MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Math. Meth. Appl. Sci. (in press)
Published online in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/mma.753
MOS subject classification: 35 C 10
An expansion theorem for two-dimensional elastic
waves and its application
Kun-Chu Chen1 and Ching-Lung Lin2, ∗, †
1 Department
2 Department
of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan
of Mathematics, National Chung Cheng University, Chia-Yi 62117, Taiwan
Communicated by P. Hagedorn
SUMMARY
We prove an Atkinson–Wilcox-type expansion for two-dimensional elastic waves in this paper. The
approach developed on the two-dimensional Helmholtz equation will be applied in the proof. When the
elastic fields are involved, the situation becomes much harder due to two wave solutions propagating
at different phase velocities. In the last section, we give an application about the reconstruction of an
obstacle from the scattering amplitude. Copyright q 2006 John Wiley & Sons, Ltd.
KEY WORDS:
Atkinson–Wilcox; Helmholtz equation
1. INTRODUCTION
Let u(x) ∈ C 2 be a solution of the scalar Helmholtz equation u + k 2 u = 0 in the exterior of the
ball with radius a>0 and satisfy Sommerfeld’s radiation condition. It is a well-known property
[1, 2] that u, in the spherical coordinates (r, , ), can be expressed as
u(r, , ) = r −1 eikr
∞
fn (, )r −n
(1)
n=0
where the series converges for r >a and converges absolutely and uniformly with respect to r, , in
the domain r >a + ε>a. The series may be differentiated term-by-term in all variables. Moreover,
the coefficients fn for n>0, can be constructed recursively from the far-field pattern f0 (, ).
Similar results for Maxwell’s equations and elastic equations in three dimensions were proved by
∗ Correspondence
†
to: Ching-Lung Lin, Department of Mathematics, National Chung Cheng University, Chia-Yi
62117, Taiwan.
E-mail: [email protected]
Contract/grant sponsor: National Science Council and National Center for Theoretical Science
Copyright q
2006 John Wiley & Sons, Ltd.
Received 23 February 2006
Accepted 5 April 2006
K.-C. CHEN AND C.-L. LIN
Wilcox [3] and by Dassios [4], respectively. In two dimensions, a convergent expansion theorem
for the scalar radiation solution was established by Karp [5]. However, a similar expansion theorem
for two-dimensional elastic waves is still missing. The present paper is an attempt to fill this gap.
In three dimensions, the way of driving expansion theorems for radiation solutions to the
Helmholtz, Maxwell’s and elastic equations relies on integral representations of radiation solutions
and the fundamental solution eikr /r to the scalar Helmholtz equation. One of the key points is that
d(r −1 exp(ikr ))/dr = r −1 exp(ikr )(ik − r −1 )
(2)
(1)
As for two dimensions, the fundamental solution for scalar Helmholtz equation is H0 (kr ),
(1)
where H0 (z) is the Hankel function of the first kind, of order zero. From Reference [6, p. 74],
we found that
(1)
(1)
d(H0 (kr ))/dr = −k H1 (kr )
(1)
(1)
where H1 (z) is the Hankel function of the first kind, of order 1. Unfortunately, H1 (z) cannot
be expressed by H0(1) (z) as we had for eikr /r in (2). So in two dimensions, if an expansion
theorem does hold, it will not be as neat as (1). In fact, Karp [5] showed that a radiation solution
u = u(r, ) ∈ C 2 to the scalar Helmholtz equation in the region r >a admits the following expansion:
(1)
u = H0 (kr )
∞
(1)
r −n Fn () + H1 (kr )
n=0
∞
r −n G n ()
(3)
n=0
where the series converges absolutely and uniformly in r a + ε>a and can be differentiated term
by term with respect to r and . The coefficients F0 and G 0 are determined from the formulas
F0 () = [ f0 () + f0 ( + )]/2
(4)
−iG 0 () = [ f0 () − f0 ( − )]/2
where f0 () is the so-called far-field pattern or amplitude. Furthermore, Fn and G n for n>0 are
constructed, respectively, from F0 and G 0 .
