Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
522
Clifford .O Bloom
Nicholas .D ffonirazaK
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Berlin.Heidelberg. New York 1976
Authors
Clifford O. Bloom
Nicholas .D ffonirazaK
Department of Mathematics
State University fo New kroY
at Buffalo
Amherst, .N .Y 14226/USA
Library of Congress Cataloging in Publieation Data
Bloom, Clifford O 1935-
The asymptotic solution of h~h-frequency radiation-
scattering problems in inhomogeneous media.
(Lecture notes in mathematics 522) ;
Includes index.
I. Radiation. .2 Scattering (Physics) 3. Asymp-
totic expansions. I. Kazarinoff, Nicholas D., joint
author. II. Title: The asymptotic solution ofh igh-
frequency radiation-scattering problems ... III. Se-
ries: Lecture notes in mathematics (Berlin) 522. ;
QA3.L28 no. 522 tQC~753 510'.8s E539'.2~ 76-17818
AMS Subject Classifications (1970): 35B40, 35B45, 35J05, 53C25,
78A05, 78A40
ISBN 3-540-0?698-0 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-387-07698-0 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
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PREFACE
These notes are based upon a series of lectures given at the University of Oxford,
Spring, 1975 by the second author. The authors thank Dr. J. .B McLeod for the oppor-
tunity for their joint work to be presented in his seminar.
These notes are primarily concerned with existence, uniqueness, a priori estimates,
and the rigorous asymptotic solution of the radiation-scattering problem:
Au+~2n(x)u = f(x) (x 6 V) ,
(p) u = u0(x) (x 6 ~v),
m-I 2
aril ~ rlUr- i~,u+--~--r u dS = 0 (Ixl = r)
R-k= r=R
for l large, Here V is the exterior of a not necessarily convex or star-shaped
body 5V C R m (m = 2 or 3) . In Chapter I we obtain new point-wise and L 2 a priori
estimates by a variation of K (cid:12)9 Friedriehs' abc-method. These estimates imply unique-
ness of the solution u of the boundary-value problem (P) above. We construct an
approximate solution (in powers of l-i ) to the problem (P) in Chapter 3. We also
construct there an approximate solution to the more. general radiation-scattering
problem where the values of a linear combination of u and its normal derivative are
prescribed on 5V . We apply the a priori point-wise estimate of Chapter I to prove
in Chapter 3 that the approximate solution to the problem )P( is an asymptotic
expansion of the exact solution as I 4 = This asymptotic approximation yields a
high-frequency asymptotic expansion of the leading term of the far field scattering
amplitude a(x,~) , where
)k)~x(a = lira (re -i~r u) .
In Chapter 2 we study the reciprocal relationship between ray systems in the
inhomogeneous medium with index of refraction n (cid:1)89 and the smoothness and asymptotic
properties of n (cid:1)89 as r ~- ~ This study, which is the first of its kind insofar
as we know, is necessary to give meaning to the computations involved in determining
the formal approximate solution of problem )P( . It is also a necessary ingredient
in using the a priori estimates from Chapter 1 to show in Chapter 3 that truncations
of the formal asymptotic series uniformly approximate the solution u in the closure
of V . The results of Chapter 2 are obtained through application of the fixed point
theorem for contractions to the integral equation form of the ray equations. In
Chapter 4 we show how results of D. E~dus imply ~xistence of u ; and we also give
an alternative existence proof~ based on his work, which may be susceptible of gener-
alization to elliptic equations of the form
V' (E(x)Vu) +~2n(x)u = f(x) ,
iv
in cases where E(x) and n(x) are not constant outside a compact set.
The authors thank Professor J. B. Keller for his encouragement and helpful
advice.
