THE UNIVERSITY OF MICHIGAN 7030-3-T LOW FREQUENCY SCATTERING FROM AN OGIVE by Ergun Ar November 1965 Contract AF 04(694)-683 Prepared for BALLISTIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE NORTON AIR FORCE BASE, CALIFORNIA

THE UNIVERSITY OF MICHIGAN 7030-3-T TABLE OF CONTENTS I. INTRODUCTION 1 II. THE STATIC GREEN'S FUNCTION OF THE FIRST KIND FOR THE OGIVE 2 2.1 The Separability of the Laplace Equation V 2 = 0 in Bispherical Coordinates 2 2.2 Definition of Ogive 3 2.3 The Green's Function 3 2.4 A Verification of the Result 13 2.5 The Regularity, the Existence and the Uniqueness 15 2.6 Summary 18 III. THE SERIES FORMS 19 3.1 Series Representation for 1/R (Residue Series) 19 3.2 Series for 1/R (Directly) 21 3.3 The Series Representation for the Green's Function of the First Kind 25 IV. THE GREEN'S FUNCTION OF THE FIRST KIND FOR THE HELMHOLTZ EQUATION 28 4.1 The Objective and the Preliminaries 28 4.2 The Proper Choice of the Eikonal f(a, j3) 30 4.3 Scattering of a Plane Wave by the Ogive (Nose-on Incidence) 34 ACKNOWLEDGMENTS 38 REFERENCES 39 APPENDIX: Orthogonality and the Normalizing Factors for the Set P m 40 sj

THE UNIVERSITY OF MICHIGAN - 7030-3-T I INTRODUCTION By means of a method developed by R. E. Kleinman (1965) of the Radiation Laboratory, it is now possible to solve iteratively the Dirichlet problem for the scalar Helmholtz equation in the regions exterior to a non-separable body imbedded in the Euclidean 3-space provided: (a) k, the complex wave number, is sufficiently small, (b) the solution of the potential Dirichlet problem for the body in question is known. In the present work we consider the low frequency scattering of a plane wave at nose-on incidence from an ogive. In Section II we find the integral representation for the static Dirichlet Green's function for the ogive. In Section II we give the series representation for this function. In Section IV, by means of the above method, we express the iterates for the scattered field explicitly in the form of integrals. In the Appendix, we indicate the orthogonality properites of the eigenfunctions arising in the problem. 1

THE UNIVERSITY OF MICHIGAN 7030-3-T II THE STATIC GREEN'S FUNCTION OF THE FIRST KIND FOR THE OGIVE 2.1 The Separability of the Laplace Equation V2 = 0 in Bispherical Coordinates Limiting ourselves to Euclidean 3-space and orthogonal curvilinear coordinate systems (u, u, u ), we give two definitions. Definition: If the assumption TT i(ui) (2.1) i=1 permites the separation of the equation V i = 0 into three ordinary differential equations, the Laplace equation is said to be simply separable. Definition: If the assumption 3 1 V (ui) (2.2) R= 1 u 2 3 i=1 R(u,u,u ) permits the separation of the equation V i2 = 0 into three ordinary differential equations, and if R t constant, the equation is said to be R-separable. Bispherical Coordinates These orthogonal curvilinear coordinates (a, P, a) are defined by csinhl coshf3- cosa c sina sin y cosh3 - cosa (2. _ csina cos0 coshp -cosa h=h c h csina f3= a cos osh-co p cos -cosa where' i i i ii i i i i i i i2

THE UNIVERSITY OF MICHIGAN 7030-3-T h2 = 2 + 2 "a \aa \9a/\Qa}' and similarly for ha and hp. In these coordinates 2 1 a 1 a / h 2,,7] ~~V 0Q = +-a hsina + 2 (2 4) h9L s0i& sin a The ranges of the variables are -oo < < < oo, 0 a Tr, r 0 4< ( 27r. The necessary and sufficient conditions for the separability of the Laplace BEnd Helmholtz) equations in various coordinate systems are given by Moon and Spencer (1952). Using their criteria for separation, we see that the Laplace equation is not simply separable in bispherical coordinates but is R-separable (whereas the Helmholtz equation is non-separable in either sense). 2.2 Definition of Ogive In the bispherical coordinates the surface a =a (const.) is a surface formed by rotating about the x-axis that part of the circle, in the x-z plane, of radius c coseca1 with center x = 0, z =c c otac1. The surface of revolution is called an ogive (Fig. 1). All the surfaces of constant c go through two points x=: c (y = z = 0): and at these points 3 = + co respectively. The surface a = 0 is the x-axis for x > c plus the sphere at infinity; the surface a = Tr/2 is the sphere of radius c with center at the origin: and the surface a = 7r is the x-axis for x < c. The exterior region we are concerned with is al a )0, oo > >-oo, 27r 3 0 >0. (there is another way of arriving at the bispherical coordinates, starting with the cylinderical coordinates, which will be considered later.) 2.3 The Green's Function The Jacobian of the transformation (2.3) is ----------------------.,,_______ _3

THE UNIVERSITY OF MICHIGAN 7030-3-T z c a positive constant (-c, 0) (c, 0) x f Kccsct a l (, c cot ( ) FIGURE 1 4.