Unlike the approach for the three-dimensional case, Karp took a different route to obtain the
expansion formula (3) by expressing u in the form
u=
∞
n=0
Hn(1) (kr )(an cos n + bn sin n)
(1)
(1)
(1)
and writing Hn (kr ) as a linear combination of H0 (kr ) and H1 (kr ) with coefficients which
are polynomials in 1/r . These polynomials are Lommel’s polynomials. We shall follow Karp’s
approach for two-dimensional elastic waves. The starting point is to decompose the elastic wave into
longitudinal part u p and the transverse part u s . We then apply Karp’s results to u p and u s , separately.
Most of efforts are devoted to driving formulas of determining coefficients in the expansion.
2. EXPANSION THEOREM FOR 2D ELASTIC WAVES
In this paper, we consider the time-harmonic elastic wave equation in two dimensions
u + ( + )∇(∇ · u) + 2 u = 0
Copyright q
2006 John Wiley & Sons, Ltd.
(5)
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
EXPANSION THEOREM FOR 2D ELASTIC WAVES
where and are Lamé constants satisfying >0, +>0 and >0 is the density. It is well-known
that u of (5) can be decomposed into u = u p + u s and u p , u s satisfy
u p + k 2p u p = 0,
u s + ks2 u s = 0
(6)
∇ · us = 0
(7)
and
∇ ⊥ · u p = 0,
where k 2p = 2 /( + 2), ks2 = 2 / and ∇ ⊥ = (−*x2 , *x1 ). Moreover, a solution to (5) is called
radiation if it satisfies the Kupradze radiation conditions
lim
r →∞
√
r (*u p /*r − ik p u p ) = 0
and
lim
r →∞
√
r (*u s /*r − iks u s ) = 0
where u p is the longitudinal field and u s is the transverse field. In other words, u is a radiation
solution of the elastic equation (5) if and only if each component of u p and u s is a radiation
solution of the scalar Helmholtz equation and condition (7) holds.
Now, we directly apply Karp’s results to each component of u p and u s to yield
Theorem 1
Let u = u(r, ) ∈ C 2 be a radiation of (5) in r >a>0. Then u admits the following convergent series
expansion:
u = u p + us
(1)
= H0 (k p r )
∞
(1)
r −n Fn () + H1 (k p r )
p
n=0
∞
(1)
+ H0 (ks r )
(1)
∞
p
n=0
r −n Fns () + H1 (ks r )
n=0
r −n G n ()
∞
r −n G sn ()
(8)
n=0
for r >a and the series converges absolutely and uniformly in r a + ε>a. It also can be differentiated term-by-term with respect to r , any number of times and the resulting series all converge
absolutely and uniformly.
(1)
From the Betti integral representation formula and the asymptotic behavior of H0 (z), we can
see that any radiation solution u of (5) has the asymptotic form
√
p
u = 2 exp(−i/4)(k p r )−1/2 exp(ik p r )u ∞ () r̂
+
√
2 exp(−i/4)(ks r )−1/2 exp(iks r )u s∞ ()ˆ + O(r −3/2 )
(9)
p
as r → ∞ uniformly in all directions . Here the pair (u ∞ , u s∞ ) is called the far-field pattern of
the radiation solution u. We observe that the far-field pattern coming from the longitudinal part
u p is normal to the unit circle, while the far-field associated with the transverse part u s is tangent
to the unit circle.
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
K.-C. CHEN AND C.-L. LIN
Our next task is to determine the coefficients in expansion (8) from the far-field pattern
p
(u ∞ , u s∞ ) of the radiation solution u. Detailed computations will be carried out in the following
section.