Amherst, New York
December~ 1975
CONTENTS
PRE FACE iii
CONTENTS v
CHAPT~ 0. INTRODUCTION I
CHAPTER .i A PRIORI BOUNDS 22
.i Introduct ion 22
,2 Geometric Preliminaries 24
.3 The Basic Inequality 27
4. A lower Bound for (Vu~) (cid:12)9 32
.5 Far-fleld Behavior of Coefficients of the ILul 2 and
lul2-terms 37
.6 The Radiation Integral 39
.7 A Priori Estimates in Weighted L2-norms 41
.8 An A Priori Estimate for lu(x,~)l 45
CHAPTER .2 GLOBAL EXISTENCE, SMOOTHNESS, AND NONFOCUSSING OF OPTICAL
PATHS IN A REFRACTIVE MEDIUM 48
.1 Introduct ion 48
.2 Ray Coordinate Systems and Convexity Relative to n % (x) 52
3. An Existence Theorem 59
4. Solution of the Ray Equations 62
.5 Existence of Ray Fields on Unbounded Domains 66
.6 First Derivatives of X and the Jacobian 77
.7 Higher Derivatives of X 78
.8 The Main Theorem 82
CHAPTER 3. A UNIFORM APPROXIMATION TO THE SOLUTION OF URSELL'S RADIATING
BODY PROBLEM 84
.i Introduction 84
2. The Ansatz 87
.3 Analysis of the A j and B j'2 87
4. The Radiation Condition 92
5. General Obstacles 92
6. An Ansatz for More General Boundary Conditions 92
CHAPTER 4. EXISTENCE OF SOLUTIONS 98
INDEX 103
CHAPTER 0
INTRODUCTION
Much effort has been devoted to studying the solutions of radiation-scattering
problems of the form
(0.I) V' (E(x)Vu) +A2n(x)u = f(x) (x 6 V)
(S) (0.2) e(x,A)u+~(x)V*(x) (cid:12)9 Vu = g(x)
(x E ~V ~ V* = unit exterior normal to 3V)
(03) aril 2 lr iAu+ ul2ds - 0
R-~ r=R
(m=2,3 ; r = Ix = Z~ (xi)2 (cid:1)89 ,
where V is the exterior of one or more scattering obstacles of finite cross-section,
E(x) is a strictly positive definite matrix that tends to the identity matrix as
r 4 ~ , n(x) is a strictly positive function that tends to 1 as r 4 = , and A
is real and positive. The inhomogeneous term in (0.i) is usually assumed to vanish
outside a compact set, or to approach zero at a prescribed rate as r ~ ~ A special
case of interest is f(x) = 6(X,Xo) and g(x) = 0 . Then the solution of Problem (S)
is a Green's function .
The interest in the scattering Problem (S) arises because the equations (0.I) -
(0.3) are a mathematical model for the propagation of time harmonic waves in an in-
homogeneous medium filling the exterior region V . If u(x,A) is a solution of (S)
with e(x,A) = el(X ) - iAce2(x ) , then u(x,A)e "iAct is a time harmonic solution of
frequency (cid:12)9 = cA of the wave equation
(0.4) V (cid:12)9 (E(x)VW) - n(X) w = f(x)e -icAt ((x~t) qVX (0~))
2 tt
c
(c = speed of propagation of signals if n(x) ~ I)
that satisfies the boundary condition
-ickt
(0.5) e l(x)W+e 2(x)W t+~(x)V*(x) "VW = g(x)e
((x,t) 6~V(cid:141) (0,=)) ,
and the radiation condition
(0.6) lim ~ rI~IWI2Hs = 0 (t > O) ,
R~= r=R '
where
~i w = Tr~w +-ik+ (m-1) 2r lJ w (m = 2,3) .
- ic~t
The time harmonic solution ue of (0.4)- (0.6) has been shown (under certain
conditions on the coefficients E(x) , n(x) , ~l(X) ~ e2(x) , ~(x) , and the source
terms f(x) and g(x) ) to be the steady state solution of the following initial-
boundary value problem.
(0.7) .7~ (E(x)VW) - n(x). 2 Wtt ~ f(x)e -i)%ct ((x,t) EVX 0,~)) ,
c
(S') (0.8) ~l(X)W+~2(x)Wt+~(x)V*(x ) (cid:12)9 ~TW = g(x)e -ilct ((x,t) E~)V(cid:141) (0,~)) ,
(0.9) W(x,0) = hl(X ) , Wt(x~0) = h2(x) (x6V) .
If f(x) (cid:12)9 hl(X) and h2(x) , n(x) - 1 and E(x) - I have compact support (and are
sufficiently smooth), then scattering theory 13, p. 164 can be applied to show that
-ic~t
the solution W(x,t) of (S') approaches ue as t 4 ~ at every point of V .
For example, D. E~dus 5~6 has proved such a result if ~2(x) = 0 , ~(x) = 0 ,
g(x) = 0 3 V is an open exterior region with a finite boundary and supp(E-I) is
compact.
