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 Chen
1and Ching-Lung Lin
2,∗,†1Department of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan 2Department 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. Copyrightq 2006 John Wiley & Sons, Ltd.
KEY WORDS: Atkinson–Wilcox; Helmholtz equation
1. INTRODUCTION
Let u(x) ∈ C2be a solution of the scalar Helmholtz equation u + k2u= 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−1eikr ∞
n=0
fn(, )r−n (1)
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: cllin@math.ccu.edu.tw
Contract/grant sponsor: National Science Council and National Center for Theoretical Science
Received 23 February 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−1exp(ikr))/dr = r−1exp(ikr)(ik − r−1) (2) As for two dimensions, the fundamental solution for scalar Helmholtz equation is H0(1)(kr), where H0(1)(z) is the Hankel function of the first kind, of order zero. From Reference [6, p. 74], we found that
d(H0(1)(kr))/dr = −k H1(1)(kr)
where H1(1)(z) is the Hankel function of the first kind, of order 1. Unfortunately, H1(1)(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, ) ∈ C2to the scalar Helmholtz equation in the region r>a admits the following expansion:
u= H0(1)(kr) ∞ n=0 r−nFn() + H1(1)(kr) ∞ n=0 r−nGn() (3)
where the series converges absolutely and uniformly in ra + ε>a and can be differentiated term by term with respect to r and . The coefficients F0 and G0 are determined from the formulas
F0() = [ f0() + f0( + )]/2
−iG0() = [ f0() − f0( − )]/2
(4) where f0() is the so-called far-field pattern or amplitude. Furthermore, Fn and Gn for n>0 are
constructed, respectively, from F0 and G0.
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)(ancos n + bnsin n)
and writing Hn(1)(kr) as a linear combination of H0(1)(kr) and H1(1)(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 upand the transverse part us. We then apply Karp’s results to upand us, 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) + 2u= 0 (5)
Copyright q 2006 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. (in press)
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
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Copyright q 2006 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. (in press)