THE UNIVERSITY OF MICHIGAN 7030-3-T a(x y,z) _ c3sina (2.5),Iv,h) - (cosh- cos a)3 Also 6(x-x)6(y-y )6(z-z) = I( 6(a-)6(- o)6(0- ) (2. 6)' Therefore, the equation to be solved is V2Gba';o ) = -4r (cosh 3- cosa)3 )-(- ( ) V G(apo N, 03a0, = f-47r 3 c (a-a )- (-/ o)a(0 - 0 ) c sinea 2.7) (2.7) where V in bispherical coordinates is given by (2.4). We find the necessary substitution for R-separability to be G = ~Vcosho3-cosa g. (2.8) With this substitution (2.7) becomes 2 2 _ _ + 1 a S)(sina g)+ 1 =-4 osh cos 2 sinG a ( a a.2 2 g =4 c sina ~a13p~sm ca 3 6(a - )6(/3- )- 6(0- ). (2.9) Equality is understood in the sense of distributions. It is sufficient for our purposes to interpret the equality (2. 5) (and any equality involving 6-functions as follows: If the equation is multiplied by an arbitrary function 0 E C(-oo, oo) and integrated from -oo to oo with 6(x) (x)dx = O(0) used to evaluate integrals involving 6-functions, then the resulting equality is correct in the ordinary sense. (Similarly, for the cases in more than one variable.) Also we shall assume the other 6-function formalism such as substitution, integration by parts, etc., and for proofs refer to, e.g., Gelfond and Schilow (1962). 5

THE UNIVERSITY OF MICHIGAN 7030-3-T We now require the periodicity of g in 0: 0 27r and expand it in a (uniformly convergent) Fourier series 00 g(a, f.;a,, ) = A gm(a O a 3) eim in 0 < 27r. (2.10) -00 ~~-a~~~oo-in Substituting (2.10) into (2.9), multiplying both sides by e in and integrating over 0 from zero to 27r (and observing the orthogonality of the set {ein ), we see that 1 -imo A - e, m 27r and (2.10) becomes 1 im(0- 0o) g(a, a,;aOa,) = g(a8;a.,,O)e. (2.11) -00 The equation satisfied by gm is 2 2 gm + gm c osa m 2 C~osra M 1 m n Vcos2h-Cosa k - + g -47r 6(a-a )(- ) a2. 2 sina csina o o fap' c sin a (2.12) We now consider the Fourier transform of g * If R is the distance between the field point p = (a, 3, a) and the source point Po = (aoo3 Po,), then 5 df3/R < o0 -OD provided p 4 p, i.e., 1/RE L2(-co, co). Since the Green's function is regular at infinity (which will be shown after the completion of the construction of G), Ge L2(-co, o). Therefore the Fourier integral theorem is valid for G in the variable j3. Or we simply note that G is a distribution and that the Fourier integral theorem is valid for distributions (Lighthill, 1960). 6

THE UNIVERSITY OF MICHIGAN 7030-3-T Ako o iv) E[gl = m(av, v; = gm((a,;acro,)e 3 d13 (-Co< v<oo) -oo and the operational property gm 2 ^/6 E -32 - v gm(ca, v;ao,) Thus taking the Fourier transform of both sides of (2.12) with respect to,3, we obtain iv4 d g d' / 2 e cosh -cos — + -- - -Iv + - + = -47r 6(Q-( ). d s2 4 gm csina o da \ si/ ^(2.13) Let ~ da 1,-,,~ cosa =, in'(2 14) d~ - - sina' then (2.13) becomes d 1- _2)-. + iv- )(iv+ )- gmco 6( -o d5 dg__7 2 2 -~2 j 0 0 (2.15) We now proceed to solve the equation (2.15) L[gj = v with L - d 1-) d+ iv- )(lv+ - ] (2.16) ivP e 1/2 v =-47r (cosh3 -) -6(- ), C 0 0 with proper regularity conditions at infinity. — m —-------- 7 m- _ —-----

THE UNIVERSITY OF MICHIGAN 7030-3-T The two independent solutions of L[g~ = 0 are g = Pm1 () and g =PivP / (-.). Taking m =0 for a moment we see that m1i - /2 Pi (cosa) = 1 - iv, 2 + i 2 222 2 2 4v +1 /2 (4v +1 )(4v+ 32) 4 1+ 2 sin a/2 + 2 2 sin a/2+... 2 2 4 (2.17) is equal to 1 for a=0 (9=1, x 2+y +z2 =O). By changing a to r -a, we have 1 1 2 Piv -/2(- ) = Piv-1/(-cosa) = 2F1( 2 - iv, ( +iv; 1; cos a/2), (2.18) Hence P. i/() is not bounded when a =r( =-1). The Wronskian:'m g'2 g2 gm1 m2gm2 mi We have (Magnus and Oberhettinger, 1949, p. 63) 2U r(v+P1+1 r v+P+2 PV) 0)- PV(9) V(9) r +)r(v1+2) ( - 2.19) pU(_ )= cos(? +/)7r#v( )- *2 sin(v+M)7rQv( ) (2.20) From (2.19) - (2.21) we obtain ]V(-<v(~- JL^(<lJ( = 2 [sin(v+,)r] 22- r,. 2_ 2) _2"? 2' 2; I2 1?01 v~p~~s~(l(2.22) ------------------— 8

THE UNIVERSITY OF MICHIGAN 7030-3-T Using Legendre's duplication formula 2z-1 1 \Fr(2z) = 22z-lr(z)r(z+ ) replacing v by iv- ~ and, by m (an integer), we finally obtain the Wronskian =M d pm d m W[~. gM 2m = din- - m Pi/ - v d- ~ n,-'/ P (-c) 1J m[,] g1, (2.23) -2 L m21S2 where J, the conjunct of g and g2, is given by r -, 2 1 J, g = * in(iv+m- )7r * - (2.24) l1 m T - 2 r(iv+' -m)) 2 The Solution of (2.15): L[g = v. m g vg m du ( g vg du "g =' + g 4 ml- (2.25) gm im2 J g 1 c (u-gm )d gm e e (os U) u)1/ r1v~ S^ ^ osp-u)m (9 46(m,.0 6(u-,)du. (2.26) The first integral vanishes for ~( while the second integral vanishes for ). 0o 0 Hence _~~~~ ~() i~

