in Cases of Large a. Two separate procedures were employed in the measurement of y. Under conditions where a was relatively large, the slotted waveguide was termi- nated in a matched load to give an approximately pure exponential mode of propagation. The value of y was then obtained from direct phase-amplitude field measurements along the axis of the guide. As a general rule, this method was employed in those cases for which the reported value of - was k greater than .005. A schematic diagram of the phase comparison setup employed in the measurement of the phase velocity appears in Figure 36. The principal limitation on the accuracy of this method arose from the occasional presence of extraneous surface wave modes. In these cases, the measured phasle and db amplitude of the fields did not bear their usual linear relationship with axial distance, and an accurate determination of y was not possible. Under conditions of single mode propagation, the experimental values of y are believed to be accurate to within about 5%. Under conditions favorable for the excitation of surface waves, in particular for 1.3 X ± W < 1. 5 X for the results in Section III, and for 0.3X < L < 0„ 5X 82 s CD \ '5 0\l 5 M >.\ -© 3 O CO o» CO •o AAA/ O X ft IpUDjg 30USJ9^9H ^A/ \ © \ \ \ & i V AAA ii — f < C O CD CD \ X \ -Q- CO m o a a; 0) x s -AAA- w -M C CO E CO 1_ CO 00 a. co CO <0 C C0 X o c CO CO CO CO CO 1_ a. 83 for the results in Section V? } the accuracy is somewhat less. 5o3 Measurement of y in Cases of Small a. For the cases of small L/W in Section IV where the reported value of 2 k was less than approximately .01, a was determined indirectly from a standing-wave-ratio measurement. For this case, the transversely slotted waveguide was terminated in a short circuit, and the input voltage- standing-wave-ratio (VSWR) was measured at a point prior to the beginning of the aperture. Since this method assumes negligible reflections at the beginning and end of the aperture, the magnitudes of these discontinuities were reduced by tapering the slot length gradually from zero to the desired L over a short distance at the aperture end points. The residual VSWR (measured with the waveguide terminated in a matched load beyond the end of the aperture.) was then found to be less than 1.03 for all cases of interest. Since the tapered endpoints contributed an unknown amount to the total radiation loss, the short circuit VSWR measurements were performed for two cases of differing lengths of the uniform portion of the aperture. In addition, conductivity losses were approximately taken into account by performing a third short circuit VSWR measurement with the aperture complete- ly covered over. Assuming that the conductivity losses are only slightly affected by the presence of the slots, and assuming that the radiation losses from the uniform portion of the aperture are directly proportional to the aperture length, these three measurements gave a corrected radiation loss for a length of uniform aperture. The