Application of the theory of generalized rays to diffractions of transient waves by a cylinder

WAVE MOTION 5 (1983) 385-398 NORTH-HOLLAND

385

A P P L I C A T I O N OF T H E T H E O R Y OF G E N E R A L I Z E D R A Y S TO D I F F R A C T I O N S OF T R A N S I E N T W A V E S BY A C Y L I N D E R YIH-HSING

PAO

a n d G e o r g e C.C. K U *

Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, N Y 14853, USA

F. Z I E G L E R Institut ]'iir Allgemeine Mechanik, Technische Universitat Wien, A 1040 Wien, Austria

Received 15 October 1982, Revised 24 February 1983

The theory of generalized rays was developed to analyze transient waves in layered media where incident circular or spherical waves are reflected and refracted by plane boundaries. The theory has been recently extended to analyze the diffraction of transient waves by a spherical or a cylindrical boundary. In this paper, the generalized ray integrals, which represent the Fourier transformed diffracted waves, are formulated for the diffraction of an incident spherical pulse by a circular cylinder. The ray integral involves a double integration with respect to two variables of wave slowness. Through a simultaneous transformation of variables, the inverse Fourier transform of these double integrals are completed by applying the Cagniard method.

1. Introduction T h e t h e o r y of g e n e r a l i z e d r a y s was d e v e l o p e d o r i g i n a l l y for a n a l y z i n g t r a n s i e n t w a v e s in l a y e r e d elastic m e d i a . T h e p r o b l e m of a h a l f - s p a c e , a m e d i u m b o u n d e d by o n e p l a n e , was first a n a l y z e d by H. L a m b (1904), the p r o b l e m of two s e m i - i n f i n i t e m e d i a in c o n t a c t b y L. C a g n i a r d (1939), a n d the p r o b l e m of a h a l f - s p a c e with an o v e r l a y i n g l a y e r b y C.L. P e k e r i s (1940). M a n y i n v e s t i g a t i o n s on single o r m u l t i l a y e r e d m e d i a f o l l o w e d . By t h e m i d d l e of 1 9 6 0 ' s t h e t h e o r y of g e n e r a l i z e d r a y s was e s s e n t i a l l y c o m p l e t e d . A s y s t e m a t i c p r e s e n t a t i o n of the t h e o r y can b e f o u n d in a r e v i e w article by P a o a n d G a j e w s k i [1] a n d in the m o n o g r a p h b y A k i a n d R i c h a r d s [2]. M a n y o t h e r r e f e r e n c e s to t h e t h e o r y can be f o u n d in t h e s e two sources. T h e t h e o r y b e g i n s with a F o u r i e r t r a n s f o r m e d or L a p l a c e t r a n s f o r m e d w a v e field g e n e r a t e d by a p o i n t s o u r c e , o r a line s o u r c e , in an infinite m e d i u m . T h e s p h e r i c a l , o r cylindrical, w a v e s e m i t t e d b y the s a m e s o u r c e in a l a y e r e d m e d i u m a r e t h e n s o r t e d into g e n e r a l i z e d r a y integrals, e a c h t r a v e l l i n g a l o n g a specific r a y p a t h f r o m t h e s o u r c e to a receiver. T h e first r a y r e a c h e s d i r e c t l y t h e r e c e i v e r , the s e c o n d o n e u n d e r g o e s o n e reflection at the p l a n e b o u n d a r y , etc. T h e i n v e r s e t r a n s f o r m , F o u r i e r o r L a p l a c e , of t h e s e r a y i n t e g r a l s a r e a c c o m p l i s h e d t h r o u g h a m a p p i n g of t h e i n t e g r a t i o n v a r i a b l e a n d an " i n s p e c t i o n " , a m e t h o d g e n e r a l l y r e f e r r e d to as the C a g n i a r d m e t h o d [3]. T h e success of a p p l y i n g the t h e o r y to t h e r e f l e c t i o n a n d r e f r a c t i o n of t r a n s i e n t s p h e r i c a l o r cylindrical w a v e s b y p l a n e b o u n d a r i e s l e a d s n a t u r a l l y to a n o t h e r a p p l i c a t i o n , t h e diffraction of t r a n s i e n t p l a n e w a v e s b y a s p h e r i c a l o r c y l i n d r i c a l b o u n d a r y . A first a t t e m p t was m a d e b y G i l b e r t a n d H e l m b e r g e r for t h e diffraction of a t o r s i o n a l w a v e b y a s p h e r e [4]. A m o r e d e t a i l e d analysis of t h e diffraction of a t w o * Now at the Naval Research Laboratory, Code 51-32, Washington, DC 20375, USA. 0165-2125/83/$3.00 O 1983, Elsevier Science Publishers B.V. (North-Holland)

