RL-863 UNIFORM ASYMPTOTIC EXPANSIONS Thomas B.A. Senior January 1991 RL-863 = RL-863

28 January 1991 1 Introduction In studying the physical optics solution for the bistatic scattering of a plane wave by an infinite, two-dimensional, perfectly conducting, S-shaped surface, it is necessary to develop a uniform asymptotic expansion of the integral r - [f'(x) cos > - sin ] ejk[c+sf(z)ldx (1) V 27 J-oo where C = cos + cos o, S = sin 6 + sin 0o (2) with 0 < 7r - $ <. The surface is y = f(x), and as a typical example consider f(x) =A e-t dt (3) where A and 1/c have dimensions (length)1. Then f'(x) = Ac e-()2 which is positive for all real x; f"(x) = -2Ac3 xe-()2 which is negative (positive) for x > (<) 0, and f'(x) = -2Ac3 (1- 2c22) e-()2 which is negative for Ilx < -7. In this particular example the surface is an odd function of x, but it is convenient to carry out the analysis for a more general surface having the following properties: f(x) is a monotonic function of x, -oo < x < oo 1

f(x) > (<)0 for x > (<) 0, implying f(O) = 0 f'(x) > 0 with max f'(x) = f'(0) f"(x) < (>) 0 for x > (<)0, implying f"(O) = 0 and f"'(x) < 0 over the range spanned by the stationary phase (SP) points for the angles >, >o of interest to us. The SP points are such that C+Sf'(x) =0 (4) implying f'(x)= - cot ( + 0), (5) and if 7r- > 2tan-1(0) - -(r-q), (6) (5) defines the two distinct SP points xl,x2 with xl > 0 and x2 < 0. However, when 7r- = 2tan-1 f'(0)- — ) (7) the two points merge at the origin, and for smaller values of 7 -- 4 the SP points correspond to pure imaginary values of x. For given r - bo the angular region (6) is that where specular contributions to the field exist, and the boundary is defined by (7). Beyond this there are no specular contributions, and only a small field is expected. We seek asymptotic evaluations of the integral expression (1) that are uniform in angle throughout the specular and non-specular regions. 2 Specular Region At angles well within the specular region there are two distinct real SP points x1, 2 at which Slf"(x)l/k >> 1, and the dominant contributions 2

to the integral come from small ranges of integration about these points. Since Clos \ - o) f'(x) cos sin - sin = - - 2 sinI (F + 0o) at the SP points, we have cosCO ( - o) 27r sin 2(q + 0) where (8) (9) I = 1 + 2 (10) with I = xl+61 eJk[Cx+S f()]dx J1-61 (11) 12 = j2 eJk[c+Sf(x)]dx. (12) x2 -62 Expanding f(x) in a Taylor series about x = xl (> 0) Consider first I1. and using (4),, ejk[Czl+Sf (l)] ej kS2f"()dx -61 ejk[Cxl +Sf(xl)] 2 Al kS{ -f"(Xl)} -^ (13) e-t2 dt where 1 A1l= - ^kS{-f"(X1)} 61. Recalling that f"(xl) < 0, the leading term in the 11 is obtained by allowing A1 - oo, giving 2 I ejk[C1k+Sf(z)l] / 2 VkS-f/"(Xl)} J (14) asymptotic expansion of 00 -00 e-t2 dt 3

and therefore 2. I 1 j ejk[Cxl+sf(rl)] e24. (15) k-V^-ff'(zO * Similarly 12 -ek[CX2+Sf(z2)] 6 ej 2 f(x2)d, (16) -62 and since f"(x2) > 0, 12 " eJk[C12+Sf (2)] 2/ 4. (1) -kSf"(x2) Hence, to the leading order, 2 7r. I- ejk[Cxl+Sf(xl)] 27r e-i VkS -f"(Sl)} +jk[C2+Sf(X2)] 2 e (18) kSf"(x2) which can be written more compactly as I _ eJk[Cx +sf/()] 27r e[arg f"(x)-] " s (19) i=1,2 IkSjf"(xI) ( where -7r < arg f"(Xi) < r. 3 Non-specular Region In this region the values of x specified by (4) are pure imaginary, and if x1 is such that Im.x1 > 0, then Im.x2 < 0 and Im.f"(xl) < O, Im.f"(x2) > 0. The path of integration runs from x = -oc to x = oo but can be deformed to pass through either xl or x2, and when this is done, the dominant contribution to the integral comes from the immediate vicinity of the point. For I1 expansion of f(x) about x1 produces an integrand (see (13)) 1 kSlf "(x )1z 2 ~_2 4