value of a determined in this way is estimated to be accurate to 84' within about 10% for the largest values of a reported (^ of the order of 01) and about 20% for the smallest For convenience, the associated value of c/v in these cases was deter- mined from the distance between two well spaced minima measured along the aperture under short circuit conditions. Since observation of the fields was possible only at discrete intervals, the exact location of the mini- mum was estimated from plots of the measured amplitude distribution. The values of c/v so derived are estimated to be accurate to within about 5%. Vis CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER INVESTIGATION The propagation constant of the transversely slotted waveguide can be effectively controlled through the variation of the transverse dimensions of the waveguide and the dimensions and spacing of the slots. Thus, for antenna applications, the aperture illumination can be adjusted by appropriately varying these parameters. The phase velocity is affected primarily by the waveguide width and to the first order is independent of the slot length and spacing and of the waveguide depth. The attenuation constant is primarily a function of the latter three parameters, being to the first order independent of the waveguide width. The normalized attenuation constant, a/k , is strongly dependent on the slot length, L, for L < A/2. Above this point, a/k is practically insensitive to further increase of L. The maximum a/k obtainable in this manner is of the order of 0.1 for a standard waveguide aspect ratio. However, in the usual case where single mode propagation is required, the existence of a surface wave, which is strongly excited for L of the order of .35A, imposes a further limitation of about .01 on the maximum value of a/k obtainable. While decreasing the waveguide depth, D, gives an increase in a/k for a fixed L, it also tends to increase excitation of the surface wave, so that this method is of little value in extending the upper range of a/k. Instead, increased a/k can be obtained by employing the slot spacing as the controlling parameter with L > A/2 , since the surface wave does not exist for L in this range. Since a/k is approximately independent of L for L > A/2, the results for the particular case of L equal to the waveguide width, W, are sufficiently general. In this case the range of a/k for variation of the slot spacing, -C, alone is about 10:1, 85 86, although a/k can be increased without limit by decreasing D. The minimum value of a/k obtainable for L=W is of the order of .01, which roughly corresponds with the upper limit imposed by the existence of the surface wave when L > A/2 • Consequently, practically any value of a/k is obtainable if all three parameters are allowed to vary. The remaining parameter, the slot width, has relatively little effect on either a/k or the velocity ratio c/v. Further investigation of the surface wave supporting properties of the structure is indicated. Of importance