386

Yih-Hsing Pao / Generalized rays for diffraction

dimensional wave by a circular cylinder was made by Chen and Pao [5], the incident wave being generated by a line source parallel to the cylinder. Both papers dealt with only scalar wave equations. Extension to the diffraction of two dimensional vector waves by a cylindrical cavity in an elastic solid was made by Chen [6]. In this paper, we extend the theory of generalized rays to the analysis of three dimensional problems of diffraction. The incident wave is generated by a point source in a fluid medium, and it is diffracted by a vacuum or rigid circular cylinder. The extension is, however, not straightforward because the generalized kay integrals formulated for the diffracted pulse are double integrals. They involve an integration with respect to the wave number, or wave slowness, along the circumferential direction, and another along the axis of the cylinder; whereas previous investigations involve only one integration with respect to one slowness variable. We are able to complete the inverse transform in this paper through a simultaneous transformation of two variables. Scattering of monochromatic plane waves by a circular cylinder or sphere is a well documented subject [7]. The early works by P. Debye (1909), B. van der Pol and H. Bremmer (1937), and W. Franz (1954) are of particular interest. The integral solutions for harmonic waves scattered by a circular cylinder or sphere formulated in their studies are precisely the generalized ray integrals for transient waves. Diffractions of transient pulses by a sphere or a circular cylinder (two dimensional waves) are discussed by Friedlander [8]. As in the case of scattering of monochromatic waves [7], he first evaluated the ray integrals by the calculus of residues. The solution for the transient disturbance near the wave front is then obtained by inverse Laplace transform. A similar approach was adopted later by Huang [9] to analyze a three dimensional problem of diffraction as discussed here. The analysis presented in this paper and in [4] and [5] bypasses the evaluation of residues. Application of the Cagniard method to the diffraction of a transient pulse by a semi-infinite screen was made earlier by Felson [10]. The corresponding generalized ray integral is the classical solution by A. Sommerfeld (1896) for an incident harmonic wave, which contains only one variable of integration.

2. Ditfraction of a spherical pulse by a circular cylinder Consider an incident pulse in a fluid medium, ~bi(r, t), generated by a point source at (r', O, O) in cylindrical coordinates (r, O, z). The transient wave is diffracted by a cylinder of radius a, the z-axis being coincident with the axis of the cylinder (Fig. 1). If the time history for the source is given by a function F(t), the incident pulse is represented by

~i(R, t) = F ( t - R / c )/R,

(2.1)

where c is the wave speed, and R the radial distance from the source,

R 2 = R ' 2 + z 2,

R'2=r2+r'2-2rr' cosO.

(2.2)

Let the waves scattered by the circular cylinder be represented by 4~S(r, t). The total wave is 4~(r, t)=4~i(r, t)+cbS(r, t),

r~a.

(2.3)

The scattered wave can be constructed from the solutions of the Fourier (or Laplace) transformed wave equation 0

1 0

1

r~ 2

0 2

"

--

~ r2 + 7 Or+r~ 0-~+ O ~ + ~2) d~s(r' O, z ; ~o) = O,

(2.4)

387

Yih-Hsing Pao / Generalized rays for diffraction Z

~ ( z r , ) 0 ., (r,0, z)

Y

¢t

-r

(r',O,O)

X

22; Fig. 1. The incident ray, reflectedray, and diffractedray. where a = w / c . In the above and the sequel, an overbar denotes the generalized Fourier transform of a causal time function, / I oo

¢(r, ~o) = Jo &(r, t) e i'°' dt.

(2.5)

The inverse generalized Fourier transform of 47(w) is 1 ¢(r, t ) = ! ~

) _,o,

~+i~

e

dw,

(2.6)

where e is a positive constant. We shall not need this formula of inversion because the inverse Fourier transform will be carried out directly by applying the Cagniard method. In terms of the Fourier transformed variables, the total wave field outside the cylinder for an initially quiescent medium is given by [1 1]

1 I~ 4;(r, O; % w) e - i ~

&(r, O, z; oJ) = 2--~w

e;(r,o; v,o,) -

iP(oJ) 8C 2

~ ~

r

,,,

[H,,

dy,

,2~

( k r > ) H m (kr<)

m=-oo

(2.7)

1 2 H ~ ~( k a ) r . l m ( k r , ) H ~ 12H~ ~(ka) --m

(kr)] e i'°

,

(2.8)

where H ~1~and H ~2~are the Hankel functions of first and second kinds respectively, and k 2 = a2_

2 = ~o2/c 2 _ 3,2.