In the case of scattering by a single obstacle of finite cross-section that does
not "trap" rays (see 13, p. 155) it is reasonable to expect that the solution W
of (S') should approach a time harmonic steady state if E(x) and n(x) are
sufficiently smooth. C. .S Morawetz 15 and R. Buchal 4 using different arguments
have shown, for solutions of the wave equation defined outside a star-shaped obstacle
in R 3 , and satisfying a Diriehlet boundary condition that
W(x~t) = u(x,~)e-i~ct+~(t -I) (x6V) as t 4 -
C. O. Bloom 3 has established an algebraic mate of approach to the steady state for
solutions of (S') ~ defined outside a star-shaped body if hl(X) and h2(x) have
compact support~
rtfi2dV<-, ~2(x) =~(x) ~0, g(x) =0
V
and provided E(x) - I , n(x) - i are sufficiently smooth (lie in CI(~) , ~= VU ~V ),
E(x) - I = O(r -I-6) (0 < 6 < )I , and n(x) - I = ~(r -I'6) as r ~
The argument of Bloom assumes the existence of a unique solution of (S) under
these conditions on E(x) and n(x) . As far as we know, such an existence-unique-
ness result has not been established. We prove the existence of a unique solution of
(S) in Chapter 4 of these notes if ~(x) ~ 0 , ~(x,l) ~ i , and E(x) ~ I . In our
proof n(x) - 1 is not required to have compact support~ but it is assumed to satisfy
the hypotheses of Theorems 7.1 and 8.1 of Chapter .I The a priori estimates estab-
lished in these theorems in~nediately yield uniqueness of solutions to (S) in this
case.
Our existence theorem is alternative to a theorem of D. M. EZdus 6. As is often
the case with a priori bounds, if they imply uniqueness9 then they imply existence as
well. The idea of our proof is to consider a sequence of modified problems (S) with
n replaced by nj , where supp(nj - )I is compact and expands to all of V as
j ~ ~ , and to show that solutions to these problems converge to a limit that is the
desired solution of (S) when E(x) ~ I , ~(x) = 0 and ~(x,l) = I . Existence
of solutions u. to the modified problems is implied by EYdus' result. Applying the
a priori estimates derived in Chapter I of these notes allows us to conclude that
these solutions form a Cauehy sequence, which converges to u(x,l).
If E(x) ~ I ~ equations (0.I) - (0.3) may govern the propagation of electromag-
netic waves of frequency w=lc in an optical medium~ where the index of refraction
is n(cid:1)89 . The solution of (0.4) - (0.6) is the amplitude of the scalar potential
of the time-harmonic electromagnetic field. Under certain conditions on the shape
and the physical properties of ~V ~ the components of the electric field intensity
and the magnetic field intensity also satisfy (0.4) - (0.6).
If E(x) = n(x)l, equations (0. )I - (0.3) govern the propagation of acoustic waves
in a slightly compressible medium of density p(x) ~ I/cn(x) ; see 8~ Chapt. I.
Under the acoustical interpretation the solution of (0.4) - (0.6) is the excess
pressure or the velocity potential of the time harmonic field. In many important
applications~ the wave length I >> i/a ~ where a is the minimum diameter of the
scattering obstacle.
In Chapter i of these notes we obtain a priori estimates for the solution
u(x~l) of the following radiation-scattering Problem:
(0.I0) Lu = f(x,l) (xs c Am;m=2,3 ; I > 0)
)P(
(0.ii) u = g(x) = u0(x) (xESV) ,
(0.12) lim ~ rl~lUl 2 : 0 ,
R-~= r=R
where
Lu = Au +k2n(x)u
If m=2(3) ~ then V is the exterior of a smooth closed curve (surface) 5V
(a smoothly embedded (m- I) sphere in Am). We assume that 5V can be illuminated
by convex surface (curve) contained in V (see Definition 2.1 of Chapter i).
We require that
)i( u0(x) E cl(sv) ,
(ii) n(x) 6 C2(~) ,
(iii) f(x,~t) 6 C(~) for every t~ > 0
(H) (iv) ~ r21fl 2 < ~ , and
V
(v) n(x) >_ n o > 0 for all x E ~ . In addition we require that ss__a r-~ ~
(vi) In(x) - i I = ~(r "p) for some p > 2 ,
(vii) Vn(x) = ~9(r "2) ,
(viii) ~i+j n(x)/~x i~x j = ~9(r -3) (i+ j = 2 , i,j > i) .
Here ~ is the closure of V .
Most of Chapter I is devoted to obtaining estimates for the "energy norms"
*vll (cid:12)9 VutL~v = ~v Iv* (cid:12)9 Vu,2)(cid:1)89
(P* = exterior unit normal to ~V ,)
1'r-lvuv= ~vr-2'Vu'2)(cid:1)89 ,
and
vllul-rI' = ~vr-21u'2) (cid:1)89
We find that as I 4 ~ (we let a = 1 for convenience)
(0.13) "*vTI vull v, r'Xvul v <_ q N(f, u 0 ; ).~
and
(0.14) vllul-rll < r2~-iN(f,u0 ; ~) ,
where F 1 and F 2 are constants that depend only on ~V and n(x) , and
N(f,u 0 ; ~) = ~Max lu01 +llU0r, V~ll vlfrll+ .
