SCATTERING BY A TORUS by Pushpamala Laurin A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1967 Doctoral Committee: Professor Otto Laporte, Chairman Professor Chiao-Min Chu Doctor Ralph E. Kleinman Associate Professor C. R. Worthington Associate Professor Alfred C. T. Wu RL-415 = RL-415

ABSTRACT Solutions of low frequency scattering of acoustic and electromagnetic waves by a torus are derived in this work using a "quasi-static" approach based on a method due to Stevenson (1953) and Kleinman (1965). The solutions are in power series, in ascending powers of k, the wave number. These series are also called the Rayleigh series and are valid for small k. Since the method is such that the solution is constructed from the solution of the potential equation, Laplace's equation is solved in toroidal coordinates for both the Dirichlet and Neumann boundary conditions. Green's function is derived for the Dirichlet case and particular problems are solved for the Neumann case. These results also have applications in fluid dynamics. Two non-zero terms in the low frequency solution expansion are explicitly derived for the cases of acoustic scattering by soft and rigid tori. Two terms in the low frequency expansions for both the electric and the magnetic fields are derived for the scattering of a normally incident plane electromagnetic wave. For this case the torus is assumed to be perfectly conducting. The far field is calculated for a small torus for the acoustic problem, with normal incidence on a soft forus, and compared with the known results for the corresponding problem of a sphere and of a disc. The radii of these bodies which give equivalent scattered far fields are calculated as a function of the radius of the torus.

ACKNOWLEDGE MENTS The author is very grateful to Dr. Ralph E. Kleinman for his continuing guidance during the period of her employment at the Radiation Laboratory in the last five years. His close attention to this thesis problem has been very valuable. The inspiring discussions and the very valuable suggestions made by her chairman, Professor Otto Laporte. have contributed much toward the completion of this work. Thanks are due to all her committee members and also to Dr. Olov Einarsson, who during his short stay in the U. S. provided the author with insight into scattering theory and who served as an excellent teacher to her. The constant encouragement and assistance of her husband, Mr. Dean Laurin, has been an important factor in finishing this work. The careful typing of the manuscript by Mrs. Katherine McWilliams is sincerely appreciated. Finally, the author is indebted to National Science Foundation Grant No. GP-6140, for providing the financial assistance through the Radiation Laboratory, The University of Michigan, Department of Electrical Engineering. The author wishes to express her thanks to Professor Ralph E. Hiatt for providing this opportunity. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.....................ii LIST OF ILLUSTRATIONS.................... iv LIST OF APPENDICES.....................................v I INTRODUCTION.I....................1 II SOLUTION OF LAPLACE IS EQUATIONS..................... 7 2.1 Toroidal Coordinate System............................ 7 2. 2 General Solution of Laplace's Equation.................. 11 2. 3 Green's Function for the Exterior Dirichlet Problem for a Torus...................................... 14 2.4 Exterior Neumann Boundary Value Problem....... 17 2.4. 1 Torus Immersed in an Infinite Incompressible Fluid Flowing Parallel to the Axis of the Torus with a Uniform Velocity v........... 17 2.4. 2 Torus Immersed in a Fluid Flowing Perpendicular to the Axis of Torus with a Uniform Velocity v............................. 23 2. 5 Green's Function...................... 27 II: QUASI-STATIC APPROXIMATION FOR LOW FREQUENCY SCATTERING......................................... 30 3. 1 The Scalar Problem.................... 30 3. 2 Far-Zone Field....................... 37 3. 3 The Vector Problem.................... 39 IV ACOUSTIC SCATTERING BY A TORUS................... 46 4.1 Plane Wave Incident Normally on a Rigid Torus 46 4.1.1 ZeroTth Order Term............... 46 4.1.2 First Order Term................ 48 4.1.3 Second Order Terms.............. 53 4.2 Plane Wave Incident Normally on a Soft Torus...... 57 4.2.1 Zerotth Order Term.............. 57 4.2.2 First Order Terms............... 59 4. 3 Far - Zone Field.....................................63 V ELECTROMAGNETIC SCATTERING FROM A TORUS...... 67 5.1 Zerolth Order Terms............................ 67 5. 2 First Order Terms............................... 71 VI SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK.. 77 BIBLIOGRAPHY.................................. 83 iii

LIST OF ILLUSTRATIONS Figure Page a (eW+ 1) 2-1 Bipolar Circles, z*: a(e....... 8 e -1 2-2 Toriodal Coordinates (r, 0,) ). Coordinate Surfaces are Torolds (r = constant), Spherical Bowls (0 = constant), and IHlf -Planes (q = constant)............. 9 2-3 Cross-Section of a Torus.................16 3-1 Geometry for Scattering by an Arbitrary Body.......31 4-1 Capacity of a Torus /Capacity of a Sphere........... 61 4-2 Equivalent Radii for Sphere and Disc in Terms of the Radius of Torus.....................66 5-1 Electromagnetic Wave Incident on a Torus..........67 B-1 Region of Application of Green's Theorem.......... 83 iv

LIST OF APPENDICES Page APPENDIX A...........................................82 APPENr1IX B............................................83 ApPENDixc C............................................87 V

I INTRODUCTION Scattering theory deals with the interaction of an obstacle in a wave field and the best known and most often studied wave motions are those associated with acoustics and electromagnetism. A standard problem in acoustic scattering involves the determination of a function D (r) which is the field scattered by a body. In particular we seek a solution of the Helmholtz equation (V2+k2)6 = 0 such that (r) - - inc(rB) or a$(r) | _ an - - an r B rB and j satisfies the Sommerfeld radiation condition (assuming harmonic time dependence, e ). Lim ( — a -ik ) 0 r -> oo r r where c is the incident field, r is the radius vector (employing spherical coordinates) and rB is a vector from the origin to a point on the boundB ary, B, of the scatterer and n is the outward normal from the scatterer. Correspondingly, in a vector problem we seek solutions of the Maxwells equations in vacuum, VxE = at V = 0 Vx = + J V' = 0 at 1

2 where E and H denote the electric and magnetic field strengths, while B = pH and D= E where E and p are the permittivity and permeability of the medium, assumed to be constants. It is of particular interest to solve for E and H, exterior to a body of Infinite conductivity and In a region of space containing no free charges and electric currents. If we assume harmonic time dependence of the wave motion, our problem can be stated as follows: Determine E and H, such that V x(VxE)-k2 E = 0 V x(VxH)-k H = 0 ^B r - rB ^ -Ir = r B A — n D = 6 where k represents the surface current density and 6 is the surface charge density. The scattered fields must, in addition, satisfy the radiation conditions: lim - - -scat) scscat r x (VxE )+ikrE =rx(VxH )+ikrfca =0 The scalar Helmholtz equation in three dimensions has only been solved In eleven coordinate systems. These eleven coordinates systems have the property of separability; viz., the wave equation is split into three second order differential equations, each equation being a function of only one variable. Then the field ~ can be written as a linear combination of terms of the form 1 = U (u)V (v)W (w) where u, v, and w represent the coordinates in an orthogonal curvilinear coordinate system. On the other hand, the vector wave equation in three d imensions is separable onlyinrectangular coordinates and spherical coordinates. In general the vector

3 problem is considerably more difficult than the scalar, since one has to solve for six scalar components of the fields E and H. Efforts have been made to extend the method of separation of variables to certain other coordinate systems where one of the coordinates can be separated. leaving a non-separable second order differential equation. This, then is reduced to a recurrence set of ordinary differential equations in one variable. In particular, this method has been applied to the wave equation in toroidal coordinates by Weston (1956). Although the solution of the equation is written in terms of toroidal wave functions, the application of boundary conditions poses some problems because the wave functions do not form a complete set. But in the limit of a very thin ring results have been obtained by Weston for the scattering of a plane electromagnetic wave. These problems have led researchers into attacking the scattering by non-separable bodies by techniques, where separability of the wave equation is not involved. In the low frequency limit (when the wavelength of the incident radiation is larger than the dimension of the scatterer), advantage has been taken of the solution of Laplace's equation, which is a limiting form of the wave equation when the wave number, k, is zero. This method of treating the scattering problem as a perturbation ot the potential equation is, of course, valid only when k is small. A detailed exposition of various methods of treating the low frequency scattering problems for both scalar and vector problems is given by Kleinman (1966). The correspondence between low frequency solutions and the static problem was recognized by Lord Rayleigh as early as 1897.. He seems to be the first to have observed that the solution of Laplace s equation constitutes the first term in an expansion for the scattered field in powers of k, when k is small. This was not pursued any further until the 1950's, when interest was revived in this subject to obtain systematic series expansion valid for small k. The advantage of such a method lies in that the potential problems, though formidable at times, still are simpler than problems in wave phenomenon. The major contributions in this area have come from Stevenson (1953). Noble (1962), and Kleinman (1965).

4 Stevenson' s method holds good for both scalar andvector problems. The method is fairly straight forwardand has been applied successfully to certain shapes including an ellipsoid of revolution. But the major disadvantage of this method comes from the fact that one has to solve a static proble m at every stage of the expansion. Every term is derived in terms of the previous term and a static problem. In spite of it, it has proved to be a powerful method for low frequency scattering problems. Noble (1962) formulates the problem in terms of integral equations and a solution for a scattering problemfor a general boundary is obtained as the perturbation of the solution of the corresponding potential problem. Each term in the low frequency expansion is the solution of an integral equation differing from term to term in the inhomogeneous part. In general the solution is obtained only as a formal inverse for successive terms and does not yield an explicit representation. Difficulties arise in carrying out the scheme except for some simple shapes. The technique proposed by Kleinman is a very elegant one and one can obtain the n th iteration in the expansion of the scattered field systematically. The method is limited to scalar problems as yet and it is limited in another sense that one would have to know the static Green' s function. Details of the method are given in his paper (1965). It was originally applied to problems with Dirichlet boundary conditions, but it has recently been extended to scattering problems with Neumann boundary conditions by Ar (1966). In this thesis a quasi-static approach based on Stevenson' s method will be used to study the scattering of plane acoustic and electromagnetic waves by a torus. This brings us to the subject of solution of Laplace's equation. For separable coordinates systems this is a fairly simple matter, but the Laplace's equation in toroidal coordinates is not separable. By removing a factor called the R-factor, the equation can be made simply separable and the solution of the equation can be written as U(u) V(v) W(w) R(u, v, w) where R is a function of the coordinates: and is not a constant. The separability of Laplace's and Helmholtz equations are discussed in great length by Moon and Spencer (1961). The work reported in the literature on the subject of toroidal coordinates have dealt only with the solution of Laplace's equation and in particular for

5 the Dirichlet case. The earliest one reported is by Hicks (1881), who has developed many interesting results involving toroidal functions with applications to some potential problems. It was followed by another paper by the same author in 1884, where he focussed his attention on the motion of a hollow vortex, where cyclic motion exists in a fluid. His work is by far one of the most detailed and valuable. The papers by Basset (1893) and Dyson (1893) are mainly concerned with the toroidal functions as such and serve as a good supplement to Hick's work. Recently, S. Loh and his coworkers have done extensive numerical work on the toroidal functions which are published in a series of papers 1959a, 1959b, 1961). Most of the theoretical work in their papers is already contained in Hick's papers. Although a good deal of work has been done on the Dirichlet or the electrostatic problems involving tores surprisingly little has been reported on the Neumann boundary problem. In the preparation of this thesis it was found that there are great difficulties in solving the Neumann boundary problems, while the Dirichlet problems can be solved in a fairly straight forward way a4 ha been rqppr1edby Iicks (1881), lMoonand$pencer (1961), Morse andFeshbach (1953), Hobson(19), Ioh (1960) etc. jiaist to ame a few. Butwith the exception of Hikp' work, no mention has been made by any of the authors about the other problem where the pormil 4eriva4ve rather than the potqntial itself is specified on the boundary. Flui dynicit ve bn intrested in the Neumann problem in connection with the vortx rings 4nd Basset (1893) and Lamb (1932) have reported some results for the above mentioned problemfor the particular case of the fluid flowing in the direction parallel to the axis of symmetry of the torus. This special case can be handled elegantly by means of a vector potential which satisfies Stoke's equation and enables one to obtain the stream-lines. But unfortunately this method is not J, Hicks gives credit to Neumann (1864) for introducing the toroidal functions for the first time to study the temperatures in a shell bounded by nonconcentric spheres.

