Waves in Random and Complex Media Vol. 18, No. 2, May 2008, 303–324 Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments Mei Song Tong and Weng Cho Chew ∗ Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA (Received 8 September 2007; final version received 7 October 2007) A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green’s tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics. 1. Introduction The study of elastic or acoustic wave behaviour requires solving the corresponding wave equa- tions. These equations can be in the form of a partial differential equation (PDE) or a boundary integral equation (BIE). It is very clear that the acoustic wave equation is a special case of the elastic wave equation and they can be solved in a unified form as a PDE [1–3]. Conventionally, however, the elastic wave BIE and the acoustic wave BIE are treated differently [4, 5]. This is because the acoustic wave BIE has a simpler form and it is unnecessary to resort to solving the full-fledged elastic wave BIE in most cases. In this work, we unify the acoustic wave BIE and elastic wave BIE and show that the acoustic wave case can be obtained as a special case of the full elastic wave BIE. Our unified BIE provides a new approach to solve those problems in a more versatile manner. This is accomplished by introducing a shielding loss for a small shear modulus in the elastic wave BIE and solving it as an elastic wave problem. This shielding loss attenuates the shear wave in the medium. The advantage of this approach is that it requires the maintenance of only one numerical code that can account for both elastic wave physics and acoustic wave physics. This is especially important for modern day computational engineering where numerical codes for complex structures often require high maintenance due to the complexity of the codes. ∗ Corresponding author. Currently on leave from UIUC to serve at The University of Hong Kong. Email: w-chew@uiuc.edu ISSN: 1745-5030 print / 1745-5049 online C  2008 Taylor & Francis DOI: 10.1080/17455030701798960 http://www.informaworld.com