Multiple Scattering of Water Waves by N Floating Bodies Using Boundary Integral Equation Methods
The use of boundary integral equation methods for solving exterior problems of scattering of time harmonic waves leads to a problem of nonuniqueness of the solutions of the relevant integral equations. This difficulty is inherent to the methods used. Various methods have been devised to overcome this problem. One of these methods consists in modifying the fundamental solution by adding to it a series of multipoles. With a mild condition on the coefficients of these multipoles, the problem of nonuniqueness is removed. In the context of water waves, Ursell [1,2] was the first to propose the modification of the fundamental solution for problems involving one floating cylindrical body. He proved uniqueness for the high frequencies. Sayer  did the same for the low frequencies. Later, Ursell  generalized the result for all frequencies. Martin  considered the case of one three-dimensional floating body. Later, Martin  treated the problem for two infinitely long floating cylinders. More recently, we extended Martin’s work  to the case of two three-dimensional floating bodies . Here, we generalize the case to N three-dimensional floating bodies.We found out that the conditions on some of the coefficients of the multipoles depend on the number, N, of the scatterers. It is worth noting that Martin  treated the case of N scattering bodies for three-dimensional acoustic waves. He found out that the conditions on all the coefficients depended on N.
KeywordsIntegral Equation Fundamental Solution Multiple Scattering Water Wave Boundary Integral Equation
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