6 applicable to studying the related problems when the fluid is flowing in any other direction. Hicks also has studied the problem of a torus moving parallel to its axis of symmetry in an infinite fluid. This was done in terms of a scalar velocity potential and considerable simplification resulted due to symmetry in the azimuthal variable. The above mentioned symmetry does not necessarily exist in the scattering problems as such, and as a first step the solution of the potential equation with non-symmetric Neumann boundary condition is derived in this thesis. But still much remains to be done in potential problems for a torus. For example, it has only been found possible. so far to solve the Neumann boundary value potential problems whenthe torus is immersed in uniform fields, at any angle. But for arbitrary sources the results are yet not suitable, because the Green's function of the second kind is known only up to a set of constants. It is worthwhile to mention at this outset that very few authors have considered the solution of wave equation in toroidal coordinates. The two papers known so far are the ones of Weston (1956) and Bond (1955). Both solutions hold good for thin rings. The latter, obtained by the method of local separation, being valid only in a limited region, Chapter II of the present work deals with the toroidal coordinates and the solution of Laplace's equation. Chapter III describes the method of low frequency expansion for the scattered fields and the derivation of the far fields. Scattering of acoustic waves by both soft and hard tores are considered and explicit results have been obtained up to the third order terms using the quasistatic approach in Chapter IV. In Chapter V the zeroeth and first order terms in the expansion of the scattered field are derived for scattering of an electromagnetic wave incident normally on a perfectly conducting torus. The derivation and discussion of Helmholtz formula is included in an appendix for the sake of completeness.

II SOLUTION OF LAPLACE'S EQUATION 2.1 TOROIDAL COORDINATE SYSTEM Toroidal coordinates are generated by rotating the bipolar coordinates obtained by the familiar transformation Z* = a w e -1 about the y - axis. (See Fig. 2-1), where z =x + iy w = u+ iv These orthogonal curvilinear coordinates rY, 0, i, are defined by the equations: sinh r0 cos cosh r-cos 0 sinh rl sin b Y= a cosh 7-cos0 sin 0 z = a cosh r1 - cos 0 where r] ranges from 0 to oo, 0 from 0 to 2 r and, from 0 to 27r. The surfaces n = constant are tores or anchor-rings with an axial circle in the x - y plane centered at the origin and of radius a coth r], having a circular cross section of radius a csch r. The surface r7 = rY defines a torus. (See Fig. 2-2). 2 2 2 2 z + (p - a coth ) = a csch ro defines a spherical bowl, and & = 0 defines a spherical bowl, 0 7

FIG. 2- 1: BIPOLAR CIRCLES, z* a(e 1) w - e -1

= const O= const '0= + 7 0= const FIG. 2- 2: TOROIDAL COORDINATES (rl, 0, ). COORDINATE SURFACES ARE TOROIDS (r7 = constant), SPHERICAL BOWLS (0 = constant), AND HALF-PLANES (q/ = constant).

10 (z2 acot 2 2 2 (z - a cot ) + p = a csc O 0 0 where =1 2 2' p =Vx + y a sinh r cosh n - cos 0 The metric coefficients are given by, h = h = r/ 0 cosh r) - cos 0 a sinh r b cosh r - cos 0 If we define a coth r = R and acsch r = r then o o and a o o= r then 2 2 a = - r o o and R o cosh r =R 0 r 0 The z - axis corresponds to rn = o and rl = oo corresponds to an infinitely thin ring of radius a. The Laplacian in toroidal coordinates can be written as follows: 2 ( _ cosh rl - cos 0) a sinh r a sinh n a 8 1 a rl cosh r1 - cos 0 G r, J 1 aQ sinh ri a Q cosh r' - cos 0 a2} rl - cos'~ aIq2J E)S si- h o Posh siilh i r[cosh

11 2. 2 GENERAL SOLUTION OF LAPLACE'S EQUATION The Laplace's equation V p2 = 0 can be written in toroidal coordinates as, a sinh r1 a l 3 r Lcosh rl - cos a n a [ sinh q 0 Lcosh r - cos 0 a~ aej0 - a24 sinh r1coshr - cos 9 a 2 (2.1) = 0 This equation is not simply separable, but if we set = osh - os F (, 0, ) the Eq. (2. 1) reduces to s --- a sinh sih r a L na) a 2F + 2 + a 2 1 sinh2 r a2F -,4,2 + -F = 0 4 (2.2) Equation (2. 2) can be separated Into three second order differential equations given by, 1 d dE sinh dT (sinh d-) + 1 3 [ - a- s sinh r1J E = 0 (2.3) (2.4) d2 a2 dO d 2 + d a 3 = 0 (2.5) Following the usual procedure F = E (i7) o (0) ~ (O)

12 2 2 and letting a = n and a = m, the solutions of the above set of equations are, (Morse and Feshbach, 1952) E = a Pn 1(cosh r) + q Q - (cosh r7) nm n - 1/2 nm n - 1/2 0(8) = C cosnO + D sin n n n (1) = A cos mV + B sin mV mn m where a,,B C, Dn, A and B represent constants. nm nm n n m m P m/ (cosh nr) and Qn 1/2 (cosh r) are called tesseral toroidal functions. n-1/2 n-/ They are Legendre functions of the first and second kind respectively, of order m and degree n - 1/2. These functions are defined by: 1 (l)mr (n+) 12 cosmu du n c1/2 r (n - m+-) J [cosh n + sinh cosu]n 12 m 12 Go Cos h mu du Q (-1)m (n+ ) coshmudu Q u/( \n+1/2 n-1/2 r(n-m+-) J O osh + sinh rh coshu] Now, we can write the complete solution of the Laplace's equations in toroidal coordinates. m(r 0. O) = (cosh - - cosm +B sinmu [C cosn 0+ sinn] [\a P 1 (cosh +) r7+ mcosh The toroidal functions were first introduced by C. Neumann (1864), and they have been studied in detail by Hicks (1881, 1884), Basset (1893), and briefly " Alternate definitions of these functions can be found in Hobson (1955) (see Appendix C).

13 by Heine (1881 ). It should be noted that neither the space interior nor exterior to the torus is simply connected, hence the direct application of these functions for potential problems involving circulation is often not possible as the potential in such cases are not always uniquely determined by their values on the surface of the torus. Equation (2. 6) is the general solution of Laplace's equation, but now we have to consider the values of these functions in the space interior and exterior to the torus. 0 < r < rl0 constitutes the exterior to the torus and r0 < rl < oo, the interior. The outer space contains the plane surface o 1 - n=0. Investigation of P /2(cosh r) and Qn 1/(cosh ) in these regions n-ve1/a2 n-1/2 reveal that as r - 0, the function Q n /2 (cosh rj) goes to infinity and as r->,co P / (cosh rn) goes to infinity. Therefore the potential n-1/2 function suitable to the exterior region of the torus, in which we will be primarily interested, is given by = cosh - cos 0 [Am cosm1 + B sinm]/ x m=0 n=O m x [ cosnO + D sinnO Pnm /(cosh ) 0 < r < (2.7) n n ~ n-l/2 - o Similarly for the interior region, as oo = cosh - cos O [A cosmy + B' sinm] x x n m m0O n=0 x [C cos n +DI sinn6] Q m (cosh r) rl < rl < oo (2.8) Ln n J n-1/2 o

14 Two of the most important types of potential problems encountered in physics are, to find the potential everywhere, (i) when the value of the potential is specified on the surface (ii) the value of the normal derivative of the potential is specified on the surface. These are commonly referred to as the Dirichlet and the Neumann boundary value problems respectively. 2. 3 GREEN'S FUNCTION FOR THE EXTERIOR DIRICHLET -PROBLE M FOR A TORUS The Green's function for the Dirichlet problem can be derived easily, but apparently, it has not been done before. So we proceed to derive this in the following way. Let GDE represent the Dirichlet Green's function. DE 1 The free space Green's function for the Laplace's equation is where R is the distance between the source point (ro, O, o ) and the field point (r, 0,,). [cosh rl - cos 0 / [cosh - cos 0 1/2 — 2 [O o -CO o J 1 a4 2 osh i7 cosh rno - ssinh sioh rl cos(l ) - cos (O - 0O) / and - can be expanded in toroidal coordinates (Hobson, 1955): R 1 1/cOS os o 0' cosh - cos 0' co o E r -m+ — —:.. C E (-1)M -- - n=O m r(n+m+ ) m2 P n-lm (osh (osh) Q 1/2 (osh ) < cos m (/ - ) cos n(0 - 0) P 1/2 (cosh o)Qn 1/2 (cosh n) r > (2.9) e and e are the Neumann numbers given by n mr 1 n0= n 2 n 0

15 We can now find a function GDE GDE 7-= +DE 4 7 RgDE where GDE = 0I -r7s and gDE is such that V2 E = 0 1 DE gDE 4 47rR r= res r7=r7s gDE regular at oo. in the sense of Kellogg (1929), i.e., r r gDE <oo and r >oco "DE lim 2 a r - < oo r->oo ar DE (r = 7s represents the surface of the torus). Matching boundary conditions we get i/oh 1- c- - --- 0 0 1 m >-m -) _cosh - cos 'j4oshno- cos7 6(1) m rn 2 gD = 4=a 0E m ' rEJ(-m +-2 DE 2. -n m- m + 4 r a.. nOm=0 F(n+m+ PI cosh r )Q I (cosh r ) cos m(i-i/ ) cos n(0-0 ) n- 1/2 h o n - cl/2 ( (shr 7s >) oPom (cosh ) n- 1/2 s The Green's function for the Dirichlet problem is then given by: cosh rl - cos 0 cosh r7 - cos 0 GE =+ 2 e En(-l) 4 7r a n = m=0 (2. 10) 1 r(n-m+ 2) 1 cos m(O - )cos n( -0 ) r n / n - )\ ns) 5 - - m no D Itrtonh n 2 - L n -1/2 - "s n-12 (coshr o) Pn /2 (coshr7) (2.11) nol o2 an - 1/rsi With the help of this Green's function, one can solve any Dirichlet potential problem (i.e., whenever the potential value is specified on the surface) by using the Green's function as the kernel of an integral representation.

16 z T = n y D FIG. 2-3: CROSS-SECTION OF A TORUS.