3. DETERMINATION OF COEFFICIENTS
To begin, we would like to put related differential operators in the polar coordinates. Let F(r, ) =
fr (r, )r̂ + f (r, )ˆ be a vector field and f (r, ) be a scalar function where r̂ = (cos , sin )t and
ˆ = (− sin , cos )t . In the polar coordinates, we know that
and
∇ · F = * fr /*r + r −1 fr + r −1 * f /*
(10)
∇ f = (* f /*r )r̂ + r −1 (* f /*)ˆ
(11)
On the other hand, we can derive that
f = * f /*r 2 + r −1 * f /*r + r −2 * f /*2
2
2
(12)
To determine the coefficients in (8), we replace u in (5) by (8). Before the computations, it is
useful to note that
(1)
(1)
H0 (z) = −H1 (z)
and
(1)
(1)
(1)
H1 (z) = H0 (z) − z −1 H1 (z)
(13)
Furthermore, we use the following notations:
p
p
p
p
p
Fn () = fn, r ()r̂ + fn, ()ˆ = fn, r r̂ + fn, ˆ
p
p
p
p
p
G n () = gn, r ()r̂ + gn, ()ˆ = gn, r r̂ + gn, ˆ
Fns () = fn,s r ()r̂ + fn,s ()ˆ = fn,s r r̂ + fn,s ˆ
s
s
s
s ˆ
ˆ
G sn () = gn,
r ()r̂ + gn, () = gn, r r̂ + gn, Making use of (13) and (8), we get from (5) that
p
p
p
p
(1)
(1)
(1)
(1)
S0, r H0 (k p r )r̂ + S0, H0 (k p r )ˆ + S1, r H1 (k p r )r̂ + S1, H1 (k p r )ˆ
(1)
(1)
(1)
(1)
s
s
s
s
ˆ
ˆ
+ S0,
r H0 (ks r )r̂ + S0, H0 (ks r ) + S1, r H1 (ks r )r̂ + S1, H1 (ks r ) = 0
Copyright q
2006 John Wiley & Sons, Ltd.
(14)
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
EXPANSION THEOREM FOR 2D ELASTIC WAVES
where
p
S0, r = ∞
n=0
{n 2r −n−2 fn, r − 2k p nr −n−1 gn, r − k 2p r −n fn, r + r −n−2 ( fn, r ) }
p
∞
+ ( + )
p
p
p
{(n 2 − 1)r −n−2 fn, r − 2k p nr −n−1 gn, r }
p
p
n=0
∞
+ ( + )
{−k 2p r −n fn, r − (n + 1)r −n−2 ( fn, ) + k p r −n−1 (gn, ) }
p
p
p
n=0
∞
+ 2
r −n fn, r
p
(15)
n=0
p
S0, = ∞
n=0
{n 2r −n−2 fn, − 2k p nr −n−1 gn, − k 2p r −n fn, + r −n−2 ( fn, ) }
p
∞
+ ( + )
p
p
p
{(1 − n)r −n−2 ( fn, r ) + k p r −n−1 (gn, r ) + r −n−2 ( fn, ) }
p
p
p
n=0
∞
+ 2
r −n fn, p
(16)
n=0
p
S1, r = ∞
{2k p nr −n−1 fn, r + (n + 1)2r −n−2 gn, r }
p
p
n=0
+
∞
{r −n−2 (gn, r ) − k 2p r −n gn, r } + 2
p
p
n=0
+ ( + )
∞
∞
r −n gn, r
p
n=0
{2nk p r −n−1 fn, r − k 2p r −n gn, r − k p r −n−1 ( fn, ) }
p
p
p
n=0
+ ( + )
∞
{n(n + 2)r −n−2 gn, r − (n + 2)r −n−2 (gn, ) }
p
p
(17)
n=0
p
S1, = ∞
n=0
+
{2k p nr −n−1 fn, + (n + 1)2r −n−2 gn, − k 2p r −n gn, }
p
∞
p
{r −n−2 (gn, ) } + 2
p
n=0
+ ( + )
∞
∞
p
r −n gn, p
n=0
{−k p r −n−1 ( fn, r ) − nr −n−2 (gn, r ) + r −n−2 (gn, ) }