(2.9)

The symbols r< (r>) denote the smaller (larger) value of r' and r. The operator 12 depends on the boundary conditions at r = a : (soft cylinder, & = 0)

Dirichlet: Neumann

:

12 = 1,

(hard cylinder, Ock/Or = O) 12 = O/O(ka).

(2.10)

Yih-Hsing Pao / Generalized rays ]:or diffraction

388

Note that the 4~ in (2.8) for 3, = 0 is precisely the solution for the scattering of a two dimensional wave which is generated by a line source with a harmonic time function [7]. Since the series in m converges slowly for large wave number k (k = a = w/c when 3' = 0), it is usually converted to a sum of integrals by applying the Poisson sum formula, see e.g. [12], ~(r,

0

; 3,, 09)

=

8C 2

H~ (kr>)Hv (kr<)

. . . .

.OH ~2)(ka!

.OHm" ( ka )

H (1) (kr')H~ '~ (kr)] e ~°+2"') dr.

(2.11)

The u-integrals can be evaluated by the calculus of residues, which has been discussed extensively in literature [7, 8]. Since the major contribution to ~(r, 0; 3,, o9) is from the first few terms of the series (n = 0, +1, +2 . . . . ) only a limited number of v-integrals are to be evaluated. We shall, however, not evaluate the residues because their values will depend on the 3,-variable. Instead, we follow the approach developed for two dimensional diffraction [5], and replace first the Hankel functions by their asymptotic expressions, and then invert the Fourier transform by the Cagniard method. We consider only the scattered wave represented by the second term of the integrand in (2.11).

3. Generalized ray integrals of the diffracted ray We discuss here the inverse transform of the second part of 4~ in (2.11) and rewrite it as 5~ 4~ (% ¢o). The q~, (3/, o9) and its inverse transforms are

~,(r,O; y, oJ)=

~ .0H(~, (

dSn(r,O,z ; 09)=

~1

) H~!~(kr')H~(kr)eiV(°+2"'~)dv,

(3.1)

oo r~.(r,O;3,,~o) e i.z d3,,

(3.2)

1 I +~+i~ ~b, (r, 0, z, t ) = ~

-~+i~ ~ ( r , 0, z; w) e-i'°t d¢o.

(3.3)

The Hankel functions in 4~, are approximated by the Debye approximate expressions [7], 2 ±i(z sin&--v& H ~ )'~2~(z) = (wz sin 4~)1/2 e

7r/4)

,

(3.4)

where ~b = cos-l(v/z) and z sin & = (z 2- v2) 1/2. The approximate expression for ~ . is

~, (r, 0 ; 3,, o9) =

f +_oo -2.0 e iG(v'k;r'O'l ~ wk (rr')~/2[1 - (v/kr)2]a/4[1 - (v/kr')Z] ~/4 dr,

(3.5)

where the phase function is 1/2

G(v,k;r, O n ) = k r [ 1 - ( ~ r ) z] - v cos

" b' _2-I 1/2

+kr'[1-(~r,) - v cos

J

12

-2ka[1-(~a)

+ 2v cos -1

+

v0.

2 1/2

] (3.6)

Yih-Hsing Pao / Generalized raysfor diffraction

389

and

O,=O+2nrr,

n=O,+l,+2 .....

The operator of (3.5) has the following value Dirichlet:

.(2 = + 1,

Neumann:

g2=-l.

(3.7)

Since the factor u/k appears repeatedly in the amplitude and phase functions, it suggests a change of variable as follows,

u = ku = u(w2/c 2 -3/2) 1/2.

(3.8)

Similarly, the 3' is changed to another variable v as is done in the case of generalized ray theory,

3" = oJv/c.

(3.9)

In terms of the new variables, (3.1) and (3.2) are combined to read +oo

-i°Jff;'(t°)f +°°-i'°vz/c 8~r2---~~ e d r . f_

~,(r,O,z;w)-

cO

E(u; r) ei''~;(u'U;r'°")/cd//,

(3.10)

oO

where

E(u ; r) = .O[(r 2 - u2)(r '2 -//2)]-1/4, (3.11)

g(u, v ; r, 0n)= ( 1 - v 2 ) l / 2 [ ( r 2 - u 2 ) l / 2 + ( r ' 2 - u 2 ) l / 2 - 2 ( a 2 - u 2 ) 1/2 - u c o s - ' ( U ) - u cos-1 (~,)+ 2u c o s - l ( u ) + u 0 , ] .