17 2.4 EXTERIOR NEUMANN BOUNDARY VALUE PROBLEM Physical problems which involve such boundary conditions frequently occur in studying fluid dynamics, magnetostatics and some steady - state heat conduction problems, However, as stated in the introduction only special case can be solved explicitly as follows and Green's function can be written up to a set of constants and in the following we treat the two cases when the torus is moving parallel to the fluid and when the flow is perpendicular to the axis of the torus. The former has been studied by Hicks (1881), but a summary of it is included here. 2.4. 1 Torus Immersed in an Infinite Incompressible Fluid Flowing Parallel to the Axis of the Torus with a Uniform Velocity v It is quite clear that there is complete symmetry in the angular variable b and hence the velocity potential V is independent of Qb. We seek a solution of the problem 2 V2V = 0 (2.12) avotal av an v na n ne a n 21 177 77=r rvrirs77= 7 and V is regular at infinity in the sense of Kellogg. The incident velocity potential V can be obtained from the velocity vector v A i asinO v = vz V ovz+ c=v - -r --- + c z v z+c=v cosh r7 - cos 0+ c where c is an integration constant. Expanding V in an orthogonal expansion

18 vi= v jf'a coh-os 0 7r OD In sinnO Q,(cosh ti) n0 n- 1/2 (2.14) The solution of (2. 12) can be written as or) V = osr-oO [ mcos mq/, + B sin mO& [Ccos nO + Dnsin no] m0o P (cosh ri) n - 1/2 (2.15) in the exterior of the torus (cf. Eq. (2. 7)) Because the problem is independent of ip (2. 15) can be written as V = \cs 7Cs & 4CncosnO+ D sin no] Pn112 (cosh YI). (2.16) Applying the boundary condition (2. 13), we get n= sinh 17 r ICcosnO+ D nsinnO P n,12(cosh ri ) 2 cOsh17 -CosO n n19 + + Cosh17- 9Is I(C cos nO +D sinnO) P' (cosh ri ) n0 -n n n.-1/2 v Vf" a 2 cosh17 - cos O Z n Q11 (cosh, 17) sin n - vVJF8a 7r Go I Os 6 n Q I (cosh n ) sin n O n -1/2 s n - (2. 17) The primes denote differentiationwith respect to 17. Since the right hand side is only a. function of sin n 0, a~ll C Is must equal zero. Dividing throughout n

19 (2.17) by F- and rearranging 7r rjsinh B P 1/2 (coshr ) - nQn 1/2(cosh rls + 2(coshri -cosO) I-= n s- 1/2 ~s [B Pn/2 (coshn) - nQ' 12(cosh r)] sinnO = 0 (2.18) n n _ 1/2 s n - 1/2 where D 7r n B = - n vF a Rearranging further, B P' + B P' - B[inh71P + 2 cosh P 1/2] (2.19) The arguments of the Legendre functions are omitted for convenience, but they must be understood to be cosh r. But + P-3/2 inhn P + 2cosh r Pt2 = (2. 20) n+1/2 + s n-1/2 sn-/a and similarly for the Q-functions, Introducing this simplification in (2.19), we obtain (B -B)Pt -(B -B )Pt =Q -t Q:n+ n n+1/2 n n -l n-3/2 =Qn+ /2 n3/ (2.21) true for n > 1 c,. with '2 1) 3/2 1 1/2 3 /2 1/2.2 for n = 1

20 If we write the successive equations in order and multiply the equations containing P and P by P and add, we get ln - 3/2 and + 1/2 n- 1/2 + r [.112 r~+ 12 r -112 r + 1/2Y But (2.23) (2. 24) Pt r Q' pt r-1/2 r+ /2 r- 1/2 r+1/2 2r+l 2 Using (2. 24) in (2. 23) gives [B n+l-B n] Pt Pt' - = 1Pt ' L n+l nj n+ 1/2 - 1/2 1 1/2 -1/2 n-1/2 n+/2 -/ -1/2 Q+ 1/2 Summing the series in (2.25) and rearranging, we get 2r+1 2 (2.25) QI B -B +1/2 n+l n P'1/ n+ 1/2 + -1/2 [1 + 1/2Q+ 12] P' p't n+ 1/2 n- 1/2 1 2 2 n Pt P Pt n+1/2 n-1/2 Let (2.26) (2. 26a) 2 Pt' [B P - Q' 2] then (2. 26) can be written as nn B -B n+/2 n+1 n Pt n+'1/2 2 + n +a 2n+ 1 L n+1/2 n- 1/2 (2.27) Rearranging we obtain B1 B 1 [(n+ 1) + -(n+ ) n-1/2 n+.1/2. -1/2 (2. 28)

21 writing n more equations in succession, and adding, B -B (n+)2+a Q n+1/2 2+ Q1/ n+1/2 ' r r- 1/2 n+1 1r 2n+ 1 P' - 2 pt 3 Pt n+1/2 r=2 4r-1 r-1 /2 (2.29) Bn is now determined up to the extent of (a or B1) and to determine o (2. 26a), follow the argument due to Hicks (1881). Since we know that the velocity potential must be finite everywhere, we can choose an a such oo 0 this series to be convergent is that the nth term of this series goes to zero as n goes to co. So weproceed as follows: (i) It must be proved that B is finite when n is large. (ii) a must be chosen such that B vanishes when n is infinite, by making the limit of B go to zero. From (2. 29) we can see that for large n, the term that should be considered is oo 2 Q1 r + a i r-1/2 r=2 4r2 1 r-1/2 and for large n (Hobson, 1955) m Imr r)1/2 e-n r (n+m+ 1) Qn-1/2 n 2s 1/2 (n+ 1) (2.30) m1 en r (n+1) n-1/2 (n7r)1/2 (2sinh 2 r(n-m+l) hence, for large n Qn- 1/2 -2 nr Pt n-1/2

22 which is highly convergent. So, r + -1/2 ) r2Q ^Qr-1/2 2 P' r=l 4r -1 r-1/2 is also convergent. Thus B lends to a finite limit for increasing n n co 2 Q Lim 2 r +a -1/2 Q-l/2 B 2 Pa (2.31) n —o n co P(. P1) nr= n r 4r1 r-1/2 -1/2 This limit must be set equal to zero, when we get a vlue for a co 2 Q' - 2, r r 1/2 -2 2 P't r1 4r -1 r-1/2 (2. 32) 1 r - 1/2 'l/2 29 Pt Pt | rl 4r 1 r-1/2 -1/2 and hence 2 co 2 B 2 n~ +a r-1/2 (2.33) n 2n-1 2 2 P' r n 4r -1 r-1/2 and the velocity potential is given by V = 1cosh r - cos e B sinn e P (cosh r7) r Bn n-1/2 where B is given by (2.33). " This value for a is unique as it is establised by Hicks., by verifying the co convergence of the series 2 B sinneP 1/(cosh r), with this value. It must o n also be mentioned that such a method of determining the coefficients has not been necessary in any other diffraction problems in the author's knowledge and seems to be rather peculiar, being necessary for this case.

23 2.4.2 Torus Immersed in a Fluid Flowing Perpendicular to the Axis of Torus with a Uniform Velocity v The boundary value problem in this case is very similar to the previous case, except that we no longer have symmetry in i. We seek a solution for the velocity potential V, V V = 0 av _ av1 an an V regular at infinity in the sense of Kellogg. The velocity potential of the uniform flow is =i sinh ri sin Q V = vy = va -c cosh r - cos 0 -va c r - sin cos n0 Q (cosh r) (2.34) Employing the boundary condition, we obtain cosh - os IAm osmo+B sinmi C ncosn0+D sinn } - 1/2 ( osha c - os 0 sin cos n 0 Q 1 (cosh r) }. (2.35) n=0 /n=0 This yields A = 0 for all m m 1 for m - 1 m 0 form 1

24 Rearranging (2. 35), we obtain ~ s) 2 coshr ] ]CcosnO- =00 C p /2cosne - inh= n- 1/2 s n- /2 n n= n+ n /2 co -P /2 C p ' cos n n=0 n-i n-3/2 va2*V' (1) +2 osh "d 008 Qi -'1/2 Q0i - n- 3/2 } - n 1 3/2 cs n (2 36) The arguments of the Legendre functions are understood tobe cosh rs and are omitted here for convenience. Equation (2. 36), in turn gives rise to a set of recurrence relations for the coefficients C n (1)) (1)' sinhrn- 1/2 + 2coshs P C -C p (1)' - C [inh sQn - 1/2 - cosh7 1/2n - 3/2 (2.37) for n = 2, 3. 4... with the initial equations, C(1) sinhs 1p (1 (1)' Co 0sinh P -1/2 + 2 coshr P 1/2 C P /2 s 1/2 va 2 F2 sinh rl Q (1 + 2 coshq Q l1) 1 } -1/2 -Q12 1/2 (2.38)

25 and -2C P + Co sinr P +' 1) (1)' PCP} -2 C P-1/2+ cl h 1/2+ 2 cosh 1/2 } C2 3/2 va2'V (1)? (1), -71/2 - 1/22 C '}T {2Q -1/2 + sinhr-a Q12+2cosh Q1/2 - Q 3/2 (2.39) C is determined to the extent of C. (We cannot employ the method of the previous section to determine C ). But it has not been found possible to express C explicitly in terms of CO or to get a sufficiently simple expression to test for convergence. Hence a different approach has to be taken. Although we do not have a symmetry in the angular variable, the symmetry about the plane z = 0 brings a simplification. Since the flow of fluid is parallel to the y - axis, the flow has 'stagnation points' at the points A,B,C, D shown in Fig. 2-3. These four points correspond to rn = Y, 0 =, = 7; 7 = s, 0 = r, / =r;; r, 0=ir, =/ 0; and =r7, 0 =0, =0O respectively. At S s these points the total velocity should be zero. This can be seen as follows. We know that the normal component of total velocity is zero everwhere on the surface. Due to the symmetry about z = 0 plane, the tangential velocity components must be zero at these points or this would give rise to circulation. There is a cancellation of the tangential components at these points thus making the total velocity zero. This can be written as, i v + V = 0 V 7r 1 s 0 ' O 0 i.e., V(V+V ) 0 07 7S nrns, o= 0 ' =0

26 We obtain the two equations (coshri + 1)3/2 oo 1) sinhr7 C( 1) P (coshr ) + sinh s s a nz n n-1/2 s 0.0 - 1/2 (oshr) = and (cosh Y - 1)3/2 (o s 0 7 C p(Cn P/2 ( cosh rs) + sinh (s a nqa/n t (coshn - i)32 OD a+ explicit --- vaa2l jQ C '. (coshr) ) = 0. (2.41) sinhr? a 0r Q n-1/2 s n These two equations wfil be consist maetIf it is possi ble to find a set of Cvs that satisfy both the equations. This set of C' s is then given by n v2.-(1) (coshr s) - vs (2.42) n=_ n=0 and C 2n+ 1 P2n+ 1/2 (coshls)= - -- - n12' ( < 243) n 0 n=0 Equations (2.42) and (2.43) seem to be the additional equations necessary to determine all the coefficients C. Although this still does not provide us with an explicit value for C, Eqs. (2.42) and (2.43) implicitly give the information needed. These conditions come from the symmetry of the problem involved and are connected with the circulation, and hence serve as additional conditions to the boundary conditions.

27 2.5 GREEN'S FUNCTION We can follow the method of the previous section to write the Green's function as follows NE 47rR NE where 2 V gNE agNE an = 0 a 1 an 4TrR Ti = Ti5 r7 =- 7 gNE is regular at infinity. Using (2.9) and (2. 7) we get 00 gNE = ccoshr - cos cosh -cos 0 n=0 m=0 em cos m (/ - o ) m 0 C cosnO + D sinn 0 P (coshr ) n n n- 1/2 where Cm satisfy the recursion relation n Cm sinhrs-P (cosh ) + 2 coshPs 1/2 (cosh)s)I -m m' n+ 1 n+ 1/2 s m nm' - C1 P (cosh 00 ) n-1 n-3/2 5 2 r(n-m+1/2) rf, F (n+ m+ 1/2) [(sinh sQn 1/2(oshn) + 2 coshQn 12 (cosh s) cos n 0 o Qn+ 12 (coshrTs)cos (n+ 1) 0 - n-1/2(cosh ) cos(n- 1)] n+for n 2, 3, for n = 2, 3,...