THE UNIVERSITY OF MICHIGAN 7030-3-T -47re (cosh- )1/ m 0 m O cJfg g Jm 9 /^ m2 l o o 2 e 0V/o )1/2 1 P (2 e ~(cosh3 - )2 r(iv+ -m) iv -/2 iv- /2o' sin(iv+m- )7r r(iv + +m) m_ ()P ( ) m 0 (o 2 2 i gm S iv - 1/2( - i v-oo' -3| _ (2. 2o) =~ _2o - e o shivo- ~o1/2 /(i+ - -m (2.27) A Representation for the Free Space Green's Function Taking the inverse Fourier transform of gC from (2.27) it follows that _gm1(__;_o10, = 2I7r ) 0dv -10 OD W1 f-31 r(iv + - min) -r (cosh3- )1/2 d e 2 c 1 1 sin(iv+ m- -)7r r(iv+ + m) -00 2 2 m m /20' (2.28)'" p - 12 ()p - 12 ) 0 0> Substituting (2.28) into (2.11) and the resulting expression into (2.8) we finally obtain the following integral representation for the inverse of the distance. " The integral is understood in the cense of Cauchy principle value. 10

THE UNIVERSITY OF MICHIGAN 7030-3-T Go 10 0 -O 1 r(iv+ -m) iviv- /2 e 2 sin(iv +m -)7r r(iv+ +2m) p m 1 ( m) iv 2~piv- 2((2.29) The Green's Function with Boundary Condition To obtain the Green's function for the Dirichlet problem, we go back to the equation (2. 15) and require that the condition g (al v;ac, P)= 0 be satisfied on the surface of the ogive a =a (const.). Using the same notation as before, we define two functions 1 m( m) 1 m2 m m in2 2 gm2() with (2.30) ml() i/ ) gm(F) P /(i) * (2.30)-2 01 02 are two linearly independent solutions of Lg =0, since g,g are, and L is a linear operator. Also, 01(.1)=0. First, we observe that ^,(E)lbB g)- y3 5) = gm (ME ( (9)-^ (9)-g (9)i ()19 -g m2 1 m gm2 m2L 1m _ (2.31) 1 2 2 11 2 I —--------------------- T1 —-------------

THE UNIVERSITY OF MICHIGAN 7030-3-T where W is the Wronskian and J is the conjunct of m',m m1 2 Proceeding as before, we find that iV o )1 /2 1 - 0 e (cosh o o ()2(o)' 1 2 0o gm 0 Cm2(eo mm' >" )lJ(6o 22 e (cosh3o - )/ (iv + 2-m) 1' __ 00. c 1 1 c sin(iv+m- I )r r(iv + 2 +m) (02(?)1( o), (2.32) and that 2e ~ ~im(-0_.o) -00 sd iv/| ( -OD m) - 00i eiv'0 0' r3+ - in-m) dv - e 2 i- P1,2(1) sin(iv+ m- 2)7r r(iv/+ +m).1 iv1 21(21 1)iv/2~) 0( i-l/2) 2 o iv- J12e Liv% 0( )-i- P iv- iv-/ 0 > (2.33) With (2.29) and (2.33) we also have the solution to the exterior Dirichlet potential problem: V2u = 0 boundary R' and u is regular at infinity (yet to be shown). 12.......

THE UNIVERSITY OF MICHIGAN 7030-3-T 2.4 A Verification of the Result In this section we shall use a different procedure for obtaining the bispherical coordinates, which is particularly suitable in treating the potential problems by considering a single (and/or double) charge layer on the boundary of the body. This will also serve as a check of the results of section 2.3. Let (x, r, 0) be the cylindrical coordinates. Then the equation to be solved is V2G(x, r,= - - 4r 6(x-x )6(r- r)6(0-) (2.34) V G(xr,0;xOrO,0O) r o o (2.34) (with boundary conditions to be imposed later). Assuming the periodicity of G in 0, with period 27r, we may expand it in a Fourier series 1 im(0- 0) G(x,r,;xr ); 27r G(xo,x,r )e (2.35) -00 Substituting into (2.34) we obtain 3~2 32 1 - a 3 2] -47r r + - (m/r ) Gm(xr;xor) = 6(x-x )6(r-r) rLax2 ar2 r ar o o Let G =-g.m (2.36) m \ fgm Then g satisfies the differential equation ~2 2 1 2 a ax2 - 4! [a2+ + ] + m <xr;x,r0) f= - 6(x-x )6(r-r). (2.37) L x 3r r Bipolar Coordinate$: In general, for the map z = f(w) = x(a, 3) + ir(a, 13) (2.38) 13

THE UNIVERSITY OF MICHIGAN 7030-3-T where f is analytic in its domain of definition, we have d l)f a x a) (rx)+ (O ) (2.39) dw| ^ yr jS/ u \a ap and ax ar ax ar =&2~~~~'~~~ ~ ~ ~ ~~(2.40) Using (2.39) and (2.40) we obtain / a2 2, dz P/2 +a2, a a (-dz a + — g(x, r) 82 12 dwIa2 2 In particular, if we consider the map l+e z = iccotW/2 = c e (2.41) 1-e with z = x+ir, w = a+i3, we obtain c sinh3 r c sina (2.42) x = -- - r= (2.42) cosh- cosa cosh- cosa and d 2 2 Wdz = c' (2.43) dw (cosh,- cosa)2 The range and the domain of definition of this map is -oo <x < oo, O<r<oo (2.44) 0 <a < T, -oo <p <oo --------------- 14