(3.12)

Note that by the changing of variables, oJ appears only in the exponents of the double integral. This is a key step for successfully applying the Cagniard method. Since the amplitude function E(u), which is independent of v, is a real, even function of u, and the integrand of the v integral is a real, even function of v, the integral can be rewritten as

¢Sn(r, O, z'; w)

-

iwP(w) 2,rrZc 3 Re I 5 cos ( c zv ) dv Io°° E(u;r) -

-

e

i,o~(,.~/cj

ou,

(3.13)

where Re means "the real part of". The cosine-function may be decomposed further into two terms, one with exp(+izwv/c) and the other with exp(-iztov/c). In view of the factor exp(-kot) adopted for the inverse transform of ~(w), the term with the first factor represents a wave moving in the direction of the +z-axis, and that with the second factor in the - z direction. Because of symmetry in z, it is sufficient to consider only the term in the +z direction. We thus obtain finally 0(3

C~n(r,O,z;w)=A(oJ)Relo

OO

dVlo E(u;r) ei[~(u'~;r'°")+~z]'°/Cdu

(3.14)

where

A (to ) = itoP (o~) / 4.rr2c 3.

(3.15)

390

Yih-Hsing Pao / Generalized rays for diffraction

In the sequel, we shall set A ( w ) = 1, which c o r r e s p o n d s to a step time function with strength 1/4-rr2c 3. T h e general result for an arbitrary A ( w ) can be o b t a i n e d by a convolution. T o shorten the derivation, we consider only the case 0, ~> 0. T h e f , ( w ) in (3.14) is a generalized ray integral which r e p r e s e n t s the p r o p a g a t i o n of the F o u r i e r t r a n s f o r m e d wave, or the h a r m o n i c wave with f r e q u e n c y w, along a specific ray path. As shall be shown later, the case of n = 0 r e p r e s e n t s a reflected ray in the lit zone or the diffracted ray in the s h a d o w zone of the circular cylinder. T h e cases of n > 0 r e p r e s e n t waves creeping a r o u n d the cylinder. T o obtain the inverse Fourier t r a n s f o r m by the C a g n i a r d m e t h o d , the standard p r o c e d u r e is to t r a n s f o r m the variables of integration by a m a p p i n g , (3.16)

ct = g(u, v; r, O~) + vz.

Unlike the two dimensional p r o b l e m of diffraction [5], the generalized ray integral in this case is a double integral of two variables, u and v. A l t h o u g h the new variable t is uniquely d e t e r m i n e d f r o m u and v by this m a p p i n g , the inverse m a p p i n g is not. T h a t is, both u and v cannot be d e t e r m i n e d uniquely for a given real value of t. A n additional constraint i n d e p e n d e n t of t must be i m p o s e d on the variables u and v. This is a c c o m p l i s h e d by a s i m u l t a n e o u s t r a n s f o r m a t i o n of two variables as discussed in the next section.

4. Simultaneous transformation of two variables

In (3.14) note that if v = 0 and A ( w ) = 1, we have

&0(r, 0 ; o)) =

t"c~ R e J0 E ( u ; r) e i'°~"'°;r'°')/c du,

(4.1)

w h e r e E ( u ; r ) is the s a m e as in (3.11), and g(u, 0; r, 0 , ) = ( r 2 - u 2 ) l / 2 + ( r ' 2 - u Z ) 1 / z - 2 ( a 2 - u 2 ) X / 2

+ u[2 cos-1 (U)-cos

1 ( ~ ) -- COS- 1 ( ~ ) "1"-0hi •

(4.2)

T h e function d~,o(r, 0; w) is i n d e p e n d e n t of z and the preceding result is the s a m e as that for two dimensional diffraction except for a constant factor contained in A (w). T h e g e o m e t r y of the phase function g(u, 0) has b e e n discussed extensively in [5, 7] and elsewhere. W e introduce first a change of variable f o r m u to r, c~(u) = g(u, 0; r, 0,).

(4.3)

T h e m a p p i n g of (3.16) is thus e x p r e s s e d as (cf. (3.12) and (4.2)) t(v) = ~-(1 - v2) 1/2 + vz/c.

(4.4)

A s i m u l t a n e o u s t r a n s f o r m a t i o n of u and v to r and t will enable us to p e r f o r m the inverse Fourier t r a n s f o r m without invoking the f o r m u l a in (2.6). 4.1. M a p p i n g o f u to "r

T h e integration for ~ , in (3.14) is for real variables u and v. T h e variable r according to the m a p p i n g is real or complex, d e p e n d i n g on the m a g n i t u d e of u relative to that of r, r' and a. In Fig. 2 and Fig. 3

Yih-Hsing Pao / Generalized rays for diffraction

Imu

Imtl

F

F~ N

N

~ / H / / .

"~"~ C"'O

391

E

•~Reu B'" c'".'.'=,f>~>~: ="';"A " B C ~''0

(a)

E '~Reu

(b)

Fig. 2. The complex u-plane and the Cagniard contour F1; (a) with the stationary point M; (b) without a stationary point. Im2

Im/"

f

. . .........................