p
p
p
(18)
n=0
s
S0,
r =
∞
n=0
s
2 −n s
{n 2r −n−2 fn,s r − 2ks nr −n−1 gn,
fn, r + r −n−2 ( fn,s r ) }
r − ks r
+ ( + )
∞
s
{(n 2 − 1)r −n−2 fn,s r − 2ks nr −n−1 gn,
r}
n=0
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
K.-C. CHEN AND C.-L. LIN
∞
+ ( + )
+ ∞
2
s
{−ks2r −n fn,s r − (n + 1)r −n−2 ( fn,s ) + ks r −n−1 (gn,
) }
n=0
r −n fn,s r
(19)
n=0
s
S0,
=
∞
n=0
s
2 −n s
{n 2r −n−2 fn,s − 2ks nr −n−1 gn,
fn, + r −n−2 ( fn,s ) }
− ks r
∞
+ ( + )
s
−n−2 s {(1 − n)r −n−2 ( fn,s r ) + ks r −n−1 (gn,
( fn, ) }
r) + r
n=0
∞
+ 2
r −n fn,s (20)
n=0
s
S1,
r =
∞
n=0
+
s
{2ks nr −n−1 fn,s r + (n + 1)2r −n−2 gn,
r}
∞
s
2 −n s
{r −n−2 (gn,
gn, r } + 2
r ) − ks r
n=0
+ ( + )
∞
∞
s
r −n gn,
r
n=0
s
−n−1 s {2nks r −n−1 fn,s r − ks2r −n gn,
( fn, ) }
r − ks r
n=0
+ ( + )
∞
s
−n−2 s
{n(n + 2)r −n−2 gn,
(gn, ) }
r − (n + 2)r
(21)
n=0
and
s
S1,
=
∞
n=0
+
s
2 −n s
{2ks nr −n−1 fn,s + (n + 1)2r −n−2 gn,
gn, }
− ks r
∞
s
2
{r −n−2 (gn,
) } + n=0
∞
+ ( + )
n=0
∞
s
r −n gn,
n=0
s
−n−2 s
{−ks r −n−1 ( fn,s r ) − nr −n−2 (gn,
(gn, ) }
r) + r
(22)
It should be noted that (14) implies
p
p
p
p
s
s
s
s
S0, r = S0, = S1, r = S1, = S0,
r = S0, = S1, r = S1, = 0
Now, collect the terms of r 0 in (16), it admits that
p
p
0 = − k 2p f0, () + 2 f0, ()
= − 2 ( + 2)−1 f0, () + 2 f0, ()
p
p
= 2 ( + )( + 2)−1 f0, ()
p
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
EXPANSION THEOREM FOR 2D ELASTIC WAVES
p
which implies f0, () = 0. Similarly, evaluate the coefficients of r 0 in (18), (19 and (21),
we have
p
p
s
f0, () = g0, () = f0,s r () = g0,
r () = 0
(23)
From (23) and (8), the components u p and u s verify the following expansions
(1)
u p = H0 (k p r )
∞
p
n=1
(1)
∞
(1)
r −n Fn () + H1 (k p r )
(1)
p
r −n G n ()
p
n=1
p
+ H0 (k p r ) f0, r ()r̂ + H1 (k p r )g0, r ()r̂
(24)
and
(1)
u s = H0 (ks r )
∞
∞
(1)
r −n Fns () + H1 (ks r )
n=1
(1)
r −n G sn ()
n=1
(1)
s
ˆ
+ H0 (k p r ) f0,s ()ˆ + H1 (k p r )g0,
()
(25)
where u p and u s also satisfy the Helmholtz equation (6). Following the same arguments in
p
p
s () are
Reference [5] to u p and u s , respectively, these terms f0, r (), g0, r (), f0,s () and g0,
related to the radiation pattern by the formulas
p
p
p
f0, r () = [u ∞ () + u ∞ ( + )]/2
p
p
p
g0, r () = i[u ∞ () − u ∞ ( + )]/2
(26)
f0,s () = [u s∞ () + u s∞ ( + )]/2