THE UNIVERSITY OF MICHIGAN 7030-3-T The z-half plane is represented on the w-strip as shown in Fig. 2. Under this map the equation (2.37) transforms to 2-4+4 ^ ^"o^ - ^2 21r - -2y a- s- 1 2 g (a,,3;a y, = dz 1a( r).. *6(a-)6 — )47) 01 2~~~~~~~~1/2 -47r (cosh 3- cosa) )6( 1/2 - 1/2 0 6(a(sina) /2 (2.45) with csinhf c sina sinhf_ o o x -- r =' 0o -cosh, -cosa rO coshf3 -cosa o o o o We now define the bispherical coordinates (a, 3, p) by rotating the bipolar coordinates (a, j) around the x-axis. As in section 2.2 the ogive is the body of revolution obtained by rotating the coordinate surface a =a (const.) around the x-axis. By proceeding as before, we solve the equation (2.45) and arrive at the same representations for 1/R and G of the previous section. Details are omitted. 2.5 The Regularity, the Existence and the Uniqueness In bispherical coordinates r x 2 z2 cosho+cosa (2.46) cosh/3-cosa Since a and j3 are real, r -co is equivalent to a — 0 and — > 0. That rG < co as r —->o is clear by inspection of (2.33). Oa~G 3=~G. a +aG. 8B (2.47) ar aa & ar a/3 ar o _ IVcoshf +cosa (cosh- cosa)3/2 ar c cosh3 sin ------------------------- 15

THE UNIVERSITY OF MICHIGAN 7030-3-T I 0 I~ ~~~~~~~~~~~~~~~~~I I~o C so o- 1 w I AIl~~~ G I Ix I *L 4 1 I 1I c~ o, I oi g.. 0 W / O, / / ~^< c o 0

THE UNIVERSITY OF MICHIGAN 7030-3-T _g = cosh +cosa (cosh - cosa) 3/2. ar c cos a sinh3 aG (,. sina aG = O (- shna- as r- oo (2. 49a) aG = o ( sinhB ) as r co (2.49b) ap3 \ cosh 3- cosr- Substituting (2.48) and (2.49) into (2.47), aG - = O(cosh - cosa) as r- co ar and hence 2 G O( r - = 0(1) as r-oco. 2-+E aG Note that r * - - oo as r- oo for an arbitrary e > 0. So the static Green's ar function (2.33) is regular at infinity (in the sense of Kellog). The existence question does not arise in our particular problem, since we have actually constructed the Green's function, and have justified every step in the process (either by providing or by indicating the proofs or referring to the proper sources). The uniqueness, of course, follows from the fact that the solution to the exterior potential problem (1) V20 = 0 (2) b regular at infinity is - - o. 17

THE UNIVERSITY OF MICHIGAN - 7030-3-T 2.6 Summary In bispherical coordinates (a, 3, 0) we have constructed the (unique) static Green's function of the first kind, i.e., a function such that (a) VG(a,/;a P,, ) = (cosh f- cosa)36(000 3.c si na c sina (b) G(a1, 3,;a,, o) = 0 (c) G is regular at infinity in the sense of Kellog. Next we shall find the series representation for (2. 33) because that form will be more suitable for generating the Green's function for the Helmholtz equation for the ogive. 18 -

THE UNIVERSITY OF MICHIGAN 7030-3-T III THE SERIES FORMS 3.1 Series Representation for 1/R (Residue Series) We have 1 _ 12 imh 1/2 R = 2c -i, e ~ (cosh - (osh -00 ivI "'I3-13 r(iv -m) ) (3.1) 1o00~~ m1~0 oo 2 2 ~ ~0-ioWith the substitution s = iv - 1/2, this becomes (cosh,B- cosa) -(cosh - cosha ) / / es sins P Eirn m r(s+l-m) pm (3 3 wic = i coshm-i). m +1 — cos )P(-cosa)P (cos ) mR 2m o (1) f~ (s+1+m) s s o aa >. a(3.2) where E =1, ~ =2 for m=1,2,3,... 0 m This, in turn, reduces by means of the addition theorem to MiO -V^ (S ) p-+| 1.i 1/2 e/ - cos1/P2 e(cs (cosh 3-cosa) (coshj -cosa ds P (cos 5 ) R 2c 0 o sin s7r -ioo- i/2 -(3.3) where cosO'=-cosaucosa -sina-sin- cos( -~ =or). o' m'....~~1

THE UNIVERSITY OF MICHIGAN 7030-3-T The only poles of the integrand are at the zeros of sin s7r; hence, they are simple and located at s=0, 1, 2,.... Furthermore, since 2 1. + P (cos ) s i sin (s+ )e + (3.4) s 7 sinO 2 4 for large s I we see that the integrand vanishes if (and only if) Re s < -1/2 as is |-> co. This condition determines which way the contour is to be closed (Fig. 3). Since co is the only limit point for the poles and the spacing between the poles remains uniform, no special analysis is required in "threading" the poles for large | s | and we proceed in the usual way to obtain the residue series. Ims |~~~~I i~eRe s.. 3 - 2 -1 1 0 2... -3 FIG. 3: THE CONTOUR AND THE POLES 20

THE UNIVERSITY OF MICHIGAN -- 7030-3-T R = - 7r (cosh -cosa) /2 (cosh -cos )1/2 e e P (cos ) R c 1 s n=l n 1 -— 1 — (3. 5) 7r COS S 7r n where s =-n, n=1,2,3,... Replacing n by n+ 1 and observing that P (cosO) = P (cos0), (3.5) be-n-i n comes -(n+ 11 = (cosh3- cosa) /2 (cosh3 -cosa ) 1/2 e (os y), R c o o n n (3.6) where cosy = -cos = cosacosa +sinasina cos(0-0 ). 3.2 Series for 1/R (Directly) We now use a well known procedure to find the series representations for 1/R directly, which will also serve as a check In bispherical coordinates, as we noted earlier, with the substitution G = icoshBcosa g, V G = 0 yields the following equation for g, 2 sina (sina ) 2 — g 0, (3.7) a2 sinc sina a +1 which, by g = A(a) B(3) 5(o), separates into the three equations 2d = -m2 0 < 27r (3.8) do d2B -2 = (n+ ~ )B, -oo<,3<ao (3.9)..... 2 1 ----------------------— 21 ------------