are methods of surface wave excitation which give high excitation efficiencies without the need for external launching devices. Alternatively, the characteristics of different periodic coupling mechanisms, e.g., closely spaced circular holes, should be studied from the standpoint of attempting to decrease the surface wave excitation. Since the pattern of a traveling wave line source is conical when c/v>l, a broadside array of slotted waveguides is ordinarily employed to give the required pattern characteristics. Thus, the effect on the propagation constant and on the individual element radiation pattern when several slotted waveguides are in close proximity should be investigated. Finally, the effects of finite wall thicknesses on the attenuation constant of the fundamental mode and on the excitation and phase velocity of the surface wave should also be considered. This parameter is of particular importance in cases of small slot length. REFERENCES 1. Goubau, G. " On the Excitation of Surface Waves," Proceedings of the IRE, Vol. 40, No. 7, July 1952; p. 865. 2. Whitmer, R.M. ' Radiation from a Dielectric Waveguide" Journal of Applied Physics, Vol. 24, No. 9, September 1953; p. 949. 3. Zucker, F.J. ' The Guiding and Radiation of Surface Waves", Proceedings of Symposium on Modern Advances in Microwave Techniques", Polytechnic Institute of Rrooklyn, 1954; p. 403. 4. Rotman, W. "A Study of Single Surface Corrugated Guides", Proceedings of the IRE, Vol. 39, No. 8, August 1951; p. 952. 5. Elliott, R.S. " On the Theory of Corrugated Plane Surfaces", Transactions of the IRE, Vol. AP 2, April 1954; p. 71. 6. Mueller, G.E. "A Broadside Dielectric Antenna," Proceedings of the IRE, Vol. 40, No. 1, January 1952; p. 71. 7. DuHamel, R.H, , and Duncan, J.W . A Technique for Controlling the Radiation from Dielectric Rod Waveguides, Technical Report No. 11, Antenna Laboratory, University of Illinois, Urbana, Illinois, 15 July 1956- 8. Rotman, W. The Channel Guide Antenna, Report No. E505b, U.S. Air Force Cambridge Research Laboratories, January 1950. 9. Rumsey, V.H. " Traveling Wave Slot An tennasy Journal of Applied Physics, Vol. 24, No. 11, November 1953; p. 1358. 10. Harrington, Roger F. '' Propagation Along A Slotted Cylinder", Journal of Applied Physics, Vol. 24, No. 11, November 1953; p. 1366. 11. Hines, Rumsey, and Walter " Traveling Wave Slot Antennas", Proceedings of the IRE, Vol. 41, No. 11, November 1953; p. 1624. 12. Kelly, K.C. and Elliott, R.S. , " Serrated Waveguide, Theory and Experiment", IRE Convention Reco r d, Part J; Antennas and Propagation, March 1955; p. 6. 13. Elliott, R.S. Serrated Waveguide, Parti: Theory, Technical Memorandum No. 354, Hughee Aircraft Co, , Culver City, Cal. 14. Brillouin, L. , Wave Propagation in Periodic Structures, Dover Publications, Inc., New York, 1953; p. 139. 15. Slater, J.C. , Microwave Electronics, D. Van Nostrand Co., Inc., New York, 1950; p. 169. 87 88 REFERENCES (CONTINUED) 16. Schelkunoff , S. A., Electromagnetic Waves, D. Van Nostrand.Co., Inc., New York, 1943; p, 413. 