s
s
s
g0,
() = i[u ∞ () − u ∞ ( + )]/2
Collecting the terms of r −1 in (16), (18), (19) and (21), it implies that
( + )k p (g0, r ) = (k 2p − 2 ) f1, = −( + 2)−1 ( + )2 f1, p
p
p
( + )k p ( f0, r ) = (2 − k 2p )g1, = ( + 2)−1 ( + )2 g1, p
p
p
s
2
2 s
−1
2 s
( + )ks (g0,
) = [( + 2)ks − ] f1, r = ( + ) f1, r
(27)
s
−1
2 s
( + )ks ( f0,s ) = [2 − ( + 2)ks2 ]g1,
r = − ( + ) g1, r
p
p
s () can be expressed
From (23), (26) and (27), we obtain that f1, (), g1, (), f1,s r () and g1,
r
p
in terms of the radiation pattern (u ∞ , u s∞ ). Now we are devoted to giving expressions for all
p
p
p
p
s . Therefore,
coefficients in terms of F0, (), G 0, (), F0,s (), G s0, (), f1, , g1, , f1,s r and g1,
r
p
s
all coefficients can be represented by means of the radiation pattern (u ∞ , u ∞ ). To begin with,
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
K.-C. CHEN AND C.-L. LIN
take (15), (17), (20) and (22) into consideration, it implies the following equalities:
( + ){(n 2 − 1) fn, r − 2k p (n + 1)gn+1, r − (n + 1)( fn, ) + k p (gn+1, ) }
p
p
p
p
+ {n 2 fn, r − 2k p (n + 1)gn+1, r + ( fn, r ) } = 0
p
p
p
(28)
( + ){2(n + 1)k p fn+1, r − k p ( fn+1, ) + n(n + 2)gn, r − (n + 2)(gn, ) }
p
p
p
p
+ {2k p (n + 1) fn+1, r + (n + 1)2 gn, r + (gn, r ) } = 0
p
p
p
(29)
s
s s {n 2 fn,s − 2ks (n + 1)gn+1,
+ ( fn, ) } + ( + ){(1 − n)( fn, r ) }
s
+ ( + ){( fn,s ) + ks (gn+1,
r) } = 0
(30)
s
2 s
s
s
{2ks (n + 1) fn+1,
+ (n + 1) gn, + (gn, ) } − ( + ){ks ( fn+1, r ) }
s
s
+ ( + ){(gn,
) − n(gn, r ) } = 0
(31)
for n0.
In view of (16), (18), (19) and (21), we have the identities in the following:
{n 2 fn, − 2k p (n + 1)gn+1, − k 2p fn+2, + ( fn, ) } + 2 fn+2, p
p
p
p
p
+ ( + ){(1 − n)( fn, r ) + k p (gn+1, r ) + ( fn, ) } = 0
p
p
p
(32)
{2k p (n + 1) fn+1, + (n + 1)2 gn, − k 2p gn+2, + (gn, ) } + 2 gn+2, p
p
p
p
p
+ ( + ){−k p ( fn+1, r ) − n(gn, r ) + (gn, ) } = 0
p
p
p
(33)
s
2 s
s ( + ){(n 2 − 1) fn,s r − 2ks (n + 1)gn+1,
r − ks fn+2, r − (n + 1)( fn, ) }
s
2 s
s
+ ( + )ks (gn+1,
) + {n fn, r − 2ks (n + 1)gn+1, r }
s
2 s
+ {( fn,s r ) − ks2 fn+2,
r } + fn+2, r = 0
(34)
s
2 s
s
2 s
2 s
{2ks (n + 1) fn+1,
r + (n + 1) gn, r + (gn, r ) − ks gn+2, r } + gn+2, r
s
2 s
s
+ ( + ){2(n + 1)ks fn+1,
r − ks gn+2, r − ks ( fn+1, ) }
s
s
+ ( + ){n(n + 2)gn,
r − (n + 2)(gn, ) } = 0
(35)
for n0.