THE UNIVERSITY OF MICHIGAN 7030-3-T 1 d dA m2A -i -a(sina — ) -r -- -n(n+ 1)A, 0 a r (3. 10) sina cb &b 2 sin a The assumptions of continuity in 0 and the boundedness of A at a = O, a = 7r restrict m to be zero or a positive integer and n to be an integer greater than or equal to m. We want to solve the nonhomogeneous equation (3.7) with right side -47T lcosEh - cosa 6(a-a )65(3- (O- ). (3.11) We have the well known complete set of (normalized) eigenfunctions for our problem in a and 0, namely,'(a,0) = 2n+e. * (n-m)! P(cosa) Cos(m0) (3.12) *nm' V m 47r (n+m)! n sin with E = 1, E =2 for m = 1, 2,.... Substituting the expansion of g n, m where B(3, o), A (a0, 0) are to be determined, into the differential equation for g and making use of the fact that the surface harmonics bm satisfy the differential nmn equation / i 8 bn sin& sina nm + nm +n(n+l) = 0 (3.14) sina au 2/) 2 nm we arrive at EA ((a(a) d- (n+ 1 )2 B(f3) = (3.15), m o o DM 2 2 47r \Icosh3B-cosa ( )^ )6(0 2csina2 6(-a-/ —)-( — 2 o 22

THE UNIVERSITY OF MICHIGAN 7030-3-T Since 5 dn 7~mn = 5mn.m \ Qmnnfm'f nmnntm' where 20 7r \d d = \ \ sina d, Q Jo0 Jo0 (3.16) Taking A (a, ) = (ao, ) (3.17) mn o 0 nm o0o we are left with the differential equation L[]=-2-n+2] = 47 coshj-cosa 6(3-3). The independent solutions of L [B] = 0 are 1 1 - ( —--— n+') 1 He, ((n+ - ) 3 Yi=e' Y2 e 23

THE UNIVERSITY OF MICHIGAN 7030-3-T -(n+ 2 )(3- f) 0___ ^ 0 0 >.._. cosh3 -cosa e' 90 B(,/o) = c. 1 3 (3.19) 0 C 2n+i -(n+ )(o-1) e 2 ~0 Substituting (3.17) and (3.19) into (3.13) and observing that G = /cosh13-cosa g, we arrive at the following series representation for the free space static Green's function in bispherical coordinates ___ _ -1 7R(a, 3, 01, osh = - \( co s oa)(cosh -cosa R (a.3,PQ O,3 ) C 0 0 o n Z Z n - m cos my)(- J pm(-'cosa)pmkcosa) n=0 m=O m (n+m)' c os n n o n=0 m=0.-(n+ 2)( 3- 0e) e E =1 0, ~ (3.20) -(n+ 1 o)3, E =2 for m = 12,3 -(n+ )(/3-/) m Ie let If we let cos y = cosacosa + sina sina cos(F - ) (3.21) then by the addition theorem n P (cos'y) = P (cosa)P (cosa) + 2 (n- m). n n n o m (n+m)' * Pm(cos Pm(cosa )cos m(p - 0o) (3.22) n n o o The above result for 1/R is expressed in the following more compact form ------------ 24 -

THE UNIVERSITY OF MICHIGAN 7030-3-T c1 ~~~~00 1s 1 1 _______ _______ ______ O:-' -(n+ ) 1I3P-t3l R = \(cosh -cos a)(cosh3 -cosa ) ) P(cos y) * e R c 0v o( o c-h7 n n=0 (3.23) which is equal to (3.6) as it should be. 3.3 The Series Representation for the Green's Function of the First Kind With the substition s = iv -1/2, our previously obtained integral representation for the Dirichlet Green's function is written as i0oo-y2 (s+I2) i 1/2 )1/2 e G = 2 (cosh - cosa) /2(cosh/3- cosa) / ds sin sr m=0 m o^ cos1^ r(s+i+ m) mr p(-)coS (o ) m * P (-cosca)P (cosa )+ S 1 P (cosa)P (cosaO. ~ PSm( m(cosa) OS O Ps(C a1 >a a. (3.24) 1 "ao We now investigate the location and the nature of the poles of the integrand of (3.24). Observations: (1) First we note that since the behavior of P (-cosa)P (cosa ) is the s s o same as Pm(-cosa ) s -. pm(cosc)Pm (cosa ) -jml s s 0 P (cosal) 25

THE UNIVERSITY OF MICHIGAN 7030-3-T for large Is I, the contour to be used for G is the same as that for 1/R. Therefore, we only need to study the integrand in the half plane Re s < -1/2. 1 r(s+l-m) (2) sins7r (s+l+m) has m poles of order two at s =-1, -2,..., -m and simple poles at s = -m-n, n = 1, 2, 3,... (3) In the s-plane P (cosa) is an entire function. Zeros of P in the s S S plane are all real and distinct.* (4) In the s-plane, the zeros of P (cosa), P (cos 3) are different for a f3; s s the zeros of P (-cosa), P (cosa) are different. S S (5) We recall that in constructing the Green's function in the variable a over the range al >a >a we used the proper combination of two functions: p.p )p m m m)p = PS (-O)P (1 )-P (e)P (-ed) = s If s is an integer, then, since pm() = (_)mpm(-), pm( ) = (-_l)mp( _e) 5 5 S 1 S 1 the total Green's function G = 0 for all a in a1 > a > 0. This means that the factor' H. M. Macdonald (1900) showed that for pM real and M > 0 P4(cos a) can have no complex zeros; all its zeros are real. He also showed that PI(cosa), " > 0, has an infinite number of distinct real zeros, and, in addition, at most 2k complex zeros, where k is the greatest integer contained in p. In our case m = 0, 1, 2... and we can exclude the possibility of complex zeros as can be seen from the relation p-m(cosa) = r(s-m+l) (l)mm(cosa) s r(s-m+ 1) s P (cosma) (-1) P (cosa) s r(s+m+l) s for integer m. 26