17. Saxon, D S Notes on Lectures by Julian Schwinger, Massachusetts Institute of Technology, Cambridge. Mass;, February 1945. 18. Morse and Feshbach, Methods of Theoretical Physics, Section 9.k, McGraw-Hill Book Co., Inc., New York, 1953. 19. Ibid, p. 1109 20. Rumsey, V.H. " Reaction Concept in Electromagnetic Theory" Physical Review, Vol. 9k, No. 6, 15 June 1954; p. 1483. 21. Woodward, P.M. " A Method of Calculating the Field Over a Plane Aperture Required to Produce a Given Polar Diagram", Journal of the IEE, Vol. 93, Part Ilia, No. 10, March 1946; p. 1554. 22. Dunbar, A.S. A Method for the Calculation of Progressive-Phase Antennas for Shaped Beams, Technical Report No. 10, Stanford Research Institute, June 1950. 23. Rumsey, V.H. Traveling Wave Slot Antennas, Report No. 486-10, Ohio State University Research Foundation, Antenna Laboratory, June 1953. 24. Montgomery, C.G. , Technique of Microwave Measurements, McGraw-Hill Book Co. , Inc., New York, 1947; p. 819. 25. Watson, G.N. Theory of Bessel Functions, The Macmillan Co., New York, 1944; p. 338. 26. Jahnke and Emde, Tables of Functions, Dover Publications, Inc., New York, 1945; p. 273. APPENDIX A PROOF OF THE STATIONARY PROPERTIES OF EQUATIONS 2-18 AND 2.-21 From Equation 2-18, h (s') h*(s) K (s/s') ds ds' = S (A-l) S S where S denotes the slot region and where s and s' have been written for y, z and y', z' , respectively. Assume h differs from g by a small amount, i.e., h(s') - g(s') t r\ (s'), h*(s) = g*(s) + n *Cs), where r\ is small in absolute value. Let K a (s/s') be the kernel when the parameter y has the value y c which satisfies Equation A-l. Thus ■ \\ [g(s') + r\ (s')Hg*(s) + n*(s)] K a (s/s') ds ds'. (A-2) Let 3Y K a = K c + ~- 6 Y + K. . where K c is the kernel when y has the correct value y c given by an exact solution of Equation 2-15, Substitution into Equation A-2, expanding, and neglecting terms of second order or greater gives = J J g(s') g*(s) [K c (s/s') + — 6y] + [g(sOn*(s) + g*(s)Ttfs') S 9 Y K c (s/s')] ds ds' ■ 6 y\\ g(s') g*(s) -^ds ds' + JJ [g(s')ri*(s) + g*(s)n(s')] K c (s/s' ) ds ds , (A-3) 89 90 since, from Equation 2-15, g(s') K r {s/a') ds' = 0. (A-4) The last term of Equation A-3, g*(s)n(s') K c (s/s') ds ds' J ))s(s) I s g(s')Tl*(s) K r (s/s') ds ds'l*, since, from Equation 2-17, K (s/s 7 ) = K* (s'/s) Substitution into Equation A-3 gives = 6 y \X g(s')g*(s) |^ ds ds' + 2 Re < \V(s) [\g(s') K c (f,Jds]dsi S ^ S S J The final term is zero by virtue of Equation A-4. Thus, assuming g(s') g*(a) ^ ds ds' f 0, we h ave 6 Y " 0, and the stationary property of Equation 2-18 is proven. The proof of the stationary property of Equation 2-21 is entirely analogous, although the z dependence has been eliminated and the problem is then two dimensional. The following is a repeat of the argument ;i'ivanced by Rumsey . 