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
EXPANSION THEOREM FOR 2D ELASTIC WAVES
More precisely, Equations (28)–(35) imply that
gn+1, r = ( + )[2k p ( + 2)(n + 1)]−1 {(n 2 − 1) fn, r − (n + 1)( fn, ) }
p
p
p
+ [2k p ( + 2)(n + 1)]−1 {n 2 fn, r + ( fn, r ) }
p
p
+ k p ( + )(gn+1, ) [2k p ( + 2)(n + 1)]−1
p
(36)
fn+1, r = −( + )[2k p ( + 2)(n + 1)]−1 {n(n + 2)gn, r − k p ( fn+1, ) }
p
p
p
− [2k p ( + 2)(n + 1)]−1 {(n + 1)2 gn, r + (gn, r ) }
p
p
− (n + 2)( + )(gn, ) [2k p ( + 2)(n + 1)]−1
p
s
gn+1,
= ( + )[2ks (n + 1)]
−1
{(1 − n)( fn,s r )
+ ( fn,s )
(37)
s
+ ks (gn+1,
r) }
+ [2ks (n + 1)]−1 {n 2 fn,s + ( fn,s ) }
(38)
s
−1
s
s
s
fn+1,
= ( + )[2ks (n + 1)] {ks ( fn+1, r ) − (gn, ) + n(gn, r ) }
s
s
− [2ks (n + 1)]−1 {(n + 1)2 gn,
+ (gn, ) }
(39)
fn+2, = −( + 2)[( + )2 ]−1 {n 2 fn, − 2k p (n + 1)gn+1, + ( fn, ) }
p
p
p
p
− ( + 2)(2 )−1 {(1 − n)( fn, r ) + k p (gn+1, r ) + ( fn, ) }
p
p
p
(40)
gn+2, = − ( + 2)[( + )2 ]−1 {2k p (n + 1) fn+1, + (n + 1)2 gn, }
p
p
p
− ( + 2)(2 )−1 {(gn, ) − k p ( fn+1, r ) − n(gn, r ) }
p
p
p
− ( + 2)(gn, ) [( + )2 ]−1
p
(41)
s
2 −1
2
s
s
fn+2,
r = ( + )[( + ) ] {(n − 1) fn, r − 2ks (n + 1)gn+1, r }
s
s + 2 [( + )2 ]−1 {n 2 fn,s r − 2ks (n + 1)gn+1,
r + ( fn, r ) }
s
s + ( + )[( + )2 ]−1 {ks (gn+1,
) − (n + 1)( fn, ) }
(42)
s
2
2 −1
s
2 s
s
gn+2,
r = [( + ) ] {2ks (n + 1) fn+1, r + (n + 1) gn, r + (gn, r ) }
s
s
+ ( + )[( + )2 ]−1 {2(n + 1)ks fn+1,
r + n(n + 2)gn, r }
s
s
− ( + )[( + )2 ]−1 {ks ( fn+1,
) + (n + 2)(gn, ) }
(43)
where n0.
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
K.-C. CHEN AND C.-L. LIN
Through the recursion formulas (36)–(43), all coefficients of u in (8) can be expressed by
p
p
p
p
s
F0, (), G 0, (), F0,s (), G s0, (), f1, , g1, , f1,s r and g1,
r inductively. Therefore, we can use
p
s
the radiation pattern (u ∞ , u ∞ ) to determine all coefficients of u in (8).