28 C0 Jsinh r P I (coshrK)+ 2 coshr nP M (cosh ri )-C MP m (coshri ) 0 12 -1/2 sj 1 1/2 s ={inhfljQrnl/(cosh ris + 2 cosh n Qrn (ch ) Q Mt?;(cosh 7s Coseo and C1n snh P7 m (coshi ) + 2coshrl ml (cosh i7 C Pml' (cosh ri m1 +1/2 12 2 3/2 - 2cmm F(-+m)nIn r Q (coshrj)+2coshn o -1/2 s 3 12.p1/2 c s>} - CosO0 - 2Q ml(cosh )Q ml(coshr-a ) cos 2e] -1/2 s 3/2 0 Similarly D Is satisfy the recurrence relations n D sinh P 'M 1/2 +2cohrPM cosh). DI ml(cosh rl n. s jr - /2\r) 5P (c 12 n+ 1 n+ 1/2 - D MP M (coshr ) 2F(n - m.+12) I (sinh rl Q (coshq + n-i n -3/2 s F(n +m.+ 1/2), s n -1/2 + 2 coshri5 Q ml (cosh rj i~ sinn60 n -i/2 s J -Q ml (cosh~ )7sin (n+-1)O0 n+-1/2 s o for n =2,~3, 4. with the initial equation

29 D ( sinhrs P2 (coshrj ) + 2cosh r P2 (c(cosh - Ds S 1/2 2 S/ S D9 1n-m (oshr ) i i= snhr P 2(coshrl )+ 2 coshr P (coshr ' (.-+m) [ 1/2 s s 1/2 s mn' sin - Q (coshr ) sin 20 0 3/2 8 0 These equations, as before, determine C ' and Dn' up to Co and m D At this point, it has not been possible to obtain C and Dm 1 0 1 for all values of m, although for m = 0 and m = 1, the constants have been determined laboriously in the previous sections. For m = 0, D1 is given by Eq. (2. 32) explicitly where -a 2 p D1 /2 1/2 and Co for m = 0 can be determined in a similar way. For m = 1, C ) is obtained from Eqs. (2. 38) through (2.43), and analogously D(1 can be obtained. In spite of many efforts it has not yet been successful to determine the coefficients C and D for all values of m. o o

III QUASI-STATIC APPROXIMATION FOR LOW FREQUENCY SCATTERING The basic assumption underlying this method is that when an acoustic or electromagnetic wave is Incident on a scattering surface, the scattered field can be expanded in a convergent series in powers of k, the wave number. The problem of determining the coefficients is then reduced to asuccession of standard' potential problems. The problem can be formulated using a method due to Kleinman (1966): 3. 1 THE SCALAR PROBLEM Let a small amplitude sound-wave propagating with a constant velocity ve be incident on the body, whose boundary is denoted by B. (The density and compressibility of the external medium is denoted by pe and m respectively.) e e The resulting disturbance, as a function of position and time, can be calculated in terms of a velocity potential I. Figure 3-1 shows the arrangement. We are concerned only with the exterior problem and the velocity potential ext satisfies the equation V2 1 0 (3.1) ext v 2 a t2 e m e v = - is the velocity of propagation. Assuming t has a harmonic e p ext time dependence e,, the Eq. (3.1) becomes, e 2 Such an expansion has been proved to exist (see Werner, 1962, Kleinman, 1965) for sufficiently small k. 30

31 z / inc e m k e e e rB B Y X A x '7 FIG. 3-1: GEOMETRY FOR SCATTERING BY AN ARBITRARY BODY.

32 The acoustically soft and hard boundaries, are generally represented by the boundary conditions =0 ext - =0 an r =rB respectively. ext represents the total field in the exterior, i.e., t= e + scat ext ext e e Hence if we consider a plane wave incident on the body, our problem becomes one of finding a s, such that e 2 2 scat (V + k) = (3.2) e e where k = e v e Lir r e -ik scat) r-> co a r - ekr The latter is the Sommerfeld radiation condition implying that the outgoing ikr waves look like f(O, 0) for large r. The subscripts and superr scripts on s and k will be dropped henceforth, because we are e e only concerned with the exterior and we can refer to the above quantities as ^ and k, without any cause for confusion. As usual, one begins to solve the problem with the Helmholtz integral representation, (Baker and Copson, 1950), which expresses the scalar solution A. See Appendix A for a discussion of Sommerfeld's radiation condition. r. 0, 0 here represent spherical coordinates.

33 of the Helmholtz equation in terms of its values and those of its normal derivatives on a closed surface. ' ikR ikR a (r) = - eiB dB (3. 3) 4 7R R a-n a The integration is carried over the surface of the scatterer and - refers to n V, and dB refers to a surface element of area. We now introduce the expansion for the incident and scattered fields, viz., co (r) = ) (F) (ik)m m= m (3.4) inc,' inc tc (r) = m (r) (ik) m=O The factor i is included in the expansion parameter for the scattered field ikR because it appears in the expansion for the incident field. e is an entire function which can be expanded as follows, ikR = (i (3. 5) I =0 Substituting (3.4) and (3. 5) in (3.3), we get I 1 m MY a(ik)m _ 1 B 1 m() (ik)m B( an (ni __4 m 4ra M=O m=O I =0 (ik)R - 1 a an m() (ik) (3. 6) Interchanging the order of summation and integration and rearranging terms, we get See Appendix B for the derivation and the physical significance of the Helmholtz formula.

34 i'm 1 f~0 O (ik) Ba f-m-1 -m-1 a I a (Y)(ik)m= - R R rn~i( )( 47r6' - n)4 an an m m=O 10 = m=a (3.7) Matching coefficients of k on both sides of the equation, we obtain 4anan m 1 m~0 1 I d. 3 i-m-1 -m-1 Q () -= 4 ( -m), ( am n R -R a m S =0. 1. 2.. (3.8) We can now introduce the boundary conditions ( B) =- nc (rB) (3.9) inc (3.10) ann - r=rB r=rB inc is a known quantity and is given by (3.4). Hence. 1 dB a R m-l1 I-m-1 _m} m dB R n R 4w U( - m+-lnR an = = 0, 1, 2... (3.11) Let us first treat the term i = 0 in Eq. (3.9). Considering the case of a soft boundary, we substitute (3. 9) in (3. 8) to get 1 inc a 1 1 1 3o dB (312) o (7) = 4 ~ dB, o an R d47 R an (3.12) B 4z B R an The first term on the right hand side is known and the second term represents an exterior potential function (for e.g. see Kellogg. 1929). In other words, if we let

35 1. 1 _ _ u (i) = — 1 - dB o 47r B R an then U (r) is a solution of v2 u (F) = O o and U (r) is regular in the sense of Kellogg. viz.. Lim a Lim 2 o rU < oo and r r | < oo r - oo o r-> oo r the boundary condition for U (7) is specified by ~ - inc Lim inc U r= - Y(rB) a 4 ~ n dB. (3.13) -o oB r —nB B RB Equations (3. 9) to (3.13) constitute a standard Dirichlet potential problem with a unique solution. Therefore U (r) is determined completely in terms of the incident field. (The integral on the right hand side must be evaluated before the limit is taken.) Hence o(Cr) is known. The succeeding terms ( e(r) can be determined as follows: Let us assume all the terms 0 f1' 1 2... up to and including l(r) are known. If we write (3.8) as (r) = UI(r) + Fe (r) (3.14) where, F 1 1 G inc ( — m-1) d F( ' 47r = (-m)'. B m nR dB 7 z, ( i f -m-1 m an d. -R - dB (3.15) 1 ( - m) I n m=0B

36 and U = dB (3.16) F (r) is a known quantity since all the I's up to and not including 4 are known. i x is known and so is R. U (r) is again a single layer distribution which obeys the Laplace's equation and the regularity conditions in the sense of Kellogg, and whose boundary value is given by. inc UI(-B) = - () - F B B Thus the determination of U is reduced to a standard potential problem and UI is known in terms of incident field and the previous term. Then A is given by Eq. (3.14). An analogous procedure holds good for the Neumann boundary condition (the hard boundary). Substituting the boundary condition (3. 10) in (3. 18), we get (f 4e 1 -) 1 d {Ra -m-1 a inc 3 a -m-1. __ man an m +m Ra (3.17) Following the same procedure as in the previous case. we can write the velocity potential At as follows: W (F) = G (r) + V1(7) (3.18) where B -1 1I( a -m-l ainc] 7r - m -1]n (3.19) n m= B

37 and V (r) = r JB (3.20) If we assume all the ( 's are known up to and not including ~, then G (r) is a known function. V I(M) is a double layer distribution, which implies V (r) is the solution of a standard exterior Neumann potential problem, satisfying the equation V2V (r) = 0 V (F) is regular in the sense of Kellogg. namely, |rV| < o r ra-r < oo r —> oo irV Io r -> r and whose boundary value is given by av2 a,, inc aG n n n - - r r r =r r=r B B B Thus V (T) is uniquely determined and hence 2 (). This method permits the evaluation of any number of terms in the low frequency expansion of the scattered field in terms of a potential solution. But the explicit determinations of the terms grow immensely in complexity and limits the ease of calculation to bodies with very simple geometry. 3.2 FAR-ZONE FIELD The far-zone scattered field, i.e. when kr >> ka, provides interesting information in practical applications. In particular, while studying the scattering cross section of objects one is primarily concerned with the far field scattered. Generally, the far field is written as ikr = - - f(0, 0) R

38 where f (0, 0) is called the far field amplitude. One can calculate readily the far scattered field when all the terms in the expansion for the scattered field are known, But when only a finite number of terms are available one has to proceed taking the far field approximation at the initial stages. r. r ikr - ik - ikr - ik r r ikR ^.. r.... B e e e R r r I (3.21) A _ A ikR ikr - ikr B ikR B e e V -R- - ikr Substituting these in Eq. (2. 6) Substituting these in Eq. (2. 6), we get (3.22) ikR ( (r) ^-; dB t lk n^ r e r B -i k r ik^ r i -e an J (3.23) But 00 (r) = I ~m (F)(ik)m m=0 -lk^. rB e OD I M=O m A m (-ik) (r rB) m! Using these, Eq. (3.23) becomes ik r O O n k (r - n ) I n k nm+ (-r) =r ('f n..(ik) a +(ik) Rs we A d =er r B =a tn ts. n+m+ Rearranging the terms, we have ikr 2o n-m i (r)e (-1) ^A -AA 4 7rr (ik (-m' Y (rB;rnr~ n _ 0{ anm }dB (3.25)

39 where _- 2= 0 Equation (2. 34) indicates that the knowledge of a finite number of terms in the near field will determine the same number of terms in the far field. 3.3 THE VECTOR PROBLEM A plane electromagnetic wave is assumed to be incident on a perfectly -inc -inc conducting body. E and H represent the incident fields vectors. We seek a solution of the Maxwell's equations V x E = ik I V x H = -ikE V E = 0 v- H = o (3.26) in the region exterior to the scatterer and subject to the boundary conditions A - f xE B - -H _ n'H r =r B ^ -inc = - nxE As -inc -- - r_ r r =rB (3.27) and the Sommerfeld radiation condition limr, r x (VxE) + ikrE = 0 r --- oo lirm ir Fr x (VxH) + ikrH = 0 r- -> co The procedure for reducing the solution of (3.26) to a set of potential problems is analogous to that of the scalar problem. The starting point is the vector analogue of the Helmholtz representation viz., the Stratton-Chu formula (Stratton, 1941), relating the field at an exterior point to their values

40 on the surface. ei kR i IkR ikR E(r)= Vx -- nx E dB+ --- nxH dB —V -— n E dB 4 B B R 4 (3.29) ik ikR A I kR ^ -- ik e ^ - - A 1 e A -- n —xHdB — 4 R n H dB B H.) 47r J R 4r J R n d 47r JB R (3.30) where the quantities R, n, B have the same meaning as in the previous section. V operates on r and the corresponding operator on rB will be denoted by VB in the future. i ikR - c - inc Expanding e, E, H, E and H in power series, we get eiR O (ik) R-1 R =0 E = E (r) (ik) =O0 0 =0, 2... (3.31) co inc = Einc ik E TE (ik). =0dary conditions can be rewritten as, and similarly for H. Now the boundary conditions can be rewritten as, A nxE r = r B n H r rB A -inc = -nxE r r =B H inc = -n H rB r = rB In addition to this. Bn - E dB = 0 B and nH dB = B

41 Incorporating all these in Eqs. (3. 29) and (3. 30). we can write 1 E (r-) = F () - -— V4 It 4 7r I I(n) = G(r) +4 Vx a- EI R dB nx HI R dB Rd ]iAnc rm-1 dB _nx 1 R dB 1 1-m (3.32) (3.33) where F (9) = - V 4w mI0 m=O m1 = m=O 1 f I. x m m' -m-1 B m.-. R- dB 1 ~ -4 m-l 1 m' SB ERm 1 dB B (3.34) 1 1 4 m m0o il m=l 1 s!B 1 A -inc!t, nxE m. n -1 AxH, R- dB -m Rm-1 dB m-1 n Rinc m-ldB I - m 1 + r = m=O m! V B (3.35) It must be noted that m-1 and, are identically zero for = 0. m=O F (r) and G () are expressed only in terms of known quantities, in view of the fact that we know EO, E.. E 1 and H, Hi. H -