THE UNIVERSITY OF MICHIGAN - 7030-3-T 1 r(s+l-m) ss r-(s + 1 +m) has in fact no residue contribution to the integral and the only contribution is from the simple poles of 1/Ps(cosa1) at the non-integer real values of s in the region Re s<-1/2. (6) Let s =sj denote the non-integer real solutions of the equation pm(cosa ) =0 in the region Re s <-1/2 of the s-plane. S We can now by means of the observations (1) through (6) write down the final residue series for G G = c (cosh3-cosa) (cosh3 -cosa )1 (-)mcosm(- )u-'(s^j) -131 r(s.+ P1-cosm P(cs )m al >a >o (3.25) 27

THE UNIVERSITY OF MICHIGAN 7030-3-T IV THE GREEN'S FUNCTION OF THE FIRST KIND FOR THE HELMHOLTZ EQUATION 4.1 The Objective and the Preliminaries We are seeking the Green's function for the surface B of the ogive a =a satisfying (a) (V2+ k2)Gk(p, p) -4 6 [(p, p, p, pV (b) Gk(PB p) =0 (4.1) lirn ( (c) the radiation condition, l r -- ikG ) = 0, uniformly in all r oo r or —1 0, r directions, where V denotes the volume exterior to the ogive surface B, p(a, 3,,) the field point, po(ao, o, 0) the surface point, pB(aol,, 0) a point on the surface of the ogive a =a. R(p, p ) is the distance between p and p, and 6 R(p, po ) has been given cosh +cosa explicitly in bispherical coordinates in the static case. We recall r= - cosh- cos and r ->oo a = a- 0, f —0. With our choice of 6-function and the radiation ikR condition, the free space Green's function is e /R, and the decomposition of Gk into singular and regular parts is ikR(p,po) Gk R(p', pP~) and (V2+k2)U =0. (4.3) k U has no singularities in the closure V, is twice differentiable, and satisfies the radiation condition. Also we note that if the Helmholtz equation is considered as reduced from the wave equation, our case corresponds to assuming the harmonic -iWt time dependence e 28

THE UNIVERSITY OF MICHIGAN 7030-3-T We now restate the representation theorem (Kleinman, 1965), which is another form of the Green's theorem, and the expansion theorem (Atkinson, 1949; Barrar and Kay; Wilcox, 1956) as applied to the ogive. Theorem 1 If (a) w(p) is defined for all pe V (b) weC (V) (c) [rw |< oo, r 2a < oo as r -> oo, then w(p) satisfied the integral equation w(p) = G(p,p') Vw(pt) dvt + \ w(PB) G(p, PB)duB J~~~ ~~~~v~~ J~B ~(4.4) where the volume element is given by Q(x, y, z) C3sina dv = dxdydz = (u d f)c dc d (d; N(aC: g:) (coshf- cosa)3 the surface element (for the surface a =a 1) by C sina do = 2 —-- dd1 (cosh - cos a ) the Laplacian V in bispherical coordinates (a, j3, 0) is as given in the static case; 8/an is the normal derivative (in the direction out of V) and is given by an 1 h = -(1(cosh - cosa) - l an h ac c Ga a Theorem 2 The field scattered from the surface of the ogive may be written as ________________________ 29 __

THE UNIVERSITY OF MICHIGAN 7030-3-T ikr, Uk(p, o) = f f /rn (4.5) n=0 where the series converges absolutely and uniformly for r >d+e, E > 0. d is the radius of the sphere obtained by rotating a = 7r/2 about the x-axis. The series may be differentiated term by term with respect to a, 3 or 0 any number of times and the resulting series all converge absolutely and uniformly. The functions f depend on ao/, O, 0, and the parameter k. Following Kleinman (1965), we want to represent the regular part Uk of the Green's function using Theorem 1. Uk is not regular at infinity as can be seen from Theorem 2. There is more than one way of making Uk regular. - ikr Although the obvious form e Uk, indicated by Theorem 2, is regular, in a particular problem such as ours the choice must be made more judicilusly to simplify the resulting equations and to enable us to carry out the integrations arising in connection with the iteration. Thus we define U = e-i( U (4.6) and call f the "eikonal" because of the apparent analogy to the corresponding entity in physical optics (e.g. Born and Wolf, 1959). 4.2 The Proper Choice of the Eikonal f(a, 3) We have IV -ikf U = e Uk (4.6) and 2 2 ikf (V2+ k2)e = 0. (4.8) Therefore, 30

THE UNIVERSITY OF MICHIGAN 7030-3-T 2ikf 2 ikf ) ikf - ikfV 2ikf" V (e U+ke U = Vo(ike UVf+e VU)+ke U =e l- (Vf oVf k2 U+eik V2 + 2ike kf oVf+ikei UV2f = or V U+2ikVUoVf+ikUV f+ [-(f oVf)k U = 0. (4.9) We see from (4.9) that the first natural simplification is achieved by setting VfbVf=l. (4.10) This is the "eikonal equation" for f. Solution of Vf 0 Vf = 1 In bispherical coordinates A V c (cosh- cos) a e a- + T (4.11) c [a &~ f3 af3 sina caja With (4.11), (4.10) becomes (af)2+ af 22 (cosh - cos a) We note that cosh3-cosa = 2sin( ) sin ) si sinh ( - ) sinh (- ) Let z = P+ia, z = -ia then 31