91 w Let y be determined from the expression pL/2 \ J^y) ■ [si £3 (y) dy = 0, -L/2 here, from Equation 2-20, [s] is the matrix (A-5) [s] - 1 1 Let E a (y) be the assumed aperture distribution which differs by only a small amount from the correct aperture distribution, E^ (y), i.e., E^ (y) - E^ (y) + 6 E . Since J a (y) is a function of y > then to the first order 9 Ja (Y C ) ia (Y) = i ( ^c) + ~~^~~ 6 Y « where, as before, Y c is the correct value of y< Equation A-5 gives to the first order Substitution into - IP *ifc (Yc> 5 Y + £a (Y c ) 'W -6 E] dy. a x 'c J c ^ c 3 Y (A-6) Since J a is proportional to 6 E, the final term of Equation A-6 is also second order. Also, from the reciprocity theorem for traveling wave line sources , a v >c \ J a ( Yr ) is] E c ( Yc ) dy ■ \ Jc <* c ) • [s] h (,r c> d y * °> 92 since J_ (y c ) is identically zero. Therefore, assuming b . u t-jkJlA dy i o. we have 6y = 0. APPENDIX B THE EVALUATION OF I n (§ A , L/ A) From Equation 2 -36., A/2 L/2 I n = l/4tt(A/L) 2 \ C cos(rty/L)cos(7iy^L)(k 2 +32/"9y2>H ( ( 2 )O n p) dydy', J. L/2 J- L/2 L/ L/ (B-i) where Let where .L/2 L/2 0n =J k2 ~ V\ p = ly - y'l- 'n " I ni + V I = 1/4* (X/L) 2 ^\ C cos(7iy/L)cosfTcyVL)(9?/3y 2 )l4 2 \3 n p) dydy'. (B-2) ^-L/2 J- L/2 Integration by parts with dv =(32/By 2 )H < 2 ) (3 n |y-y'|) ■ -(#/3y 3y) H Q (2) (Pjy-y'D, u = cos(rcy/L) cosCny'/L) gives L/2 f L/2 I„ 2 - (-l/4n)(X/L) 2 ^ (3 n p) J.Tt/2 J-n/2 ./_ t\2/A.i2 cos y cos y'-(X^/4L z ) sin y sin y (B-4) dydy. Since p is a function only of y-y' Equation B-4' may be further simplified by first integrating with respect to y' with y-y held constant, and then integrating over all possible values of y - y. If r= y - y, then >/2)-r I n = 1/nV H ( 2) (3 n L |r|/n) dr V [cos y / cos(y+r)-H Jo J-Tt/2 -(X 2 /4L 2 ) sin y' s i n ( y'+r )] dy+ r t r /2 + l/n\ H Q ( 2) (B n L | r I/tc) dr \ [cos y'cos(y / +r) + Jo J-fn/2)-r -(A 2 /4L 2 ) sin y' sin(y'+r)] dy. (B-5) i .• cos y' cos (y-i-r)= cos (-y ) cos (-y'-r), sin y'sin (y + r)=sin (-y') sin ( -y -r), 95 it is easily verified that the two terms of Equation B-5 are equal and K (K/2)-r J n = 2/n\ H (2) (3 n Lr/rc) dr V [cos y / cos(y+r)+ ° ~ K/2 (A 2 /4L 2 )sin y sin(y+r)] dy' . ( B -6,) Performing the indicated integration with respect to y gives r n =C H ^) (3 n Lr/n) -^o (1- -X 2 /4L 2 ) cos r + i/rc (1 + X 2 /4L 2 ) sin r dr. (B-7) For n/0, Equation B-7 may be approximately evaluated in closed form. From Equation 2-34', 3 n can be written 3n - " J°n - ~ jk/CnAAe - y/k) 2 - 1, where o R is approximately real and positive for n / 0. With this substitution, rl < 2 ) (3 n p) ■ j (2 A) K (a n p) Let Then, from Equation B-7, Tt/a tc/c n L. I n = j (2a/Tt)C K Q (u) [l-(X 2 /4L 2 )] [UauA)]cos 2rc, ;iri'J thus a - 1/3. (B-13) 97 Under this condition, and for the argument indicated in Equation B-12, F is given approximately by F ( 1 | J a )^1.085 ± ,165 ji Also ± ia.- 1 \ 3/4' s e +j(3Tl/40 [l+j (1 + l/2)a-(l + l/8)a 2 +j(l + l/16,)a 3 + \± J* + 1 | -? e ±j (3TC/40 [l+j i„ 5a . 1,125 a 2 ], arc cosh ( + ja) = In (a + /a 2 + 1) ± j (rc/2). Since /l + a 2 ^l + (a/2) 2 , then 2 ( a+ | 2 ) arc cosh (±ja)— a + ^ - -- ^ — + -. . ± j - 2 2 2 ~ a . a + j a 2 J 2 and sinh [arc cosh (+ja)] &■ + j cosh f a - a /2] , • L3/2 r /j.