42 The unknown term in the right hand side of (3, 32) is the gradient of a function which we know to be an exterior potential function. Let A - 1 e X Ui - J n - - dB (3.36) then V2U = 0 Ul N regular at infinity in the sense of Kellog nx VU = -nx Ei + F = r=rB r r=rB which can be solved in a conventional way. Hence El can be determined. But the determination of the HI is a considerably more difficult taskbecause it is not clear that the unknown term in the right hand side of (3. 33) is the gradient of a potential function. In order to make it possible to solve for this function in terms of a potential function, we introduce a function g1 (r) such that 1 nxH 4 V x I R dB + g(F) = VV (3.37) 47 R' To solve for g (r) which satisfies (3.37), we proceed as follows: A 1 V nXH 4 V R dB + Vxg(rF) = 0 (3.38) 471r R Using the vector identity Vx(Vx A) = V(V- A)-V A and 1 1 V- -v R B R (3. 38) can be rearranged to give

43 Vx g,(F) = 4 v xH dB 4r R B x id (3.39) = -1- V R dB 4r B R I > 0 Q = 0 = 0 (3.40) Stevenson (1953) has developed a method for finding particular solutions of equations of the type V x F = f when f is a gradient of a scalar potential function. We have from (3.40) Vxg VUe where - 1 4 1-1 n -ER- 1 R dB (3.41) Stevenson, at this point, introduces an interior potential function U (r) defined when 7 is interior to B, (This function is defined purely for analytical convenience and does not have a physical significance in this problem) such that V 2 () = 0 r interior to B. A i n VU (F) r rB A e = n vu7 (r) r rB (3.42) This can be solved for in a conventional way. have It is also necessary to n VU dB = 0 B

44 But this, clearly, is true. The particular solution of gI (i) then is g() = 4 V x V — () dV + dv (3 43) which can be rearranged to give e (B! () - x R VdB (3.43) B Now we can determine VI (i) by solving a potential problem because 1 VrfnxHl 47 R dB + = VV and V V 0, with the value for V~ on the surface given by n VV ^ -nx V~( n- UG) | from this we get H. Although the above method is a systematic way of obtaining terms of all orders, certain simplification result when one solves for the first term. From Maxwell's equations using the expansions (3. 31), it is easy to see Vx E - H n and VxH = -E p >0 p p-l p p - 1 In particular for p = 1 V x E = H V xH = -E 1 o 1 o But H and E are known to be gradients of scalar functions and hence we o o have new equations of the type V x E = VU V x H - VV 1 0 1 o J6

45 and particular solutions of these equations can be obtained by using (3.43) which is a solution of equation of the type V x F = VU. Since arbitrary gradient functions can be added to these solutions. they are required to satisfy the boundary conditions,viz.. A inc A inc x E1I -n x E and nx H -n x H r = r r = rB r = r B r B B rrB B In this chapter the theory of approximating a diffraction problem, at low frequencies, in terms of a series of potential problems was discussed. No assumption as to the shape of the diffracting body was made. In the next chapter we apply this method to the problem of diffraction by a torus.

IV ACOUSTIC SCATTERING BY A TORUS In this chapter we shall apply the method described in Chapter III to the scattering of a plane acoustic wave incident normally on a torus. 4. 1 PLANE WAVE INCIDENT NORMALLY ON A RIGID TORUS The incident wave is propagating down the negative z - axis, and taken to be of unit amplitude. inc o inc k -ik z (-z _ (ik) x =O and hence inc (-z) we now proceed to obtain the first three terms, directly from the Eqs. (3.17) to (3.19) of the previous section. 4.1.1 Zero'th Order Term -, (r) = G (r) + V (r) (4.2) o 0 0 G(F) R an (4. 3) o 4r B R n o inc Ainc But =1 and =. Therefore G (F) 0. 0 on V (F) = \ o Ln 4r J odBnR is a single layer distribution and a solution of Laplace's equation, given by 46

47 00 V (F) = cosh r - cos n=O I A cos m + B sinm m= m C cosnO+D sin 0] P m (cosh r) n n n - 1/2 (4.4) A B. C and D are constants to be deterrined by the boundary condim m n nI1 tions. V (F) is regular in the sense of Kellogg can be easily verified Since rcosh r + cos 0 1/2 r - a L --- — r a cosh r77 - cos 1 (cosh r - cos 0)1/ a 1 (coshr -cos )1/2 a 8r a 1/2 sinh r7 cosO 8 -- - a 1/2 cosh r7 sin 0 - Su(cosh rt+ cos 0) (cosh 7+ cos 0) g Substituting these along with (4.4) gives r --- oo G o lim 2 ao r —> o. 1~ ar lim IrV < oo r -> 0 | r1 —> [r | O lim 2 avo 0<co o (4.5) Boundary condition on V (r) is o av 0 a n r1=n.:77 a inc -O an r= s aG o an 17=fl s (4.6) But inc 1 o1 O and hence inc -- an This gives V 0. o and also G = 0 o

48 Therefore. the first term'in the expansion for the scattered field o = 0 o 4.1.2 First Order Term This term 1 is given by l(r) - Gl(r) + V (r) where Gf= La inc 1 r a inc Gl( 4 r an o 4dB+ R an 1 B B But a inc an 0 an and 1 4r Gli(): 447r" 1 a inc R an dB (4.8) From (4.1) we have inc a sin e 1 = - =- cosh r - cos and inc an cosh r1 - cos 0 a sinh r sin 0 coshr - cos 0 a inc a-r 77 The surface element of area d B is given by

49 a sinh n dB = 2 —. dO doB (cosh r - cos O0B B s B and - is given by (2. 9). Substituting for all these quantities in (4. 8) and R integrating over PB gives a value of 2 7r for the case m = 0 and vanishes for all values m 0. o2 (r) - Vcosh -cosO n 1/2 (cosh(oshrsinh s 2 ir sin cosn (0- OB ) S-BB dO B (4.9) 0 o (cosh 7 -cos ) / s B But sinh0 40 (coshn -cos 2 ihs pp /2 s (4.10) (the prime on Q 1/2 refers to a differentiation with respect to n7). Finally using (4. 10) n (4. 9) and Integrating, yields G (r)= a - cosh s -cos 0 sinhr 1 nP /2(coshr)Q 2(cosh ri 1 n r so= n n-i / n-o/2 s Qn l (coshri) sin n (4.11) n-1/2 s V (F)), again. Is given by V (r) = Icosh - cos 0 (AcosmI+B slnm C cosnO+D sinn cos0) 1 I A-nI m m n n m=0 nO PQn (cosh n) (4.12) n-1/2

50 with the condition av1 1a a G1 (F) ~r =s ooahn o ~ se = -a oshn-cose 2nsinn9Q' i/g(coshr) W (4.14) Examining (4.11) and (4.14~ we see clearly that the expressions are independent of ~ and therefore the only non zero coefficient of cos mi is for mO. i.e., A == 1 and all the B ' s are identically zero. By inspectionwe can also see that all the C s are zero, since Gi(r) and V1(r) n nan are only series of sin ne and not cosn0. Hence, sinc a i sin - 7 Dn c shn O jSinneQP -2(cosh s ) + - - n- 1/ 2vcoshn -cos 0 = n (4.14) Examining (4. 11) and (4. 141 we see clearly that the expressions are independent of / and therefore the only non zero coefficient of cos mo is for m =0, i.e., A = 1 and all the B? s axe identically zero. By ino m spection we can also bee that all the C s axe zero, since G (F) and V 1M) are only series of sin n and not cos n n. Hence sinhr s co 7 n sinnO 0 a Q 1/2 (ccsh s) + 2ycosh s - co s O' = Q I (cosho 7r si n^-l/( s 1/2(coshsQ'n-l(^ ) + cososs D si (osh cos 0 (oshr) nI Q n - 1/2 s an1I sinhr n1 n na2f' Z~nsinnO [-1/2 (CO(h r) - +Vcoshr-Cose19 nsinnO Q i2 (coshrs - - n- h -lPhn-P (cosh s)Q' (cosh- )Q- (coshri 7r sn- 1/2 s n(4.15) (4.15)

51 After simplification we obtain (the arguments of the Legendre functions are omitted,, but should be understood to be coshri ) S D1 [ / - p/ 2 - 2] 2 3/2 r 3/2 -1/2 1 sinhnLs l/2 1/2/2 3-/2 1/2 2 -2P1/2Q3/2Q2 32 for n 1 (4.16) n+l/2 [n+1 n-3/2'n n - 37r n+ 12 Q-32] 3+ a4'1"nh [Pn+1/2Qn-1/2Qn-l/2 n-3/2Qn-1/2Qn-2] 3 (n+ 1) sinh n+2n+1/2 n+ 1/2 - a4 (-1) sinh P -3 for n>1 (4.17) 3 (- )stS n - 3/Qn - 3/2Qn - 3/2 (4 17) This can be summed to become: D -D=. r(2 (n2+ 3) 1) P+ 12 P } 7r +n+ n - 1/2 n+ 1/2 a+ a4 W sinh 1 s {nQ(n+ 1)2Qn Q + a 4 sinhr p1/2P /21/2Q1/2 (4.18) 3r P' P' n+ 1/2 n-1/2 where '= 2 [D1 1/2 Q1/2] /2 (4.19

52 Writing successive equations from n = n, n - 1. 1 and adding together, we get, using (4.19) 2 a4n 2 a22'V2 (n+1)2+3 Qn+1/2 + 2 r3 +f 1-21/2 n + 1 - 2n++ P2 P' 2P t-f 3 sinh 1/2 1/2 n+ 1/2Qn+ 1/2 1/2 12 2 1 Pr+1/2 -1/2 4.20) In order to find 1, (i.e.. D1) we employ the same method as in (2.41) due to Hicks. i.e., we find the limit of D as n - oo and set the limit equal to n zero in order to have Vl(r) finite everywhere. The series co 2 r +f r-1/2, 2 pt (4.21) r=l 4r -1 r-1/2 converges and the series oo P, 1 Pt Pt r=l1 r+ 1/2 r-1/2 also tends to a finite limit. Hence f3 is given by - 2sinh riQ1/9Q^ + -1/2 2 (4.22) 1/9Q~,V9 (. 2 13

53 and D is then n r 2n+1 P' - r r2 r-1/2 u a2~ vn2+f 9n-1/2_ a4V210 r r - 1/2 n 7r 2n+l P' A7 2 P' n- 1/2 r=n 4r-1 r-1/2 +4 P P P' Q IQ nQ + - - (4 23) - 1/2Qn- 1/2 P(4 23) n r+ 1/2 r- 1/2 Finally, we obtain from 1(r) = G(r) + Vl(r) 00 (nh n21c P I(coshn )Q (os l(r) Oa 2 osh -cosh6s nE P (cosh e) s (COs ps) x 1 3nr= nn -1/2 n -1/2 s x Qt 1/(coshs ) sinn n n-1/2 + /cosh r1 - cos 0 Dn sin n 0 Pn- 1/2 (coshr) (4.24) n=0 where D is given by (4.23) and (4. 22). 4.1. 3 Second Order Terms We shall proceed in an exactly similar manner as we did in Section 4.1. 2 to determine Q2' the third term coefficient in the expansion of the scattered field. (2(r) G2(f) + V2(r) 2 2 2

54 1 1 a R dB+ 1 1 8' - [o-n RdB 4r2!Jo an Since a r inc an 1I dB+ 1 1 inc dB dB+~R'~(~ dB B (4. 25) = 0 and inc 0o o = 1 we get G2(+) = 4 J-a 1c dB + I 1 4 7r R a Iinc d an x2 dB (4. 26) inc 2 2 2 2 z2 a sin2 0 2 (cosh r - cos 0)2 an an sin 0 sinhr7 =a 2 (coshri - cos 0) a inc ' 3n sin 6 sinh rn cosh n - cos 0 1 -R is given by (2.9). Substituting these in (4.26) we get 2 G ) a osh r - cos 2() = 27r Go n=0E nOn pn-1/2 (coshrl) Qn- 1/2(coshrs) sinh rs S cos n 0 x (n - 5/2) - (o/2 ( ) + 2 r(n+ 3/2) (3) (cosk r(n -9/2) n- n-5/2 + 11/) (3). (coshr ) +T (n - 1/2) 'n+ 3/2 s (4.27)