_

=ReT

=Re?

"'"""" D t n ( r rCY ~ _ (a)

(h)

Fig. 3. Mapping of the positive real and imaginary u axes; (a) with the stationary point M, (b) without a stationary point. we show the complex u-planes and complex r-planes respectively. The real axis of u, A B C D E , is m a p p e d to the curve A B C D E in the z-plane by the transformation (4.3); and the imaginary axis A F of the u-plane is m a p p e d to the curve A F i n the r-plane. This intricate mapping is determined by first noting that the g(u, 0) function is multivalued at u = +a, +r', +r. In Fig. 2, we use again r< (r>) to denote the smaller (larger) of r' and r. The function is rendered single valued by the branch cuts D ' C ' , B ' B , CD. The origin A on the u-plane is m a p p e d to where czA = g(0, 0) = r + r ' - 2 a . F r o m A to B ( u < a ) , g(u, 0) is real valued, and from B to E, g(u, 0) is complex, approaching asymptotically to l u [ [ 0 , - i l n ( r r ' / a 2 ) ] (0, > 0 ) as lu]~co. The mapping of the imaginary u-axis, A F , in the z-plane is denoted as AF. It approaches lul[ln(rr'/a2)+iO,], which is perpendicular to the asymptote for A/~. In case that__r has a stationary point, which is discussed later in connection with the reflected ray, a branch cut M M ' (Fig. 3a) is introduced. As a result, the first quadrant of the u-plane is uniquely m a p p e d onto the sector F A E in the z-plane. This ensures the uniqueness of the inverse mapping from z to u,

u = g-l(cr, 0; r, 0,). In view of the complexity of the (4.2) and (4.3), this inverse mapping can only be done numerically.

(4.5)

Yih-Hsing Pao / Generalized rays for diffraction

392

4.1. I. Reflected ray B e t w e e n A ( u = 0) a n d B ( u = a) in Fig. 2, the function g(u, 0) m a y have a s t a t i o n a r y point. F o r 0, > 0, we e x a m i n e the c o n d i t i o n

(j

d g ( u , 0) _ 0, - c o s - l ( u / r ) - c o s - ~ ( u / r ') + 2 c o s - l ( u / a ) du

-

-

z

O.

(4.6)

T h e r e exists a r o o t for the a b o v e e q u a t i o n if a n d o n l y if 0.=0+2nw
l(a/r')+cos-l(a/r).

(4.7)

Since the p r i n c i p a l values of a r c c o s i n e f u n c t i o n s a r e t a k e n a n d 0 > 0, this c o n d i t i o n can o n l y be satisfied w h e n n = 0. T h e critical angle is 0 = 0b, a n d

Ob = COS-1 (a/r) + cos-l(a/r ')

(4.8)

defines the b o u n d a r y b e t w e e n the lit a n d s h a d o w zone. L e t the r o o t of (4.6) for n = 0 be u = d , which is d e n o t e d by M a n d ~ t in the u - p l a n e a n d r - p l a n e r e s p e c t i v e l y , that is, 0 - cos -1 (d/r) - cos-l(d/r ') + 2 cos 1(d/a ) = 0.

(4.9)

T h e v a l u e for r at ]~t is

czM = cg(d, O) = (r2 - a2) l/2 + (r'2- d2) l / 2 - 2 ( a 2 - d2) 1/2.

(4.10)

A s s h o w n in Fig. 4, t h e s t a t i o n a r y v a l u e u = d is the d i s t a n c e f r o m the origin of the c o o r d i n a t e s to the p r o j e c t i o n s of the i n c i d e n t a n d r e f l e c t e d ray, a n d rM is the travel t i m e of the w a v e f r o m the s o u r c e (r', 0, 0) to a r e c e i v e r at (r, 0, 0) t h r o u g h o n e reflection. B e c a u s e of the e x t r e m a l p o i n t M , the m a p p i n g of A B o n t o A B a l o n g the real axis of r is m u l t i v a l u e d . A b r a n c h cut M M ' is i n t r o d u c e d a n d the p o i n t / 3 is s e p a r a t e d f r o m A b y the b r a n c h cut.

½

Fig. 4. Geometry of the reflected ray.

Yih-Hsing Pao / Generalized rays for diffraction

393

4.1.2. Diffracted ray When n > 0, or when n = 0 but 0 > 0b, there is no stationary point between A B , and u = a at the point B is the extremal point, beyond which cr = g(u, 0) is complex valued. We set at u = a, C

dg(u,O;a, On) 0 n - 2 c o s

l(a/r)-2cos

l(a/r')=AO~

(4.11)

du

where don = zl0 + 2n ~r and/tO is an angle. The value for r a t / ~ is crB = (r 2 - a 2)1/2 + ( r ' - a 2)1/2 + aAO,.