THE UNIVERSITY OF MICHIGAN 7030-3-T af.af.af a- = ~ -l Iola azz az af af af ap3 Oz Oz f a2 af 2 Of aOf \a,/ \ az Oz The equation to be solved is af af c2 Oz Oa 2 2 az az 4 sinh2(z/2) sinh 2(/2) or af af c c(4.12) 2 2O'2 (4.12) a" a~z 2 sinh2(z/2) 2 sinh2(z/2) Assume f = g(z) + g(z), then c a 2a" = — c ~-z' coth(z/2) 2 a, = 2 a2 = -c coth(z/2) 2 sinh 2/2 OZ. g(z) = - coth(z/2) g(z) = - 2 coth(z/2) f = + coth ( ) + coth (-} c > 0 (4.13) 32

THE UNIVERSITY OF MICHIGAN 7030-3-T From r =,cosh3+cosa and (4.13) it is immediately seen that cosh3- cos a f = O(r) as r->oo (a- 0, 3->0). (4.14) From Theorem 2, ikr n rUk eikr f/rn n=O -' -ikf and with U = e k' = ik(r - f) frn rU = e /r n=O so that |rUi < o as r-)co. (4.15) Also 2 au 2 a -ikf 2 a e nl r =r are Uk r r 28 Ar - (l/r) as r -ao; therefore jr2 a <co as r-oo. (4.16) or Equations (4.15) and (4.16) show that -ikf U = e Uk is regular at infinity. 33

THE UNIVERSITY OF MICHIGAN 7030-3-T We also note that V 2f = 0; therefore (4.9) yields 2 V U = -2ikVU Vf (4.17) But -ikf -ikf -ikf VU = V(e U) = -ike UkVf+e VUk; therefore ~%. -ik[ -ikf -ikf -ikf VU oVf = -ike Uk(Vf oVf)+e VUk o Vf = -ike Uk +e fVUo Vf So we have 2-k 2 -ikf V2U = -2(k Uk+ikVUk oVf)e. (4.18) Now U satisfies the hypothesis of Theorem 1. Taking w(p) = U in (4.4), we have U(p, p) = Sdv' G (p, p9V U(p, po)+ d U(pB, po) a Go(P, B) (4.19) where V U is given by (4.17) or (4.18), or U(pp ) = -2ik dv' Go(p, p- Vf o VU(p',p) + da U(PB ) P n o(P' P) Ui(p, ) = - 2 o an o( PB J~~~~~~~~~~V JB ~(4.20) 4.3 Scattering of a Plane Wave by the Ogive (Nose-on Incidence) We write (4.20) in the operator form: (0) U = K U+UU (4.21) where 34

THE UNIVERSITY OF MICHIGAN 7030-3-T K - -2ik \ dvG (p,p')Vf V (4.22) TV (0) a U(0) da U(PB, po) 3n Go(P'PB)' (4.23) B The iterates are given by N U(N) = Kn U(0) (4.24) U'N eu "U'O' (4.24) n=0 or by N) = Ko U(N-)+U(0), >1. (4.25) On the surface of the ogive a =a1, U(PBP)=-ikf(a1, j3) U(pB, po) =e (PB, PO) and the sum of the scattered field Uk and the incident field Uinc vanish on the surface; therefore (4.23) becomes U(0) =- e ikf( uinc(P G(p) p)dan (4.26) If we assume the incident field to be a plane wave propagating in the direction of the negative x-axis, that is, if uinc - ikx(a,3) (4.27) csinhl 3x(,c) = smh, cosh3 - cosa 35 --

, THE UNIVERSITY OF MICHIGAN 7030-3-T then we may assume m =0 in the expression for the static Green's function (s- ) G 7, 1/2 co sa /2" e ( j)1I-0I 0 (cosh3- cosa)2 (cosh,3- COl)l/2 Z ~ns J J P (-cosa).-* d~-7 —---; p (cosa)P (cosa ). (4.28) ds Ps (cosla S S. o ds s. 1 ra 1 rJ1 a Observing that - =- = —(coshB- cosa) and that P (cosca ) = 0 an h &a c or s. from (4.28) we obtain Go 7r _ h 3/2 1/2 0 1 1((coshcoscosa) an 2 sina (cosh- (cosh-os ) a=a1 c (S +1 0 I P (-cosa1) (jid S. 1_ r?2r O 2. c sina1 \ d = \ d2 do JB JO -oD (cosh3-cosal) we obtain P (-cosa ) U(a ) = -2r sin a (coshp -cosa )si2 a. d- a Co d [( Jd (CoS 1'4 s(Co 3 ds 3 s.1 J j ( 4.29) ubstituting ( P (cos) d (into (4.26) and noting that30) 36 sin B 0 3 J-oo (coshf-cos)l I —------------------— oo (cosh13-cos —l) — 3/2' -'

THE UNIVERSITY OF MICHIGAN 7030-3-T where f(al,3) = oth(, + cot X(a$ -)c sinhj_ x1 ) = cosh 3- cosal and we recall that s. are the real zeros of P (cosa ), (s. <-1/2). 3 sj 1 j (N) The iterates U(N) for N >1, given by (4.22) and (4.25) are U -2ik dv' G (p, p')Vf(a',') o VU(Nl )(',')+U(0) (4.31) JV where SPr27 r T OD 3 cd' sina' dv = \ doS dt' d_ c sin J -v J oJoJ -OD (coshP' - cosa')3 37

THE UNIVERSITY OF MICHIGAN 7030-3-T ACKNOWLEDGMENTS The research described in this report is an extension of a method developed in 1964 by R. E. Kleinman, under Contract AF 19(604)-6655, for studying the low frequency scattering of a class of shapes. The author wishes to acknowledge the guidance and suggestions of Dr. Kleinman, in the early phases of this work, in adapting his method to the problem of low frequency scattering from an ogive. The author is also indebted to R. F. Goodrich who made effective suggestions for simplifying the solution to the problem. The development of the solution proceeds from basic mathematical considerations and was facilitated by the use of results obtained under NSF Grant GP-4581. -------------- 33-38