4. APPLICATION TO INVERSE PROBLEMS
¯ is
Let a sound-hard obstacle ∈ R2 be an open subset with C 2 boundary . We assume that R2 \
connected. Furthermore, we suppose that is contained in the open ball B R = B(0, R). It should
be noted that may consist of finitely many bounded domains. We are cared about the scattering
problem for the inhomogeneous isotropic elasticity system. Define the elastic tensor C = (Ci jkl ) by
Ci jkl = (x)i j kl + (x) jl ik + (x)il jk
(44)
where (x)>0 and (x) + (x)>0. Moreover, C(x) = C for |x|>R, where C is a homogeneous
isotropic elastic tensor.
To apply an Atkinson–Wilcox-type expansion for two-dimensional elastic wave, we consider
an inverse scattering problem in the following. Let u(x) ∈ C 2 satisfy
⎧
¯
Lu + 2 u = f in R2 \
⎪
⎪
⎪
⎪
⎨ T (D, )u = 0 on (45)
u(x) = u p (x) + u s (x)
⎪
⎪
⎪
⎪
⎩ lim √r (*u p /*r − ik u p ) = 0 and lim √r (*u s /*r − ik u s ) = 0
p
s
r →∞
r →∞
¯ and T (D, ) is boundary traction operator
where Lu = div(C(x)∇u), f ∈ L 2comp (R2 \)
defined by
(T (D, ))ik = jl Ci jkl j *l
with being the unit outer normal of . Moreover, for any matrix E = (E kl ), we define that
(C E)i j = kl Ci jkl E kl
4.1. Inverse problem
p
Reconstruct from the far-field pattern (u ∞ , u s∞ ) of the radiation solution u to (45) at a
fixed >0.
To apply an Atkinson–Wilcox-type expansion for two-dimensional elastic wave derived above,
we shall reconstruct by means of the far-field pattern. Some three-dimensional obstacle scattering
problems were proved in [7, 8]. Nevertheless, the situation becomes much harder when elastic
fields are involved, since we must deal with two wave solutions propagating at different phase
velocities. Dassios and Rigou [9] established some basic results containing a Runge’s-type theorem
in three-dimensional elastic scattering. Recently, Alves and Kress [10] applied the so-called linear
sampling methods to three-dimensional elastic obstacle scattering. Using the same idea, Arens
[11] established some results in the two-dimensional inverse elastic wave scattering. On the other
hand, the probe method predicts whether a point descending from the boundary of the body
along a given path hits the discontinuity surface and when it occurs. This method was applied to
reconstruct the obstacles in References [7, 8]. However, the approaches developed in References
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
EXPANSION THEOREM FOR 2D ELASTIC WAVES
[7, 8] essentially rely on an Atkinson–Wilcox-type expansion for three-dimensional Helmholtz
equation. Having an Atkinson–Wilcox-type expansion for two-dimensional elastic wave at hand,
the same arguments in References [7, 8, 12, 13] can be applied to the two-dimensional inverse elastic
wave scattering.
Algorithm of the reconstruction
Step 1: We assume that 0 is not a Dirichlet eigenvalue of L + 2 in B R . Having an Atkinson–
Wilcox-type expansion for two-dimensional elastic wave derived in Section 3, we can solve u p and
p
u s in R2 \ B R from the far-field pattern (u ∞ , u s∞ ) of the radiation solution u to (45). Therefore,
s
the Green function G(x, y) of u on |x| = |y| = R can be determined, where G(x, y) satisfies
G(x, y) f (y) dy
u s (x) =
¯
R2 \
Step 2: As the same arguments in Reference [13] where the inhomogeneous anisotropic elasticity
system is considered, we show that the Dirichlet-to-Neumann map on *B R can be constructed
by the measurements G(x, y) on |x| = |y| = R. For the reader’s convenience, the similar notations in Reference [13] will be used. Define the Dirichlet-to-Neumann map : H 1/2 (*B R ) →
H −1/2 (*B R ) by
(g) = T (D, x/x)v|* B R
where v is the solution of
⎧
¯
⎪
Lv + 2 v = 0
in B R \
⎪
⎨
T (D, )v = 0
on ⎪
⎪
⎩ v = g ∈ H 1/2 (*B ) on *B
R
R
Let v e be the solution of
⎧
⎪
Lv e + 2 v e = 0
⎪
⎨
(46)
in R2 \B R
v e = g ∈ H 1/2 (*B R )
⎪
⎪
⎩ ve
on *B R
(47)
satisfies the radiation conditions
Therefore, we can define the Dirichlet-to-Neumann map e : H 1/2 (*B R ) → H −1/2 (*B R ) by
e (g) = T (D, x/x)v|* B R
−1 (R2 \)
¯ by
Now for g ∈ H 1/2 (*B R ), define Mg ∈ Hcomp
1
¯
Mg , = g, |* B R ∀ ∈ Hcomp
(R2 \)
On the other hand, let vg be the scattering solution of (45) with the source term—Mg . Define
g(x) = vg |* B R
i.e.