~V
We use (0.13) and (0.14) to derive an upper bound for the field strength
lu(x,~)l that holds uniformly on V as ~-~ = (cid:12)9
(0.15) )X,x(ul T <_ r31(l+m)/2 r(l-m)/2N(f3u0 ; I) ,
where F 3 is constant that depends only on ~V and n(x) .
The estimates we obtain for the L 2 norms of u/r and Vu/r also imply an
upper bound on the energy ~0(ue-i~ct) of the function ue "i%ct that is contained
in the region V(R0) between the boundary ~V of the scattering obstacle and a sphere
of radius R 0 ; namely,
~0(ue - ck~i )t _ r 3
(0.16) , < N(f, u 0 ; ~) ,
~r+~r
where r 3 = 2( Min r -2) . As we mentioned above, these same estimates
V(R )0
immediately imply uniqueness of u .
In Chapters 2 and 3 we consider the following Problem )U( :
t~__eL u(x~) be the solution of equation .0( i0) subject to the radiation condition
(0.12) ~ and the boundary condition
(0.17) ~(x,k)u+3(x,X)v*(x ) .Vu = g(x,k) (x E ~V) .
Construct an asymptotic approximation uM(x~ ) of u(xj~) such that
(0.18) )k~,x(u -uM(x,~ ) = @~ )mP-l(~z+M- r 2 J (M > (cid:1)89
uniformly in x (x E ~) .
We use the notation S for the closure of a set S . We call this problem the Ursell
radiating body problem; see F. Ursell 19.
We apply the a priori estimate (0.15) to solve problem )U( for a general class
of scattering obstacles in the case ~ ~ 0 , ~(x,~) m i , under physically reason-
able hypotheses on )i( the smoothness of ~V , gC = u 0) and f , and (ii) the
asymptotic behavior as r ~ = of f , n and derivatives of these functions. We
assume for simplicity that f(x,~) = fo(X) and g(x) = u0(x ) , where fo(X) and
Uo(X) are independent of ~ . The asymptotic approximations we obtain satisfy (0.18)
=e-i~ r
for positive integer values of M . Note also that mil~-r u M is an asymptotic
expansion of the scattering amplitude of u . The function UM(X,l ) is constructed
to satisfy the radiation condition (0.12), the boundary condition (0.ii)~ and to have
the property that as ~ ~ ~ ~_ _ (m~3~
(0.19) L UM(X, k) = f0(x) +@ M r (x E~;m=2,3) .
To get (0.18) we apply the point-wise estimate (0.15) to u-u M . Under conditions
similar to those conditions that we impose on 5V , n(x) ~ g and f to construct
uM(x,~) in the case ~ ~ I , ~ ~ 0 our method can be applied to yield a function
~(x,l) that satisfies (0.i0), (0.12) the boundary condition (0.17), and which also
has the property that * ~_M _ (~3))
(0.20) LUM(X,l ) = f(x,l)+~ r (x EV;m=2,3) .
We describe the procedure in Chapter 3 assuming that )i( ~(x,k) , ~(x,l) are
sufficiently smooth in x , (ii) ~(x,k) = ~l(X,k) - ik~2(x ), (iii) DPf(x,~) ,
DPg(x,l) , DP~l(X,l) , DP~2(x ) , DP~(x,l) = ~(i) as l~ - , lPI =0,1,2,3 .... ,
(iv) ~l(X,~) , ~(x) , ~(x~k) are of constant sign on the subset of ~V contained
in suppg U supp f ; see Chapter ,3 Section 6 of these notes. But unless ~ m 0 ,
we have no a priori estimate available that can be used to prove that the function
uM(x,l) , which is an approximate solution of the boundary value problem .0( I0),
(0.12) and (0.17), is also an asymptotic approximation of u(x,k) as I ~ ~ in the
sense that (0.18) holds.
In the case n(x) ~ I we require that the subset of 5V contained in the
support of the radiating sources f0(x) and g(x) consist of a finite number of
disjoint, locally convex "patches" S i (i= I~2,...,K) joined together so that 5V
is smooth; see Fig. 0.I. In addition~ we impose the condition that )i( each
straight line ray (since n(x) ~ I ) emanating orthogonally from the patch S i ex-
tends to infinity without again meeting 5V . If all of ~V is contained in the
support of f or g, then the above requirements are satisfied if and only if ~V
is convex; see Fig. 0.2. In order to apply the a priori estimate (0.15) to u-u M
we also need to postulate that 5V can be illuminated from the exterior.
In the case n(x) ~ I we impose analogous restrictions on 5V . The portion
of ~V contained in the support of f or g should consist of a finite number of
disjoint patches S i that are "locally convex relative to n(cid:1)89 "~ and joined
together to form a smooth surface (curve) in R3 (R )2 . For a patch .S to be
i