whereas 12 gives (see (16)) e kSIf"(x 2)\ x2 The points x1, x2 are now saddle points, and clearly x2 is at a lower level than x1. Accordingly, the saddle point appropriate for a steepest descent evaluation is x2 and I ejk[Cz2+Sf(X2)] 2 ek{-"(x2)}x2dx -62 (20) _ k[Cx2+Sf(X2)] I e-t22 2 = [+kS -jf"(x2)} J dt where A2 = kS{-jf"(x2)} 62. (21) 2 The leading term in the asymptotic expansion is obtained by allowing A2 oo and is I- ek[Cx2+Sf(x2)] 2 (2) kS{-jf"(x2)} where x2 is such that Im.x2 < 0. With the previous definition of arg f"(xi), (22) can be written alternatively as 2wT2 2(23) I c~ eik[Cx2+Sf(x2)] I 2e gf "(2)_] 2 3 kSl/f"(x2) and the result is equivalent to retaining only that term in (19) corresponding to the SP point whose imaginary part is negative. 4 Boundary The boundary separating the specular and non-specular regions is defined by (7), and corresponds to the merger of x1 and x2 at the origin where f"(x) = 0. Accordingly, (19) and (23) fail when the observation point 5

is on the boundary, but it is a trivial matter to develop an asymptotic approximation valid in this case. Since there is now only a single SP point at x = 0, I _s ek[c+sf(x)]dx (24) J-8 and by expanding f(x) in a Taylor series about x = 0, we obtain I eksf(~) J eji3f"'()dx (25) J-s where we have used the fact that f'(0) = f"(O) = 0. By assumption f"'(O) < 0 and hence I jkSf(o) /j e-Jdt kSf{-f'"(O)} J-. where 1 A= ^kS{-f"'(O)} 6. (26) On allowing A - oo the leading term in the asymptotic expansion is found to be 2 0 I e~ ekSf(0) e-dt kS {-f"'() )} C d that is kS,j{-f((0) 27rAi(0) (27) where Ai(z) 2= - e-( )dt (28) is the Airy integral of the first kind. We note that [Abramowitz and Stegun, 1964] Ai(0) = 0.355028.... 6

5 Uniform Asymptotic Expansion, Specular Region The asymptotic expressions (19) and (23) in the specular and non-specular regions respectively are valid only in those parts of the regions which are bounded away from the boundary at which (27) applies, and we now seek expressions which are uniform in angle and which match (27) into (19) and (23). We consider the specular region first. Since f"(x) vanishes at the SP point(s) corresponding to the boundary, it is necessary to retain an additional term in the Taylor series expansion of f(x) about x = x1 and x2. In the case of I1, (13) is replaced by - jk[Cxl +Sf(xi)] j61 +E1 * kS (Y-1 )2 [f"( xi)+ 3 (Y-E )f M(xi)] dy Choosing =1 = the exponent in the integrand becomes [.1k,1 3 {f (x)} 2 {f"(X)}3] 3 2 3 x1 - f"'(x1) 3 {f"'(x)}2 and therefore I jk[Cx:l+Sf[x ] (x )]+ k3S f()}3/{f "'( X,)}2. 1 + /fI(Jlf)...(xl) -j kS I[ Y 3fJ,(rxl)_ ^ f,(l)j2/1flt(xi d/ (30) -61 +f "(xi)/f 'f"(Xl) Recognizing that f"(xi) < 0 and f"(xi) < 0, this can be written as,1 ejk[Cxi+Sf(xl)]-iJ-),' [ 2 A' I1+ 1 Al/f (l lyfI//(xl)lI t3 ) J —+ f "( ( ))/f'"((z ^ l [_l..t_6~.ft(Xl)/f/tt(Xl)] 7

where A = [2 kS {- (l)}] 61 2 kS 3 '1 -2 { — f'"(lx))12 (31) [2{-f"'(Xl)} ] (31) To ensure that the range of integration does not include negative values of t we now choose f"(xi) 1 = ft(= 6l) Then, for Slf"'(xl)l/k >> 1 the leading term in the asymptotic expansion is T jk[Cxi+Sf(xi)] [kS{i( }] -3 j ei( —t)dt, (32) ~-Y _-Ylt)dt (32) -kS-f'(zx)} ' ' which can be expressed in terms of Airy integrals as 1 2 1 T ei^k[Cxi+Sf(xi)] [ { ] kS{-f"l(Xl)} 7r{Ai(-7) - jBi(-71) + jHi(-7i)} (33) where 1 r00 i t3 \ 1 00 Bi(z) =- sin -+ zt) dt +- e +dt (34) 7 Jo 3 7 is the Airy integral of the second kind and [Abramowitz and Stegun, 1964] 1 /00 t3 Hi(z) - - e —+Ztdt. (35) The treatment of the integral 12 is similar. In place of (16) we now have 2 ejk[CX2+Sf(X2)] _62 e j2[f(,X2)+xf (X)]x -62 8