- m^ ±j (3te/4') (i + 3 2\ smh°/ z [arc cosh (±ja)]— e 4 Upon substitution of these results, Equation B-ll becomes C \ u K Q (u) cos au du — 2 l (l-j 1. 5a- 1.125a 2 ) (1.085 - .165ia) + (l+j 1 .5a-1.1. 25a fyl.085+.165ia) 3/2" j (1 + 2. a 2 ) (1 + 3. a 2 ) v. 4' 4' -4L (1.085 - 2.27 a 2 ). (B-14) 98 From Equation B-9, 00 K Q (u) cos au du^| (1 - | 2 ). (B-15) Also, since arc. sinh a = In (/ 1+a 2 + a) — In (l+a+ a — ) 2 *a (l - ai), 2 then, from Equation B-10, K Q (u) sin au du^a(l - a 2 ). (B-16.) Collecting terms gives approximately _« I ^ -r/2 J [l-U 2 /4L 2 )] (1 - a 2 /2) - (A 2 /4L 2 )(4/7i 2 )(a-a 3 ). (B-17) J' or Ini> from Equation B-8, 4 J n 4 . 1/a \ K Q (u/a) J [1-(X 2 /4I 2 )](1 u/?t)cos u + l/n[l+(\ 2 /4L 2 )]sin ul du. I 01 I a | < ] /-> and 4m (a) ' 4', I- (u/a)| <2/^72 /R J-^-O'^U- , 99 and \l n | <] 2 |a| ^ e - u /l a l/2<(2u/n) [1-(A 2 /4L 2 )] + 2/rt(\ 2 /4L 2 )i du = 2 j2|a|A [|a|(7i+|a|)e- 7i: /kl/2[l.(X2/4 L 2) + | a | e - (n/ |a |/2 (X 2 /4L 2)] < 2>/27T^[l/3(r C 4/3) e - 3n //^[l-(X 2 /4L 2 )]+l/3|a|e- 3 y^ +( / 35)3 ^(X 2 /4L 2 )] - it/2 [l-(\ 2 /4L 2 )] (8.56.) 10" 4 + (X 2 /4L 2 )(4/rt 2 ) |a| (2.72) lO" 3 ] . Thus J n is negligible compared with I_ and I n *j(2a/iO 1^ ja i[I- (\ 2 /4L 2 )] [l-(a 2 /2)] + (X 2 /4L 2 ) (4/rc 2 ) (a-a 3 )i . With the substitution a = rt/c n L, I n ^j[l-(A 2 /4L 2 )] -S- [l-^ 2 X2a n 2 L 2 )] +J (A 2 /4L 2 )[4/(a n L) 2 - (47t/(o n L) 4 ] = j/klj [l-(X 2 /4L 2 )](l^ n )[l-(l/8)(X 2 /L 2 )](k/a n ) 2 + ^ + j(X 3 /L 3 )[(l/2Tt 2 )(k/a n ) 2 -(l/8rr 3 )U 2 /L 2 )(k/c n ) 4 A. (B-18) This is Equation 2-37. Because the limitation on a n stated in Equation B-13 is violated for n= 0, a similar evaluation of Equation B-7 for this case is not possible. Instead the power series representation of H Q ^ '((3p) may be used and the resulting series integrated term by term. Substitution of the power series and the collection of similar terms gives I [(3A),(LA)] = Y (-l) 1 [(a/k)(L/X)] 2i + x2 /4'L 2 (r 2i - s 2i ). i R 2i : 2/rc [Q 2l Y 1/r - (t 2l .u 2l )-^ 2 /4L 2 )(t 2l - ^.^ (B _ 19) r=j Y = Euler's constant = ,5772 ..., and where r 2l = l/(i!) 2 V (u 2l A) sin u du, -'o r K s 2i = l/(i!) 2 \ u 21 (1 - u/Tt) cos u du, (B-20a) (B-20b) t 2 j l/(il) z \ (u 2i A) sin u In u du, o (B-20c) n u 2i " l/(i!) \ u 21 cos u In u du, (B-20d) These integrals are in standard form and are readily evaluated through repeated application of the appropriate reduction formula The results are r n - - ^ni 12JJ1 2i n (i!)2 y (- D r ^ , , L'r-b <2D! (i!)2 r i-l 5"* Lii! « 2r ' (1- 2i±i_) + _2JL2i±U ft (2r+Dl 2r-2 rc (B-21a) (B-21b) 101 fc 2: n (i!) 2 ,r „2t ±=DLJ*L + ci (rt) - y + (2r)! u 2i i" 1 ir ff 2r .T 7 (-D r * 2r V^ (JL + 1 } + V 7 ( JL + _J_) Z_^ ( 2r )i ,Z-^ 2 J 2J" 1 ^ 2r 2r - 1 r=0 j=r+l r = l (B-2lc) (-l)i LJ2x±1_ ln K + (.di n 2 ^-l g + (i!) 2 { i-l kl (2r-l ' ^ 2r - 1 (2i+l )! 