55 Also inc an 2zr ycos hri - cos 6' sinh 2'O O p(n+ 7/2) (3) - Q (cosh r) 2(o(3) csh) P(n+1V2) (3) r(n+ L2) +_____ (cst)rn1 + 2Q (cosh~i r1 n32 n+3/2 r(n 1/2) n-5/2 r(n-9/2) (4. 28) V (2) once again is a soluflon of Laplace's equation which satisfies the 2 regtilarity conditions and the boundary conditions av (i) 2 an 7= P inc 2 an ti = rlT aG2 (j) an rl:- 7 (4. 29) Let us abbreviate a Vi2' L F'(n+ 72) (3) (3) 2 hon5))+2/2(coshr?)+2Qn+312(cosh r ) r(n+ 12) f'(n- 12) (3) (n_+ 3/2) + 2Q (coshi 7 F(n+3/2) n -5/2 5 r'I"(n-9/2)J -K n (4.30) Equation (4. 29) takes the form cosh i -cos 7 Y7(A cos mp+B sin m)(CCcos nO+D sinn 6) s M0 n m i n n ml p m (cosh r7 ) n-1/2 sr + sinhrls 2 cosh i -cos&19 5 f'Z (Acosmsm+B sinnmV(C cosnO+D sinnO9) M —O0n= O pm (cosh n ) n -1/2 5

56 o00 cosh rl -cos e E K p (cosh 1)Q- (cosh) cos n e s nn n-1/2 s sinh r1 oo cs EK-P 1/2(coshrs)Q 1/2(coshrs)cos n0. 2~cosh rs-cos n n - 1/2Osh - 1/2 (4.31) We shall omit the details of getting the coefficients since the arguments and procedures are identical to the previous case. After rearrangement and simplification we get C *C + PFC Pt -, PQ Kl Pt 1P Cn+ 1 o+ P-1/2[ -3/2 1/Il 1 P 1 P2:- r + 1/2 r- 1/2 n r + p - p, (r-s+l)K P'- P ( /2 r r+ 1/2 r- 1/2 s1 s1/2 s+ 1/2s + 1/2 s + with C given by / 1 1 r= r+ 1/2 r-1/2 'P' P' Z2'r P K - (r-s+ 1) P' P r 1 r+ 1/2 r-1/2 -1/2 1/2 s + 1/2 ss-1/2sl/2s+1/2 +1o -13 s- -1/ 2 s s p r (4. 33) r r -'P 1 12P,1112Q? 1K 1 +- _ P Q'K- 0. =ss-1/2 S-.3/2s-s3/2sl-1 (4.33)

57 Thus (2(r) is given by )=osh - co P (cosh r)Q /(cosh r )K cos n0 n0 n n - 1/2 n- /2 s n + V0 C cnP (cosh)-co 0 C cosn 0 n n-1/2 n=0 (4.34) where K and C are given by (4.30) and (4. 32) respectively, 4.2 PLANE WAVE INCIDENT NORMALLY ON A SOFT TORUS We start with Eq. (3. 14) and (3.15) to obtain the zeroeth and the first order terms in the expansion of the scattered field when a plane acoustic wave is incident normally on a soft torus. 4.2. 1 Zero'th Order Term o(r) = Fo(r) + U(r) where (4.35) (4.36) F (F) = - S o 47r iB an () 4 7r B R OB 1 -dB R a o dB an and Uo(r) is such that V U (r) = 0 U (r) regular at infinity 0 (4.37) U rl = = - F (r) n-=r inc ( - o ( -) 27=r) (4.38)

58 Solving (4. 37) we get for U0F (. and F0 (7) U (1) =]cosh ri - cos 0 (Acosmo/+BmsinmqI)(CncosnO~Dnsinn6) 0 n = 0n=O m0mn Pm (coshriq) and n - 1/2 Fo()= 0 Also inc 0 - 1 Matching boundary conditions., loshn i-cos 01 IAmcos mt'+ B sinm Lcosnn+Dsinnj (cosh r ) = - 1 n -1/2 s and 1 giving thereby A =-1 m= 0 In 7r 00 X7cosnOQ n-112 (cosh ri ) n-0 =0 m ~0 _2Qn -12(cosli -a) B - Ofor all mand C __ n1 m n wr Pn (cosh rl) U0(i) = _ 'V(coshri - cos O' Qn-1i2 c sI fl (cosh ri) cos no o~n0 nrl/2(cosh is) n-1/ Hence

59 Substituting for this in (4. 35), the first term is found to be Q() = 2(coshr - cos 0)' n=0 n=O Q n (cosh r ) cos n 0 & ---L4L — h p n- (cosh ri) (cosh ri) n-1/2 (n-1/2 ) (4.39) 4.2.2 First Order Terms (F) F ( f) + U (r) 1 1 1 (4.40) where F () = - 1 1 I,inc 8 1 47 J Bo an R dB - a d "4~1^ " (4..41) U () = - |dB. 1 4r J R an The incident part is inc a sin 0 1 = coshr- - cos 0 This may also be expanded as follows: (4.42) inc and nc 0' = 1 1 7r n= 1 Q 1/2(cosh r) n sin n 0 Substituting these in (4.41) one obtains for F (F) F) = 3 acosh - cos nP -12(csh r)sinh rsQ 1/2(cosh ) 1 3n/r n-12 Q=1 Qn I/2(cosh s) sin n n- 1i/2 s a -, = n 0 Qn 1/2(coshns) n- /2 s (4.43)

60 a U (r) is once again a solution of V U (F) = 0, whose boundary conditions are Up1) (F) F(F) rlU ): (r) i - F1(r) r The quantity co Q (cosh ) a n Pn- 1/2(cosh 7s) is actually the measure of the capacity of the torus, since the capacity KT of the torus with respect to infinity is defined by KT V S dB T v an where V is the constant potential on the torus and E is the free space permittivity. Substituting for these values, we obtain for the capacity KT, the value 8.a- 1/2(cosh n ) K 3 8ac c... T =8 e n P /2(coshr ) n=0.-1/2 s It ts interesting to compare this with the capacity of a sphere whose radius is. a and is carrying a total charge distributed uniformly, K 47r a Sph Then the ratio is given by Qn -/2(cosh s) 2 n -1/2s081 70 En P (cosh rl) n=0 n-1/2 s and is plotted in Fig. 4-1 as a function of sechrl 0

... 61 2. 6 -2.4 -2. 2 2.0 -1.8 -1.6 -1.4 -1.2 -1.0 0.1.2.3.4.5.6 r 1 R0 cosh FIG. 4-1: CAPACITY OF A TORUS / CAPACITY OF A SPHERE

62 Matching boundary conditions oshrs cos (A cos os m/+B sin mi/)(C cosn + D sinn0) s = m m n n m P 1(coshr ) n- 1/2 ~s 7rKT - 8 4 a C os ccos - cos x 3 ir s 00 x nP (cosh s)sinh s 7 Q n (cosh ns)Q /(cosh )sin n n=l + 2 a 7r G0 cosh r -cos0 Q n/2(coshrs )n sin n. s n= n-1/2 s (4.44) Solving for the coefficients Am, B, C and D we get IIm m n n A = 1 m = 0 for m= 0 B = 0 for m = 0, 1. 2. m m /0 KT C = n 4 f2 Qn - 12(cosh -as) 1/2(cosh n ) n = 0 1.2... D = n n 2 Qn- 1/2(ccsh ns) r Pn 1/2(cosh r/s) - 3 a sinh r Qn/2 (cosh r ) n sn-1/2 s Q /2(cosh Us) } n = 0,1,2...

63 Thus the solution for U (r) is given by, ___ -K T Q-V/(coshr )7 U (F) =oshrn-cos (coshn) C)osO + + a(n QJi nV12co s) - 1 asinh rnQ' 2/(cosh r)Q 2(cosh r) V Q -P 1/2(cosh s) 3, 3n/2o 1s2-1o ln-1/2(o s s) sinn 0 - 1/2 (cosh r) (4.45) n -11/2 s n) =osh-K Q n-1/2(cshs)) + P112(c —hr) sinne P,(coshr,) - 8 (4.46) 7r P -1/2(coshs) } n-1/2 8o We can continue to obtain higher order terms this way, but the complexity in the functions Involved not only make it formidable, but also it becomes more difficult to derive meaningful results from them. 4.3 FAR-ZONE FIELD The expressions obtained for the scattered fields for the scattering of an acoustic wave by a hard and a soft torus are in a rather complicated form but can be readily utilized for computer calculations. Since it is beyond the scope of this work to do numerical computations we shall look at the analytical expressions and tryto compare this vwith some known results say, for a sphere and for a disc. In order to do this we shall look at the far-zone field for a particular case, viz., the soft torus. The soft torus is chosen because the results in this case are relatively simpler to deal with.

64 Equation (3. 25) can be directly used to obtain the far zone field i.e.. ikr oDI) n n ( i )n-m Qi) 7r r (kr) -n rB - am dB n=r m=_ B when 1 = 0 and all the quantities have the same meaning as in Chapter III. Examining the term n = 0 ikr d ()~^- - _ dB 4 7r r B an From (4.39), we know j2(cosh -Qsc- osr sco, h^ co n -0 1/2(coshr ) n-/ ~>o E /2(cosh t) Pn_- /2(osh 77)cos n O Therefore, ikr O Q 2(coshs) ikr K - a / - 1 I e T er e- - - - e T (4 47) r; ' 0 n p 1(coshs ) r 8 r n- 1/2 s where 2 2 2 a =R - r o o A plot of the quantity KT - 1/2(cosh r) o T pn- P /2(cosh7s) against sech ri is shown in Fig. 4-1. S This quantity is already shown to correspond to the capacity of a torus. So, the first terms in the far scattered field for a soft torus, in terms of the first terms in the near scattered field is always a measure of the capacity of the body under consideration. It is interesting to compare this termwith that of a disc and a sphere (Senior, 1960). ikr 2 e 2 bdisc(r) r f ad ikr e p (re) r a sphere r s

65 where ad and a are the radii of the disc and the sphere respectively. Figure 4-2 enables us to estimate the equivalent sizes of the discs and spheres which would give size to the same far zone scattered field. Figure radius o 4-2 is a plot of against To illustrate this. let us take a R R O O disc whose radius is. 6 R thus the size of the torus which gives the same far r scattered field has - =.125. Similarly for a sphere whose radius is O r.6 R the size of the torus which gives the same return is R = 0.5. But o R r o for large values of -, the relative sizes do not make much difference, o as is expected. Unfortunately, no results are available for scattering of an acoustic wave by a torus. But it is expected that at a future time numerical computations done with these analytical expressions will give systematic information (for both the soft and the rigid torus) for scattering as a function of r and/or R.

1.2 - Disc Oa --- 10 - Sphere radius R.6-.4 - -.1.1 I.2.3.4.5.6 r /R FIG. 4-2: EQUIVALENT RADIT FOR SPHERE AND DISC IN TERMS OF THE RADIUS OF TORUS.