(4.12)

As shown in Fig. 5, cru is the distance of the diffracted ray path and A0n is the central angle subtended by the ray creeping around the cylinder. For n = 0, 0b < 0o < w, the receiver is in the shadow zone. For n > 0, the receiver may be in either the lit zone or shadow zone, and n indicates the n u m b e r that the creeping ray raps around the circular cylinder. R

Fig. 5. Geometry of the diffracted ray. In the absence of a stationary point, the segment A B along the real u-axis is m a p p e d to the segment A B on the real r-axis as shown in Fig. 3b. The remaining u-axis (u > a ) is m a p p e d to the curve B E in the complex r-plane.

4.2. Inverse mapping frorn r to u With the mapping defined by (4.3), and the inverse mapping by (4.5), the generalized ray integral of (3.14) is transformed to co

d)n(r,O, + z ; o ~ ) = R e I0 dv Ix ~ E ( r ; r ) ei[(1 v2v/2,+v~l,o/c(du~ \d-rr] 0 dr,

(4.13)

where (du/dr)0 is the derivative of u with respect to r according to (4.5). Since the integrand, E(du/dr)o, has no singular point in the sector bounded by N B E and the circular arc EN, and the integrand vanishes along the arc E N as the radius approaches infinity, the path of integration B E can be replaced by B N according to Cauchy's theorem. The path A B E in the preceding integral is thus changed to A M N in Fig. 3a for the case with a stationary point, and to A B N in Fig. 3b without a stationary point. In either case, the variable r is real valued. Through the inverse mapping (4.5), the real r-axis is m a p p e d to A M N curve (Fig. 2a), or A B N curve (Fig. 2b) on the complex u-plane. We designate both curves by fix, and transform the integration of d r back to that of du

~n (r, O, +z ; ol) = Re lo dv lr E ( u ; r ) e i[(1 1

v2)l/2"r(u)+vz]~°/C d u .

(4.14)

Yih-Hsing Pao / Generalized rays for diffraction

394

This transformation can also be obtained directly from ~ , in (3.14) by noting that E ( u , r) is regular in the region bounded by N M E N or N B E N , and it vanishes along the infinite circular arc E N . The intermediate step in (4.13) proves that along the path F~, the variable ~'(u) is always real positive. 4.3. M a p p i n g o f v to t

Once the integration of real u is replaced by that of complex u along the contour FI, the variable r is always real valued. The mapping from v to t, (4.4), thus contains all real variables. The inverse mapping fromttovis v = [ctz + c ' r ( c 2 ~ ' 2 + z 2 - c 2 t Z ) l / 2 ] ( c 2 r 2 + z 2 )

(4.15)

-1.

In either mapping or inverse mapping, ~- is treated as a real parameter. The new variable t is still complex when v > 1. We shall deform the path of integration for v so as to render t real-valued. Shown in Fig. 6a is complex v-plane and the integration of ~ , is along the real axis A B C . The points B and B ' at which v = +1, are two branch points and they are connected by a branch cut on the real v-axis. The real v-axis is m a p p e d by (4.4) to curve A B C o n the complex t-plane as shown in Fig. 6b. Again the mapping onto A B may be multivalued. The stationary point M is determined by solving dt dv

-~-v z ( l - v 2 ) 1/2+-=0"c

(4.16)

Imv

Imt

D' N

...... ~ " " ~

(a)

c

-- Rev

/Ff Fff,

--Ret

(b)

Fig. 6. The mapping of v to t and the inverse mapping; (a) complex v-plane and the Cagniard contour ['2; (b) complex t-plane.

The root is

z/c

v = VM = [~2 + (Z/C)211/2 =--sin AM.

(4.1 7)

This root exists for all values of r and z. Since both r and z / c are real and ~"> 0, it is always possible to define a real angle )tM = sin -1 vM.

Yih-HsingPao/ Generalizedraysfor diffraction

395

At v = VM, we have t = tM, and tM = ~'(1 -- V2)1/2 +

VMZ/C = [r z + (Z/C)211/2.

(4.18)

The geometrical relation between AM, ~', and tM is shown in Fig. 7. This relation shows that for a wave moving along a ray path with helical angle AM, the minimal travel time along the helical path is t~t, that along the projection on the z = 0 plane is r, and that along the z-axis is z/c. The ray path shown in Fig. 4 is precisely the projection on the z = 0 plane of a three-dimensional reflected ray as shematically indicated in Fig. 1. The paths shown in Fig. 5 are the projections of a helical diffracted ray in the shadow zone, or a creeping ray with helical path in the lit zone.