2r (2i)! -1) i-l _1 + 2r±2 + y (X + U__) 2i 2r+l ^ +1 2 J 2 J +1 _ + 2 2i±L Tt (_L + _i_) 2r 2r+l l_ r : + 2_i±2 _ Si ( rt ) + 71 + Ci(rt) - y Tt (B-21d) where Tt TT Tt 1 Si (Tt) = \ SliLJC dx s ) x 11! 3-3! 5-5! Jl Ci (tt) cos_JL dx - 7 + ln Tt - -l£- + -E- x 2-2! 4-4'! £.= for i - 0, = 1 for i / 0. An alternative form (infinite series expansion) for the coefficients may be obtained by substituting the power series for sin u and cos u in Equations B-20a~d. The results are r 2 = (-1) 1 " 1 (Mil 1 -2 (-D r rc 2r , (B-22a) TT (i!) z r ^ (2r) 102 00 ti =(-l) i+1 i2iU Y Vl) r t?*+1 (1 _ 2i f 1 ) i (i!) 2 £r* (2r + 1)! 2r + 2 (B-22b) . . (-D 1+1 (2i)! ' 2l rt d!)2 ( l) r Ti2r pi (2r)! In rt 2r j=2i+l J L 1 (B-22c) u 2i , (.l)i + l I2i) ,r ff 2r+l 2r+l i^lLji^IL. I (i . 2i + l )(ln „ . V 7 l) (i!) 2 ^1 (2r+l>! L 2r + 2 J= %r +2 J 1 + 2i + 2 2i+l (2r+2) 2 • (B-22d) In some cases, for large i, because of their rapid convergence, Equations B-22 are more readily evaluated than the corresponding Equations B-21 . Values of the coefficients for i up to 10 are tabulated in Table I Table I. Coefficients for I Q (|, L) K A 2i + s 2i "2i '2i fc 2i+ u 2i '2i - u 2i 1.27324 -8.07572 (io-i) 1.21487 1 1.19022 2.54648 3.50990 (io-i) 2.20546. 2 7.22338 (10" 1 ) 3 57068 4.02443 (io-i) 3.13587 2.83234 (10-J) 2.40779 1.95932 (10"1) 2.22433 4'. 7.72136 (10-2) 9.91318 do-}) 5.92386. dO" 2 ) 9.51924' do-}) 5 1.54708 (10-2) 2.77969 (io-i) 1.28820 dO" 2 ) 2.74205 (io-i) 6 2.37413 (10" 3 ) 5.67262 (10-2) 2.07623 (10" 3 ) 5.71178 (10-2) 7 2.87979 (10"J) 8.81818 dO" 3 ) 2. 6 J 044' (10" 4 ') 9.01986. dO' 3 ) 8 2 .'83059 M0" 5 ) 1 07992 dO" 3 ) 2.63714" (10' s ) 1.11840 dO" 3 ) 2.30023 do* 6 ') 1 .06.933 (10" 4 ') 2.18994' do- 6 ) 1.11857 do- 4 ') 1 '7123 (10 -7 ) 8 74088 dO" 5 ) 1 52188 do- 7 ) 9.21884 dO' 6 ') APPENDIX C APPROXIMATIONS FOR THE SERIES S3 AND S 5 IN EQUATION 3-3 We have sin 2 ( nnd . jdj n 3 (i . L i 1)3 — n X k sin 2 (DM + yd) ■ I 2 n 3 (i + 1 £ 1)3 n A k ■ (C-l) With the relations (1 ± x) -3 = 1 + 3x + 6x 2 T 10x 3 + sin 2 [(n7td/4)-(yd/2)] - sin 2 [(nnd/^)+( Y d/2)] = -(1/4) sin (2nTtd/4). •sin yd, sin 2 [(nnd/^)-(Yd/2)] + sin 2 [ (n7id/<£) + ( yd/2 )] =1 -cos yd + 2 cos yd. >sin^ (nKd/ / t) / Equation C-l becomes approximately f 7 1 - cos yd + 2 cos yd sin 2 ^ ~ S 3 * I 1 + 6 (£ 1)2 _1_ + 15( i i)4' _L X k X k n' 3/4' (^ *) sin yd X k 00 sin 2fflSl 1 +LQ(i X) 2 JL + 7^X)4X 3 X k \k n 4' (C-2) Several of the series in Equation C-2 are relatively slowly convergent for small values of d/-t. The following empirically modified asymptotic expressions which are accurate to within about 5% for d/'i <_ 1/2 are 103 104 used: ■*, sin 2 ^ -$-s ( nd)2 ln 2_1M + 3.2 (i) 4 ' 3 'u Ttd K, f? sin 2 ^ ^£ (2) sin 2 ^ (l 4' 'v 562/1), r^ sin 2 nai -£ (3) sin 2 ^ (1 - .340 i) 5 -to- sin 2 vnd ~^— - C (V-2) sin^ a, V > 6, sin 2nrcd -^ - r (30 sin 2M (1 .680 d), .4' 'V *M sin &M * _^ I (v-1) sin 2M, v "t v > 6, where r (v) is the Riemann zeta function , 26 ' having the following values ecv) 2 1.64S 3 1.202 4 1.082 5 ] . 037 6. 