V ELECTROMAGNETIC SCATTERING FROM A TORUS Let us consider now the problem of scattering of a linearly polarized plane electromagnetic wave by a perfectly conducting torus. The incident wave is propagating down the z-ais ith the electric field polarized parallel to x-axis and the magnetic field parallel to the y-axis. - itr;i H, IZ T- IS X FIG. 5-1: ELECTROMAGNETIC WAVE INCI TORUS. inc ^ -ikz I (ik inc inc X =i e = E; E (i'';E~ E DENT ON A (-z) I' x (5.1) ~ =U -inc A ^-ikz (ikI -nc H = -i e = (i y =0 inc (-z) A i y.9 II! (5.2) We shall follow the method in Chapter III to derive the zeroeth and first order terms in the expansion for the scattered field in powers of k. 5.1 ZEROtTH ORDER TERMS -inc A = A l-cososcosh r sinhrlsin A E x ch - os-i cos - i cosh cscos- i sin i o x ri cosh ri- cos 0 0 coshr17 -cos 0 r (5.3) 67

68 The scattered electric field to this order is E = VV o o where V is an external harmonic function to be determined under the con0 ditions 2 V = 0 (5.4) A -inc x VV = -nx (5.5) 0 0 r=nr-s rln-n av V regular at infinity and dS =0 o Jan Solution of (5.4) is given by (2.6) and (5.5) can then be written as oshr7 -cos0 S S -i l a sinO e (A cos m+B sinm) (C cosn0+D sinne) 2 ca Z cm m n n Pn-1/2(cosh -0) [cosh - cos 0 3/2 + s- 2 (A cosm+B sinm)(-nC sinn0+nD cos n) a.- I m I n n P 1/2(coshr)] 3/2 r(cosh r -cos )/2 + - a sinh /( —mA sin m + mB cos m ) (C cos nO+ D sin n0) 0 a sinnh ) m m n n P 1/2(cosh sinh ts sin O = cosh r-cosO cosV - ie sin (5.6) which gives A = -1 for m 1 m = 0 for m 1

69 a 2 = ' -1 -2(coshr) P' /2(cosh ns) n T1/2 D = 0 for all n n Thus -' Q 1/2(coshs) V= - cosh - cos o P-/ cscos n P ( (COh) '-cos q (cossh n- P 12(cosh n) (5.7) and E = VV gives the first electric field term. O O To determine the first term in the series for the scattered magnetic field, we write H = VU o o The excitation is given by H = - VUin 0 0 inc A 1 - cosh rl cos 0 A sinh rsin 0 A -VU =- y i -- in-' cosh - cos - s in +i cosW o y y cosh? -cos 0 cosh7-cosO sn i 2 (5.8) U satisfies V U 0 0 0 A V A 7-inc n VU = -n ' 0 0 Y] = 7 7 s and the radiation condition. From the boundary conditions we obtain sinh YAs oo oo 2 ' L '(A cos mO+B sin m) (C cos n0+D sin n 0) P (cosh ) 2a n Om= m m n n n-1/2 s (cosh r -cos 0) co oD + Y Y?(A cosmV+B sinmi/)(C cosn0+D sinn0) a m m n n n-Om 1 - cosh r1 cos 0- 1/2 cosh1 - cosO sni. (5.9) S

70 Expanding the right hand side of (5. 9) in a series over toroidal functions. we solve for the coefficients, A = 0 for all m m 1 for m= 1 B m 0 for m 0 D - 0 for all n n and C is given by (1) (1)(1)1 C sinh (1 P () (cosh )+ 2coshrn P (cosh1 ) -C P cosh ) s s n-1/2 s n+ 1 n+1/2 s - C P ( 32(cosh s) =2 { sinhr1 Q 1 2(cosh )+2coshr Q '2(cosh n-1 n-3/2 s K s n-1/2 s s n-1/2 s (5.10) 2 -2 Q(1)' 2.~. a (1), r n+1/2 n-3/23,4 with the initial equations Co [sinhrsp 1 (/2 + 2 coshs P (1/ ) 2- 1 P1 ) 2 -1/2 L — 1 1s- /2 and c [inh s 1/2 + 2cosh lsP1/2 2 -2)1/2 - c23/2 = (11) () (1), 1 (1) = ra (sinhQ /2 cosh /2) + 2Q 1/2 Q3/2 (5.12) Now C is determined up to C. To determine C, we proceed in a way analogous to the fluid flow problem in section 2.4, since the flow of a fluid around a rigid torus is analogous to the magnetostatic problem for a perfectly conducting body.

71 By virtue of this, the magnetic field at the points A, B, C, D, shown in Fig. 2-3, must be zero, which gives rise to a condition on the C's viz. C P (cosh = - — a Qn- 12(cosh). (5.13) Thus CO can be determined from Eq. (5.13) in addition to Eqs. (5. 10) through (5.12). Therefore, U and hence H is completely known. 5.2 FIRST ORDER TERMS The next higher order terms in k can be solved for by using the equations Vx E = H and Vx H -E p >0 (5.14) p p-1 p p-1 thus V x E = H and V x H - E and H and E are both 1 o 1 o o o gradients of scalar functions and therefore Eq. ( 3.. 43) can be utilized in solving for E1 and H1 directly. V x E1 = H = V ycoshr-cos0 C sin /cosn0P /(cosh r L n=0 nnWhere C is given by (5. 10) through (5.13). Using (ue - i n V x J- dB where U and U) refer to the exterior and interior potential functions the same as in Section (3.3).

72 Omitting all the cumbersome details, we obtain the quantities E1I and H 1 given by A. ~(l) (I osr 2) 1 r2'aQ (1) (oh El Iof,n =0 [n n-i1/2 c sh rl) r n -1/2 (c shJlS r'(n- /)Qi (coh1 i s)h En P (n+ 3/2) Qn -1/2 (c shrsf n h rs+ + I~ [(sinhnr- cosh ri )l -(sinhrn -coshrl) ]+ sin 0bsin ne9P112(coshri7) A + 1 coh cs 6 I (1) 2~ (1)1 7r [C P 11(coshr77) -_ 21Q (Coshrn n 0n -V s 7r p-i/2 s EF-(n - 1'2)p ~(i) (coshr0 Q (1) (cosh 7 )cos q/'cos n6 (5.15) and similarly VxHi =E = + [ co v[ cohcs&Z 1 0 Qf (cosh r 2fa n-i1/2 5 7r Pt (coshrn ) n -1/2 5 and using (3.43) cosocosn9 Pn i/2 (CoshriO [ 2~a + Al Q? (cosh i Lr nj n-1/2 s - HI I i ohjcsO n ~n r (n~+3/2) n-1/2 5 ++ [(sinhri7-coshri7) li-ni _(sinhri7-coshri7) 1 n] { -n sinhrqs cos~cosn& P ()(cosh n) n- 1/2 & I Q I +i caosh 7- cosO0 7r14fLl +AJ Q1/ (cosh r (5. 16) E F' (n-i 12) n I"(n +3/2) P (co sh) sinV sin n6 P~ (cosh r) n-1/2 5cs -)n-1/2

73 where A is given by: = L[12 s n12s n - 1/2 + n1 n+1/2 n+ n+1/2 a2 '-2 2 n- 1/2 (1)' + sinh nl2 p (1) n -/1/2 /.-1/2 -/ - 7 p -/ (coshr ) - P M2 f(cosh) ) n+ 1/2 n-/2 3 n-3/2 This can be solved, as usual, up to A, and A remains to be determined. In order to obtain this, the static Neumann Green's function must be obtained complete with all the coefficients. (5. 15) and (5.16) represent general solutions of E and H that satsify Eq.(5. 14). Now any arbitrary gradient of a scalar (say VS1 and VS2) can be added to (5.15) and (5.16) and still have the-: satisfy (5. 14). These arbitrary functions then should be fixed by the boundary values, which E1 and H1 should satisfy viz., n x QE- +E 1 )| = ~ (5.17) ^s n x (EI1 =0 (5.18) 2 2 inc a (1- Cos 9 cosh r)in asin sinin asinh sin, E =- z = - i +i cosh-cos) 1 x r (cosh rl - cos0) 2 (cosh r?-cos 0) (5.19) inc= = - asin 0(1 -cos coshtrl) i assinh rsin ^ asncosV 1 y (o z- 2 - 2 a(coshn -cos 1 Y (cosh r- - cos 0)2 (cosh -cos)2 ) cos (5.20) If we let r (1) 2 2 a,(1) ] (n-12) (1) = 7rC P1/2(cosh l) (cosh (n+32 (1 ) n [ n - 1/2 r n- 1/2(cosh - 1/2 s) (5.21)

74 n = sinhr1 + - (sinhri - cosh7 )( - I (sinhis - coshn ) ] s s 4s s s OA (1) (5 22) E =i an n] sini/ sinl Pn-1/2 (cosh1i) = n n- 1/2 c + a a cosocosnOP (coshns)+1 VS +1 V S +1 V S -r0 (5.23) where A A V 1 = i V S+i V S + i V s 1 77 7 0 1 V 0 1 is the arbitrary gradient of a scalar that is added to E1. Hence A + inc n x (E + ) = 0 r s V S = ~ 171 asinh r sin 0 cos _ c S = 2 - cosh s-cos a cosscosnO (cosh r -cos 0) n p (cosh s) n - 1/2 s a sinh rl sin / sin 0 o VS -- (coshr cos 0 1 a (13 )sinO' slnn0 0 1 (cosh -cos 0) - h 0 n P (coshn ) (5.24) n - 1/2 s Similarly for H we can add VS2, and solve for it using ^ - -inc n (H+ 1) 0 r =r's we get 2 a sin 0 (1 - cosh n cos 0) V S -.........- (5.; 25) (cosh r1- cos 0) VS2 = 0 02 V S = 0 2

75 This technique of obtaining higher order terms in a low frequency expansion can be continued, but at this point it is considered to be not so important to get the higher order terms. but rather to derive some meaning from these complicated expressions. The only result known for the scatterin of an electromagnetic wave by a perfectly conducting torus is that due to Weston (1956). He has derived the far -zone scattered field for a very thin ring. In order to compare our result with his, we first have to derive the far-zone field for our case. We use a form for this field, given by Kleinman (1966): ikr co n -scat e 2 n ) A n-m 4,- r r x (ic) -- (-r- r B E 47rr nO r (n- m) JB B n=O M- B A A A B A A -B A B -r x r C 11 r - n xH + r x r E xL E + r enx E B Yo Po m B o m o 0 m + /I r n Hr dB. (5.26) 0o o oB m This expression is a power series expansion for the far field in powers of o(i. e., k) in terms of the near field terms. This also gives n far fieldterms when n near field terms are known. If we are interested only in the first term (i.e., n = 0) -scat e 2 A A A A B A A B bct r x i;4-rxr ltA JIA r nxH +rxri E c nE + 47rr rB Jo o o o Bo o o A A B A- ^ B. + r i nxEO + lo - -rn Ho dB (5.2 7 00 o oIoo B of Since (5. 27) involves E and H and in as one E is known explicitly, but H is known only implicitly, we cannot yet obtain a suitable far-zone scattered field to compare with Weston. It is expected that some numerical work will be done to be able to compare with experimental results for a ring.

76 The low frequency expansion terms is carried out here for the case of the electric field polarized in the plane of incidence, but the other polarization can be used without any difficulty. The scattering cross-section is also a quantity of interest which can be obtained from the far field. The scattering cross-section is defined by lirn 2 s ca 47rr - r ->-oo W o when W is the scattered power density and W, the incident power density at the scatterer. at the scatterer.

VI SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK The main results of this work contain the derivation of the solution of the potential equation for a torus placed in a uniform field in directions parallel to and perpendicular to the axis of the torus. From this one can solve the corresponding problem for a torus placed in a uniform field in an arbitrary direction. Further these results have been used to construct the solutions of scattering of acoustic and electromagnetic waves using the techniques proposed by Kleinman (1965) and Stevenson (1953). The solutions are in a power series, in ascending powers of k, the wave number. This series is often called the Rayleigh series and is valid for small k or low frequencies. Two non-zero terms are derived in the scattered field expansion for the acoustic problem for rigid and soft toroids and for the electromagnetic problem for a perfectly conducting torus. The process could be carried to higher order, but the complexity in the forms of the terms increase progressively thus making it more difficult to derive meaningful results from them. The far field is calculated for the simpler case of the acoustic problem i.e.. for an acoustic wave incident normally on a soft torus, and compared with the known results for the corresponding problem of a disc and a sphere. The equivalent radii for the latter are plotted in terms of the radius of the torus, to obtain the same scattered field, No attempt has been made to do any numerical computation, but the formulae can readily be used to get results from an electronic digital computer. Also, no attempt has been made to estimate the radius of convergence of these solutions. * The exterior Neumann problem for a point source excitation can only be solved yet up to a set of constants, which still need to be determined. The The series for the scattered field converges for I|k sufficiently small, that is, there exists some number kol > 0, such that the series converges for Ikl < Ikol. Ikol in turn determines the radius of convergence. This has been estimated only for very special surfaces so far (e.g., Senior and Darling, 1965). 77

78 procedure of determining the scattered fields will apply directly to all R - separable bodies (for definition see Chapter I), for e.g., flat ring, washer, special bowl, ogive etc. In contrast to a torus which is a smooth body. these bodies have edges., and it is expected that this might help in solving for the unknown set of constants, because in addition to boundary and radiation conditions the scattered fields should also satisfy edge condition. Once the Green's function of the second kind is completely explicitly found, one could use a method to find the scattered fields by an iteration technique due to Kleinman; however, it is hard to predict whether the method will render itself convenient at this stage, since when applied to ogive (Ar, 1966) the results could only be found in terms of integrals which could not be evaluated easily. Torus is typical of a class of R-separable bodies and any work done on this will certainly lead a way to solve the problems involving other R-separable bodies.