g(x) = −
Copyright q
2006 John Wiley & Sons, Ltd.
* BR
G(x, y)g(y) ds,
x ∈ *B R
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma
K.-C. CHEN AND C.-L. LIN
The following key lemma verifies that the Dirichlet-to-Neumann map can be constructed by
G(x, y) on |x| = |y| = R.
Lemma 2 (Nakamura et al. [13, Lemma 5.3])
− e is injective and ( − e ) = I .
It should be noted that is determined by G(x, y) and e can also be constructed. With the
aid of Lemma 2, we can get by
= e − −1
Step 3: We convert our problem to construct by the Dirichlet-to-Neumann map . The
problem is the same as Inverse Problem 2 on p. 209 of Reference [8] and Section 5.2 on p. 608
of Reference [13]. We will not repeat the proof again and refer the readers to the above articles.
ACKNOWLEDGEMENTS
We would like to thank Professor Jenn-Nan Wang for bringing the problem to our attention and for
many stimulating discussions. The authors are partially supported by the National Science Council
and National Center for Theoretical Science of Taiwan.
REFERENCES
1. Atkinson F-V. On Sommerfeld’s ‘radiation condition’. Philosophical Magazine 1949; 40:645–651.
2. Wilcox C-H. A generalization of theorems of Rellich and Atkinson. Proceedings of the American Mathematical
Society 1956; 70:271–276.
3. Wilcox C-H. An expansion theorem for electromagnetic fields. Communications on Pure and Applied Mathematics
1956; 9:115-134.
4. Dassios G. The Atkinson–Wilcox expansion theorem for elastic waves. Quarterly Journal of Applied Mathematics
1988; 46:285–299.
5. Karp S-N. A convergent ‘farfield’ expansion for two-dimensional radiation functions. Communications on Pure
and Applied Mathematics 1961; 19:427–434.
6. Watson G-N. Theory of Bessel functions. Cambridge University Press: Cambridge, MA, 1957.
7. Ikehata M. Reconstruction of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems
1998; 14:949–954.
8. Ikehata M. Reconstruction of an obstacle from boundary measurement. Wave Motion 1999; 30:205–223.
9. Dassios G, Rigou Z. On the density of traction tracesin scattering of elastic waves. SIAM Journal on Applied
Mathematics 1993; 53(1):141–153.
10. Alves C-J-S, Kress R. On the far-field operator in elastic obstacle scattering. IMA Journal of Applied Mathematics
2002; 67(1):1–21.
11. Arens T. Linear sampling methods for 2D inverse elastic wave scattering. Inverse Problems 2001; 17:1445–1464.
12. Ikehata M. Reconstruction of inclusion from boundary measurement. Journal of Inverse Ill-posed Problem 2002;
10:37–65.
13. Nakamura G, Uhlman G, Wang J-N. Reconstruction of cracks in an inhomogeneous anisotropic medium using
point source. Advance in Applied Mathematics 2005; 34:591–615.
Copyright q
2006 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. (in press)
DOI: 10.1002/mma