T Fig. 7. The helical ray path and the helical angle ,~M. A branch line MM' is introduced along the real t-axis to render the mapping of v to t single valued. The first quadrant of the complex v -plane is thus m a p p e d uniquely onto the sector DABCD, and the inverse mapping is also unique. The real v-axis is m a p p e d to the curve AMBC. Again this complex contour is deformed to line AMN along the real t-axis by applying the Cauchy integral theorem. The integral for 0~, of (4.14) is transformed to o0

d~n(r,O,+z;to)=Re I dtlr (~t) E(u;r) ei'°'du, A

(4.19)

1

where (Ov/Ot), is derived from (4.15) by keeping r constant. The lower limit of integration is tA = ~', which is determined from (4.4) by setting v = 0. As in the previous inverse mapping of z to u, the segment AMN can be inversely m a p p e d to the AMN curve in the complex v-plane. The curve is denoted by F2 in Fig. 6a. The ~n is thus transformed to a double integral along two complex contours

~,(r,O, +z;to)=Re lr dV fr E(u;r) ei'°"u'~:"°"~du. 2

(4.20)

1

Both t and r are assured of real valueness for all values of u and v by these mappings. Since (4.19) is the most convenient form for the inverse Fourier transform, we shall not need this last form of double-integral.

5. Transient diffracted waves

Through a simultaneous mapping, the original generalized ray integral, (3.14) is brought nearly to the form of a Fourier transform. By extending the lower limit from real tA to 0, (4.19) can be written as

~,(r,O,+z;~o)=Re

Io

H(t-r)E(u;r) ~-~ du.

d t e i"~' 1

"r

(5.1)

396

Yih-Hsing Pao / Generalized rays for diffraction

In the above, the

E(u; r) is defined in (3.11), (5.2)

E ( U ; r) = ,O[(r 2 -- u 2 ) ( r '2 --/4 2)] 1/4,

v is given by (4.15) U = [CtZ -'}-CT(C2T2q-Z2--C2t2)I/2](C2T2q-Z2)

and

(5.3)

-1

(c~v/Ot)T is derived by differentiation, (av/at)~ = c[z -cry/(1 - v2) I/=] 1.

(5.4)

The real variable r is given by (4.2) and (4.3), C7" = ( r 2 - - U 2 ) 1 / 2 .+_ (r'2

+ u [2 c o s - '

_

u 2)1/2 _

2(a 2 __ U 2 ) 1 / 2

(u/a) - cos 1(u/r) - cos l(u/r') + 0 + 2n ~r].

(5.5)

The contour /~1 begins at A, the origin of the complex u-plane (Fig. 2). It runs above the branch cut on the real axis, and turns at either point M (Fig. 2a) or point B (Fig. 2b) to the first quadrant. Note that if the inverse Fourier transform of 4~n (r, 0, +z ; w) is ¢b, (r, 0, +z ; t) and the latter is a causal function, they must be related according to (2.5), c~

d~"(r'O' +z;°J)=£~

&n(r'O' +z't) ei'°' dt"

Comparing the above with the 4~, in (5.1), we find

~n(r,O, + z ; t ) = R e

H(t-r)E(u;r)

-~

i

du.

(5.6)

,r

The shifting of Re inside the integral of dt is permitted because &~ (t) is causal. We thus have accomplished the inverse Fourier transform by "inspection." Although the path F1 extends to infinity, the integral is finite because of the step function. After the substitution of (5.2)-(5.5) into (5.6), the integrand is a function of u with parameters r, 0, z, n. On the other hand, u is related to r by (5.5) and the integral vanishes when r > t due to the step function H(t - r ) . Therefore the upper limit of integration along the F1 contour u (r)]T ,, and the complex integral in (5.6) can be expressed as u(r)

cb,(r,O,+z;t)=Re f

E(u;r)(O~)

aA

du.

(5.7)

r

Along the real axis from A to either M or B, the integrand E(Ov/Ot), is pure imaginary. The wave field &, (t) thus vanishes for u < uM, uB, or r < rM, rB which are given by (4.10) and (4.12) respectively. This implies that for a fixed observer at r, 0, z and for a particular n u m b e r n, the ~b, (r, 0, z, t) changes with the real time t. As t increases, the time r which is the travelling time along the projection of the helical path also increases. H e n c e the observer will not receive any signal for r < rM or rB. F r o m (4.18), we find the arrival time along the helical path for the n-ray, t v , = [(rM.~)= + For example,

(z/c)=]1/2.

(5.8)

tMo = [r 2 + (z/c)2] 1/2 is the minimum travel time of the reflected ray shown in Figs. 1 and

Yih-Hsing Pao / Generalized rays for diffraction

397

4. Equation (5.7) is now written as

&.(r, O, +z, t) = H ( t - t M , )

E ( u ; r) •t M , B

du.