BIBLIOGRAPHY Ar. Ergun, (1966), "Low Frequency Scattering from an Ogive", The University of Michigan, Radiation Laboratory Report No. 7030-3-T, November 1965 (being published in Quart. Appl. Math.). Ar, Ergun, and R. E Kleinman (1966), "The Neumann Problem for the Helmholtz Equation", being published in the Arch. Rat. Mech. and Anal. Baker, B. B., and E. T. Copson (1950), The Mathematical Theory of the Huygen's Principle, Oxford, Clarendon Press, Second Edition. Bassett, A.B. (1889), Treatise on Hydrodynamics, Deighton, Bell and Company, Cambridge, England. Bassett, A.B. (1893), "On Toroidal Functions", Amer. J. Math., XV, pp. 287-302. Blank, A.A., K. O. Freidrichs and H. Grad (1957), "Notes on MagnetoHydronamics, V, Theory of Maxwell's Equations without Displacement Current", New York University, Report No. NYU - 6486 - V. Bond, S (1955), "The Current Distribution on a Toridal Antenna", M.A. Sc., Thesis, University of Toronto. Carter, G.W., and S.C Loh, (1958), "The Approximate Calculation of the Electric Field Between a Rod and a Concentric Ring by Means of Toroidal Functions", J. Inst. Elect. Engrs., Part IV, pp. 13-17. Dyson, F.W. (1893), "The Potential of an Anchor Ring", Phil. Trans. Roy. Soc., London, 184A, pp. 43-95. Erdleyi, Magnus, Oberhettinger and Tricomi, (1953), Higher Transcendental Functions, I, Bateman Manuscript Project, Mc-Graw Hill, New York. Grobner, W. and N Hofreiter (1957), Integraltafel, Tell I unbestimmte Integrale, Tell II bestimmte Integrale, Wien: Springer-verlag. Heine, E. (1881), Anwendungen der Kugel Functionen, 2, (Reimer, Berlin). Hicks, W. M. (1881), "On Toroidal Functions", Phil. Trans. Roy. Soc., London Part III, 31, pp. 609-652. Hicks, W. M. (1884), "On the Steady Motion and Small Vibrations of a Hollow Vortex", Phil. Trans. Roy. Soc., 35, pp. 161-195. Hobson, E W. (1955), The Theory of Spherical and Ellipsoidal Harmonics Chelsea Publishing Company, New York. Kellogg, O. D. (1929), Foundations of Potential Theory, Springer- Verlag, Berlin. Kleinman, R E. (1965), "The Dirichlet Problem for the Helmholtz Equation", Arch. Rat. Mech. Anal., 18, No. 3, pp. 205-229. 79

80 BIBLIOGRAPHY (continued) Kleinman, R.E. (1965), "Low Frequency Solution of Three-Dimensional Scattering Problems", The University of Michigan, Radiation Laboratory, Report No. 7133-4-T. Kleinman, R. E. (1966), "Low Frequency Methods in Classical Scattering Theory", Laboratory of Electromagnetic Theory, Report No. NB18, The Technical University of Denmark, Lyngby. Lamb, H. (1932), Hydrodynamics, Dover Publications, New York. Loh, S.H. (1959), "On Toroidal Functions", Can. J. Phys., 37, pp. 619-634. Loh, S.C. (1959), "The Calculation of the Electric Potential and the Capacity of a Tore by Means of Toroidal Functions", Can. J. Phys., 37, pp. 698-702. Loh, S. C. (1961), "Uncharged Conducting Toroidal Ring in a Uniform Electric Field", Can. J. Phys., 39, 1961. Magnus, W. and F. Oberhettinger (1949), Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea Publishing Company, New York. Moon, P.H. and D.E. Spencer (1960), Field Theory for Engineers, Van Nostrand Company, New York. Moon, P.H. and D.E. Spencer (1961), Field Theory Handbook, SpringerVerlag, Berlin. Morse, P. M. and L. Feshbach (1952), Methods of Theoretical Physics, Part II, McGraw-Hill Company, New York. Noble, B. (1962), "Integral Equation Perturbation Methods in Low-Frequency Diffraction", Electromagnetic Waves, edited by R. Langer The University of Wisconsin Press, Madison. Neumann, C. (1864), "Allgemeine Losung des Problems iiber den stationaren Temperaturzustand eines homogen en Korpers welcher von irgend zwei nicht conzentrischen Kugelflachen begrenzt wird: Schmidt, Halle, 1864. LordRayleigh (1897),"On the Incidence of Aerial and Electric Waves upon Small Obstacles in the Form of Ellipsoids or Elliptic Cylinders and on the Passage of Electric Waves Through a Circular Aparture in a Conducting Screen", Phil. Mag., XLIV pp. 28-52. Senior, T. B.A. (1960), "Scalar Diffraction by a Prolate Spheroid at Low Frequencies", Can.J. Phys., 38, pp. 1632-1641. Senior, T B.A. and D A. Darling (1965), "Low Frequency Expansions for Scattering by Separable and Non-Separable Bodies" J. Acoust. Soc. Amer., 37, No. 2, pp. 228-234.

81 BIBLIOGRAPHY (continued) Snow, C. (1952),"Hypergeometric and Legendre Functions with Applications to IntegralEquations of Potential Theory,' Appl. Math., NBS, Series 19. Stevenson, A. F. (1953), "Solution of Electromagnetic Scattering Problems as Power Series in the Ratio Dimension of Scatterer/ wavelength", J. Appl. Phys., 24, No. 9, pp. 1134-1142. Stevenson, A. F. (1954), "Notes on the Existance and Determination of a Vector Potential", Quart. Appl. Math., V, No. 2. pp. 194-197. Van Bladel, J. (1964), Electromagnetic Fields, McGraw-Hill Company, New York. Werner, P. (1962), "Radwertproblems der Mathematischen Akustic", Arch. Rat. Mech. Anal., 10, pp. 29-66. Werner, P. (1963), "Beugungs problems der Mathematischen Akustik", Arch. Rat. Mech. Anal., 12, pp. 155-184. Werner, P. (1966), "On the Behavior of Stationary Electromagnetic Wave Fields for Small Frequencies", J. Math. Anal. Appl., 15, p. 447-496. Werner, P. (1966), "On an Integral Equation in Electromagnetic Diffraction Theory", J. Math. Anal. Appl., 14, p. 445-462. Weston, V.H. (1956), "Solutions of Toroidal Wave Equations and Their. Applications", Ph.D. Thesis, University of Toronto.

APPENDIX A Sommerfeld, In 1912, was the first to discuss the conditions. viz., R (| < k R - co (A. 1) (R aR -ik) > 0 R —>o. (A. 2) Equations (A. 1) is called the "condition of finiteness" (Endlich keitsbedlngung) while (A. 2) is the important "t radiation condition" (Ausstrahlungsbedlngung). From the mathemnatical point of view, these equations are Important because this enables one to find a unique solution to the Helmholtz equation I. e.. a solution of a Helmholtz equation which Itself or whose normal derivative takes on prescribed value on a surface and which satisfies (A. 1) and (A. 2), is necessarily the only solution. These conditions specify the behavior at infinity of the wave function and in particular the condition of finiteness states that - 0 as R -> co and the radiation condition Implies that $ must behave like outgoing spherical wave at infinity and the sources written cannot give rise to incoming waves at infinity. All physical fields generated by finite source distributions must satisfy these conditions. 82

APPENDIX B HELMHOLTZ FORMULA Helmholtz formula expresses a scalar solution of the reduced wave equation 2 2 (V + k )u = 0 (B.1) au in terms of the boundary values of u or In order to. derive this an formula, let us suppose u is a solution of (B. 1) in a domain V bounded by a closed surface S. and internally by S1, and u together with its first and second derivatives is continuous in V. Let v be another function defined throughout in V with the same continuity properties as u. Now we can use Green's identity, which states 2 2 av au JV S+S an n2 1 O qS FIG. B-l: REGION OF APPLICATION OF GREEN'S THEOREM a denotes a differentiation along an normal drawn outward from V. an Let v satisfy (B. 1) except at a point P within V, where it is singular. If we choose v, in particular, to be spherically symmetric, given by ikR(P, P ) 0 1 e v = - R(P,P) (B.3) where P and P are both inside V, then v corresponds to spherical waves emanating from P0 since (B.2) is not valid unless P is excluded from V, 83

84 and let us define the newvolume by Vt which represents the volume bounded externally by S and internally by S and S, which is the surface of a small 1 o sphere bounding P. The Green's identity becomes fS+S +S o 1 u av u = (u- -v — ) dS = 0 an n (B.4) S is a sphere of radius p and -- = -. Also o an ap d2 Is the solid angle subtended by dS at P. Also an = e- (ik - p )/p and is of the order 2 2 P But dS is of the order p as p -> 0 and the integral 0 dS =p d 2 where as p -> 0. S 0 u k-v dS is independent or p. Hence p can be made very small i.e., we can choose p -> 0. But SI 0 v -S dS = 0(p) an 0 as p -> 0 and constitutes nothing. However, lim P -- 0 S 0 a v u (po) u dp = - an o 47r L dQ = -u(P ) J o (B. 5) and so Eq. (B. 4) gives (P) = (u an van ( S O (B. 6)

85 We can drop the subscripts on P and we get ikR(P, P ) ikR(P, P) au s s au(P ) dS e e s u[ -u(P ) s dS s 47rR(P, P) 47rR(P.P ) an S+ S S s where P is a point on the bounding surface. Thus Eq. (B. 6) express u at P in V in terms of u and a- on the bounding surfaces of v. A an similar formula could be easily derived if there are source distributions within a small volume. Equation (B. 6) is termed the Helmholtz formula. In many problems, we are interested in the field outside the surface S which contain the sources. Then there are no sources outside S and we expect the surface integral over S to vanish. In order to verify this let us use the Helmholtz formula on two functions u and v with no singularities outside S, satisfying the wave equation a the same continuity conditions as Sefore. If we suppose S is a large sphere of radius R, centre P and has S interior to S. If ikR e v = 47TR on S, we get u(P) = [ (P P) n u(P) - u(P) dS )s L- U(P.) ( s -n s + S [u(P ) v(P, P) -v(P, P) u(P) dS s -R s s s S then the integral over S takes the form dI I d0s sin OsR u(Ps)Rs -R -ikv -RvR [u.iku 0 0 s s Rs

86 In order to have this integral vanish, we have to impose restrictions dictated byphysical considerations and hence u must process the property Rlim IrR bounded R -> OD I r [R - iku1 - - in order to be a physically meaningful solution if there are no sources at infinity. (That these conditions are 'stronger' than necessary has been shown by some authors who have termed the first condition superfluous). Thus, u(P) J (u a v a ) dS an an P outside S 1

AP PENDIX C Some useful relations involving the toroidal harmonics: -m 2- -2prlm -(n +12)7 n -1/2 (oh)=F(M+1)(l - m+ n, I.+ m; 2m+ 1l- e-27 m (-1)m 2m r(n+ m+1~2)i' m -(n+ m+ 1/2) 7 Qn -12(coah ri)rl1)= (sinh) me x 1 1 -2 rl x F ('+ m. n+m+ —;n+ 1;e ) 2 1 2, 2 pm Z n - 1/2 PA (Z) -v -1 =( 1)11(z2 - O/ (z2_ 1) m/2 dm dz m d-n (Z) dzm n -1/2 Qn - 1/2 (Z) = P1,(z ) V p-/(z) = P(P-.+Al) P (z)-2 -/je sin/jiri Qm(zJ W(P,Q)P= P — (cosh )Q m dP (cosh r)= (-1)r F(n+ m+ 1/2) F(n -m+ 1/2) sinh rl 87