1 017 7 1 008 8 1 004 105 Since yd < n/4', then sin yd ~ yd *- Y *l' , 2 cos yd — 1 - ' Y ' 2 With these approximations, and upon dropping all terms involving (— ^) to the fourth power or greater, S3 is given approximately by S3 - 2k 2 U/l) 2 Vp 2i (i I) 21 , (C-3) where P = In (2.23-t/Ttd) + (3.2/tc 2 ) (d/^) 2 , p 2 = 1.20 + (7.2/n 2 )sin 2 (7td/^)[l - .34_ 2 - " 3 (1 " ^) 5 (1 + £ *) 5 A k A k With the approximation (1 + xf 5 a 1 + 5x + 15x 2 +35x 3 + 70x 4 ', where x = (^ ^) , Sr is given approximately by A k ° 106 ■I r - / 1 ^5 i-s given a^pi'i K r i^O where S 5 -4<^)2 £ q 2l (4l)2i i (c-5) q - (l/2)aAd) 2 sin 2 (rid/4), q 2 = U5/2ri 2 )(4/d) 2 sin 2 (7id/^) + (l/2)cos 2 (nd/^) + (5/8KH4/d)»in(2nd/4) 1 q 4 . i (35/ri 2 )(4/d) 2 sin 2 (rid/4) + ( 15/2) cos 2 (rid/4) + (35/8ri)(4/d)sin(2rid/4) + - .416. n (d/4) sin (2rid/4). 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Kelleher Melpar, Inc. 3000 Arlington Blvd. Falls Church, Virginia Electronics Research Laboratory Stanford University Stanford, California Attn: Dr. F, E. Terraan Dr. R.E. Beam Microwave Laboratory Northwestern University Evanston, Illinois ERRATA Technical Report No 14 CLOSELY SPACED TRANSVERSE SLOTS in RECTANGULAR WAVEGUIDE by Richard F Hyneman Contract AF33 (6160 -3220 Project No 6.(7-4600) Task 40572 Page 6. Equation following (2-1) should read: G(x,y,z) - G_(x,y,z-^) 15 Tne first sentence of the second paragraph starting on Page 15 should read " the case of an aperture of finite extent, S " 17 Equation 2-30 should read p L/2 r * pd/2 Hy^ = (l/j^)[k 2 ^ 2 /9 2 y 2 ](=j/2^)\ \) \ cos (ny'/U ^L/2 ifcoo ^d/2 eJYn z h o ( 2 ) ((3 n p) dz'l dy'. 18 The closing brace } follows the final term of Equation 2-33 In Equation 2-33. tne differentials should be dy dy instead of dy dz' 19 The closing brace } follows the final term of Equation 2-35, In Equation 2-35: 5 m n = for m 1 when n ■- = 1 otherwise 22 The third equation is Equation 2-40, the fifth is Equation 2-41, 52 The last term of Equation 3 16. should be "1*215 (X 2 /4W 2 )" In the Equation following 3-16 s the closing brace } follows the final term 58 Equation 4-1 should read 9 r_» cot y. D = 1 (1 - L 2 /w 2 r k D 1-(X 2 /4W 2 ) cos 2 (tiL/2W) ERRATA (continued) Page 60 The second equation on Page 60 should read (i+1) 3 = k/i - (yA)? +1 ■ k/i - [( oY /k) + a^y/k)] 2 70 Figure 30: D/W = 466. 71 The next -to-last equation should read: I (-J g , L ) - - J 2.J(1 -273) [y + In f L] - .807 + 1.215 - o k A re J k A 76. For Figure 33c: L/A = 356. 89 The final term of tne equation preceding Equation A-3 should read: -[ g (sV (s) + g* (s) r\ (s')] K c (s/sO}ds ds^ 90 The final term of the fourth equation should read: .. . . + 2Re)\ ty*(s) [\ g(s') K c (s/s 1 ) ds' ] ds 1 91 In the second equation: s E 1 -1 95 A plus (+) sign snould precede the second line of Equation B-6. 96. The next-to-last equation should read: c n L > 3tt . 98 The closing brace follows the final term of Equation B-17. 100 Equation B-20d should read: re ^ u^ 1 (1 - — ) cos u In u du. u 2i (i!)' 105 Equation C-3 should read 2 E i = o