Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions (original) (raw)
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Proceedings Mathematical Sciences, 1996
The Green's function solution of the Helmholtz's equation for acoustic scattering by hard surfaces and radiation by vibrating surfaces, lead in both the cases, to a hyper singular surface boundary integral equation. Considering a general open surface, a simple proof has been given to show that the integral is to be interpreted like the Hadmard finite part of a divergent integral in one variable. The equation is reformulated as a Cauchy principal value integral equation, but also containing the potential at the control point. It is amenable to numerical treatment by conventional methods. An alternative formulation in the better known form, containing the tangential derivative of the potential is also given. The two dimensional problem for an open arc is separately treated for its simpler feature.
Mathematical Methods in the Applied Sciences
We consider the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second-order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of singular localised boundary-domain integral equations. Fredholm properties of the corresponding localised boundary-domain integral operator are studied and its invertibility is established in appropriate Sobolev-Slobodetskii (Bessel potential) spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.
arXiv (Cornell University), 2018
We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of {\it localised boundary-domain singular integral equations}. Fredholm properties of the corresponding {\it localised boundary-domain integral operator} is studied and its invertibility is established in appropriate Sobolev-Slobodetskii and Bessel potential spaces,which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.
Existence and uniqueness of solutions for acoustic scattering over infinite obstacles
Journal of Mathematical Analysis and Applications, 2008
The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂ D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models threedimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.
Direct Boundary Integral Equations Method for Acoustic Problems in Unbounded Domains
We investigate some aspects of the so called direct boundary integral equation method in acoustic scattering theory. It is well known that by the direct approach the uniquely solvable exterior boundary value problems for the Helmholtz equation can not be reduced to the boundary integral equations which are uniquely solvable for arbitrary value of the frequency parameter. This implies that for such resonant frequencies the corresponding integral operators are not invertible and consequently solutions to the nonhomogeneous integral equations are not defined uniquely. They are defined modulo a linear combination of the elements of the null spaces of the corresponding integral operators. In the paper, it is shown that among the infinitely many solutions of the corresponding integral equations there is only one solution which has a physical meaning and corresponds either to the boundary trace of the unique solution to the exterior problem or to the boundary trace of its normal derivative...
A direct boundary integral equation method for the acoustic scattering problem
Engineering Analysis with Boundary Elements, 1993
This paper presents a direct boundary integral equation method for solving the exterior Neumann problem of the Helmholtz equation which is a mathematical formulation of the acoustic scattering problem at a perfectly hard body. It is proved that the integral equation obtained from the Helmholtz representation is equivalent to the original boundary value problem and it has a unique solution in a suitable Sobolev space in the framework of pseuo-differential operators. Moreover, the numerical treatment and error estimate are given by using a Galerkin approximation.
Journal of Mathematical Physics, 1998
wherek i ϭk(sin i ,0,Ϫcos i ), i istheangleofincidence͑itistheanglebetweenthedirectionof propagationandthenegativez-axis͒, xϭrxϭr͑sin cos,sin sin,cos ͒, and(r,,) are spherical polar coordinates: xϭr sin cos, yϭr sin sin and zϭr cos. All thefieldsu tot ,u inc ,andusatisfytheHelmholtzequation,