(5.9)

"r

The complex integral in (5.9) can be evaluated numerically as follows: Select a set of values r, 0, +z for a receiver in either the shadow zone or the lit zone, and start with the n = 0 ray. The lower limit of integration M is fixed by substituting rM into (5.5) and solving for uM ; and B is simply u = a. At a specific time t > tMO, we set r = t in (5.5) and solve for u (t), which defines a point on the F1 curve as the upper limit of integration. The integral can then be evaluated numerically by any complex algorithm. We then proceed to the cases of n = 1, 2 . . . . until the end of duration of observation. Since all rays arrive in successive order with known travel times, only a finite n u m b e r of integrals need to be evaluated. The total diffracted wave (the second part of 2.11) for A ( w ) = iwP(oo)/4w2c 3= 1 is a finite sum of all 6 , (t),

CS(r,O, + z , t ) = Y. H ( t - t M , ) n =0

E(u,r) aM, B

du.

(5.10)

"r

6. Conclusion In this paper, we have shown the application of the generalized ray theory to the solution of a three dimensional problem of diffraction. The Fourier transformed wave field diffracted by a circular cylinder is expressed as generalized ray integrals with respect to two variables, the wave n u m b e r u along the circumferential direction and the wave n u m b e r y along the axis of the cylinder. Inverse Fourier transforms of these double integrals are accomplished by a sequence of change of integration variables: (1) The change of wave numbers u and 3, to u and v, respectively, by (3.8) and (3.9). (2) The simultaneous mapping of u and v to r and t by (4.3) and (4.4). The first step eliminates the dependence of the frequency w in the integrand except that in the exponential function. The second step reduces the double integral to an integration of u along the Cagniard contour F1 and another integration of the real variable of time t. The inverse Fourier transforms of the generalized ray integrals are then obtained by 'inspection'. Only solutions for the diffracted rays are discussed here. In the lit zone, they can be combined with the expressions of the incident pulse, (2.1), to yield the total wave. Because of the asymptotic representation of the Hankel function, these solutions are expected to be valid at and shortly after the wave front. In the shadow zone, the role of the incident wave term ¢i remains to be re-examined. It is believed that the contribution of the incident wave term to the total wave field in the shadow zone as discussed in [5] is erroneously large when a receiver is at a large distance from the circular cylinder. Some numerical results for the reflected waves are given in [11]. Those results and an alternative simultaneous transformation of variables will be discussed in a subsequent report.

Acknowledgment This research was supported by a grant of the National Science Foundation awarded to the College of Engineering of Cornell University. The third author (FZ) conducted this research while visiting Cornell University in 1977 under the sponsorship of the same grant.

398

Yih-Hsing Pao / Generalized rays for diffraction

References [1] Y.H. Pao and R. Gajewski, "The generalized ray-theory and transient elastic waves in layered media", Physical Acoustics XIII, Academic Press, New York (1977). [2] K. Aki and P.G. Richards, Quantitative Seismology, Freeman, San Francisco (1980), Vol. 1, Ch. 6. [3] A.N. Cagniard, Reflection and Refraction of Progressive Seismic Waves, Translated by Flinn and Dix, McGraw-Hill, New York (1962). [4] F. Gilbert and D.V. Helmberger, "Generalized ray theory in a layered sphere", Geophysical J. Royal Astronom. Soc. 27, 57-80 (1972). [5] P. Chen and Y.H. Pao, "The diffraction of sound pulses by a circular cylinder", J. Math. Phys. 18 (12), 2397-2406 (1977). [6] P. Chen, "Diffraction of sound pulses and acoustic emission in a hollow elastic cylinder", Ph.D. Dissertation, Cornell University, Ithaca, NY (1978). [7] H. Honl, A.W. Maue and K. Westpfahl, "Theorie der Beugung", in: S. Flugge, Ed., Handbuch der Physik 25/1, Springer, Berlin (1961). [8] F.G. Friedlander, Sound Pulses, Cambridge Univ. Press, New York (1958). [9] H. Huan'g, "Scattering of spherical pressure pulses by a hard cylinder", J. Acoust. Soc. Amer. 58, 310-317 (1975). [10] L.B. Felson, "Transient solutions for a class of diffraction problems", Quart. Appl. Math. 23, 154-169 (1965). [11] C.-C. Ku, "Diffraction of spherical sound pulses by a circular cylinder", PhD.Thesis, Cornell University, Ithaca, NY (1981). [12] E.C. Titchmarsh, Theory of Fourier Integrals, Oxford University Press, London (1948).