which can be written as I 2 ejk[cz2+Sf(X2)]+j kS{Sf("(2)3/{ff.'(x2)}2 12 3 62+f+"(X2)/f'(z2) j S[y 3yfll (x2)-y f"(2)}2/fl"(X2)]dy (37) J-62 +f 11(2)/f ft( 2) Recognizing that f"(X2) > 0 but f'1(X2) < 0 we choose f"(x2) 62 - -f( ) (> 0) to ensure that the integration is confined to negative values of y, and then 12 ik[CX2+Sf(X2)] {-(1)] 2 -2A2 e-(- -2)dt kS {-f/'(x 1)} where A2 = [kS {-f' (x2)} 62 (38) 72 =2 {- ( )2 { (x2)}2 (39) The leading term in the asymptotic expansion is obtained by allowing A2 - oo, giving 12 k[CX + Sf (X2)]2 [ { )-]t) dt (40) kS {-f "(xZ2)} oo which can be expressed in terms of Airy integrals as S{ f(\ 2 3 } 2 12 N ik[0x2+Sfl)] [kS{ —"(x2)}j e(3X2 kS {-f,()I} *~ {Ai(-72) + jBi(-72) - jHi(-2)} (41) The desired uniform asymptotic expression for I is therefore [ ek +()]kS {-f"'(i)}] e ' 7r {Ai(-71) - jBi(-71) 19 1 2 3 +jHif_-, )} + ek[CX2 +Sf (X]2)] +Jks{ 7-f,,,"'(x)) ej3r2 {Ai(-72) + jBi(-72) - jHi(-2)} (42) 9

where 71 and 72 are given in (31) and (39) respectively. On the boundary of the specular region, 1 = x2 = 0, and since f"(0) = 0 we have 71 == 72 = 0 there. The expression (42) for I then reduces to that in (27). Well within the specular region, 71,72 >> 1, and for >> 1 7r {Ai(-7) F jBiz( —) ~ jHi(-7)} - --— e (43) 7Z [Abramowitz and Stegun, 1964]. When this is substituted into (42), we recover (18) precisely. 6 Uniform Asymptotic Expansion, Non-specular Region The final task is to develop a uniform expression which matches (27) into the formula (23) in the non-specular region, and the procedure is similar to that given above. The relevant SP point is now the saddle point x2, and by including an additional term in the Taylor series expansion of f(x) about x = x2 we again obtain (37). However, f"(x2) is now pure imaginary with Im.f(x2) > 0, and therefore I ek[Cx2+Sf(X2)] 2 {} (-^2) rJ2 6 7 ~2) 3 (_t3 -Y2~ e- t)dt S{-f'"( 2)} I '[-, ' ] -j(,3-'2t)dt where A2 is given in (38) and 2 72 = - 2 x)}2 {-jf"(x2)}2 < 0. (44) An asymptotic expression is obtained by allowing A2 -- oo, and the result is I _c e/ikk[c2+Sf(2)] ___(- e ( 10

which can be expressed in terms of an Airy integral as I ' ejk[c2+sf(X2)] [kS f()}] (-2 e3 27rAi(-7) (45) with 72 as shown in (44). On the boundary between the non-specular and specular regions, x2 = 0 with f(x2) = f"(x2) = 0. Thus, 72 = 0 and (45) is then identical to (27). On the other hand, well within the non-specular region where 72 < 0, the Airy integral can be replaced by the leading term in its asymptotic expansion, viz A Z )2)(-1-2, (46) Ai(-72) - 2,/ (-72)-4 e-(-'2), (46) and when this is inserted into (45) we recover (22). 7 Summary In the specular region a uniform asymptotic expression valid up to and including the boundary with the non-specular region is 1 z 3k[Cx{+Sf (x)] 2 ],T2yi i=1,2 kSlf "'(Z xi) l 7r {Ai(-7y) F jBi( —7) ~ jHi(-7y)} (47) with 2 3 = [kS ()2] " 2 (48) and the upper (lower) signs for xi > (<)0. The analogous result for the non-specular region is I e k[C2+S() Sf"'(z2)1 eI3 ]Az( —) (49) _kSlf'"(x) e(')27Ai ( ')2)| 11

with 72- = kSIf( x2 2 - x (50) where x2 is the pure imaginary saddle point having Im.x2 < 0 and Irn.f"(x2) > 0. Reference Abramowitz, M., and I.A. Stegun (1964), Handbook of Mathematical Functions, National Bureau of Standard, pp. 446-7. 12