THE UNIVERSITY OF MICHIGAN 7030-3-T REFERENCES Atkinson, F.V. (1949) Philos. Mag. XL, Series 7, 645-651. Barrar, R.B. and A. F. Kay (no date) "A Series Development of the Wave Equation in Powers of 1/r", Internal Memorandum, Technical Research Group, Inc., Syosset, New York. Born, M. and E. Wolf (1959) Principles of Optics (Pergamon Press, New York). Gelfond, I. M. and G. E. Schilow (1962) Verallgemeinerte Funktionen (Distributionen) I (Veb. Deutscher Verlag Der Wissenshaften, Berlin) (Translation from the Russian). Kellog, O.D. (1953) Foundations of Potential Theory (Dover Publications, Inc., New York). Kleinman, R.E. (1965) Arch. Rational Mech. Anal. 18, 205-229. Lighthill, J.J. (1960) Fourier Analysis and Generalized Functions (Cambridge University Press, Cambridge). Macdonald, H.M. (1900) Proc. London Math. Soc. (1), XXXI, p. 264. Magnus, W. and F. Oberhettinger (1949) Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea Publishing Co., New York). Moon, P. and D.E. Spencer (1952) J. Franklin Inst. 254, p. 227. Wilcox, C.H. (1956) Proc. Amer. Math. Soc. 7, 271-276. 39

THE UNIVERSITY OF MICHIGAN 7030-3-T APPENDIX m ORTHOGONALITY AND THE NORMALIZING FACTORS FOR THE SETP I In the usual manner we obtain 1 1 (s.-s.)(s.+s.+ 1) P I(~)P (5)d2 = I I ( i3 j)( @ s, S, { Si d, s PSj ll S (A.1) Since P (cosa ) = P (cosa ) = 0, we conclude that S. 1 5. 1 S. i 1 P pSId = 0 if si s. for m=0,1,2,... s. s. 1 J To evaluate the integral for the case s. = s., we consider the Taylor expansion of P around P substitute the result into (A. 1) and let s. - s. to obtain sj si 1 3 i 1 Squaring (A. 3) and integrating, S~1.~ 1 d(A. 4) d 1 _14 1 dPI ( I (A 2 ) PS.() d9 (1- d + 2. - d r 2 S;d 40

THE UNIVERSITY OF MICHIGAN 7030-3-T Integrating the first two terms by parts, T~P d dPm(,,1 ", 2 m d ( 2 S. -m ({ d9 2 (9) ( 1 d9 (A-5) equation, we obtain _1. 1 2 S [Pm+1(e]2 dt = (sm)(sd+m+)S [Pm ( 2jd(.(A.6) ]>o 1 \1o i1 -j[Pi] d = I.( d+1 [P(0. (A.7) Substituting the value of the first integral on the right from the differential equation, we obtain y 1 12 1 1 1 Now we iterate (A. 6) to obtain P1,,2.2 r(s.+m+1 d) 2. d1 F(s - -m + 1) i i1 1 i?! ^~~~~s0, s5 s. 1 ] 41 S. S 1-2 dp l) dPs l7 r(s.+ +l) 2s.+1 dse dC r(s.-m+l) 1 3 --------------------— 41 ------------

UNCLASSIFIE D Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) I. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan UNCLASSIFIED Department of Electrical Engineering 2b GROUP Radiation Laboratory. Ann Arbor, Michiga 3. REPORT TITLE Low Frequency Scattering from an Ogive 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Technical Report 5. AUTHOR(S) (Last name, first name, initial) Ar, Ergun 6. REPO RT DATE 7a. TOTAL NO. OP PAGES 7b. NO. OF REFS November 1965 41 11 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) AF 04(694)-683 7030-3-T b. PROJECT NO. c. 9b. OTHER REPORT NO(S) (A ny other numbers that may be assigned this report) d. 10. A VA IL ABILITY/LIMITATION NOTICES Qualified requestors may obtain copies of this report from Defense Documentation Center (DDC) Cameron Station, Alexandria, Virginia 22314 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY USAF Ballistic Systems Division Norton AFB, California 92409 13. ABSTRACT By means of a method developed by R. E. Kleinman of the Radiation Laboratory, it is now possible to solve iteratively the Dirichlet problem for the scalar Helmholtz equation in the regions exterior to a non-separable body imbedded in the Euclidean 3-space provided (1) k, the complex wave number,i s sufficiently small, and (2) the solution of the potential Dirichlet problem for the body in question is known. In the present work we consider the low frequency scattering of a plane wave at nose-on incidence from an ogive. Drr FORM 1A73 DD, JAN 64 1 /73UNCLASSIFIED Security Classification

UNCLASSIFIED Security Classification 14. LINK A LINK B LINK CKEY WORDS ROLE WT ROLE WT ROLE WT Low frequency Ogive INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Departmenftof De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report. report from DDC. 2a. REPORT SECURITY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whethert b C nt au ize "Restricted Data" is included. Marking is to be in accord- report by DDC is not authorized. ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di- users shall request through rective 5200.10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional.. markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. report directly from DDC Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified.,, If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of." report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial. -tory notes. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE: Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though 7a. TOTAL NUMBER OF PAGES; The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information, be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number,. wr r tcic iu subproject number, system numbers, task number, etc. 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers, task number, etc. or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical conassigned any other repcrt numbers (either by the originator text. The assignment of links, rules, and weights is optional. or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those TTN.T.A TARTFTE D Security Classification

UNIVERSITY OF MICHIGAN 3 9015 02493 8253