Layer potentials and boundary value problems for the helmholtz equation in the complement of a thin obstacle

Abstract

Our purpose is to show in a precise manner the mathematical approach of the problem of the acoustic diffraction by an infinitely thin screen. The classical equations of acoustics are transformed into integral equations. The sound field diffracted by the obstacle is described by a double layer potential, the density of which is equal to the step of the potential across the screen. To avoid the mathematical difficulties, the infinitely thin screen will be considered as the limit of a sequence of obstacles with finite thickness. On such obstacles, the operators can be defined, thanks to the theory of Pseudo‐differential operators and Pseudo‐Poisson kernels. Then a limiting process is used and gives existence and unicity of the desired solutions in Sobolev spaces.

On the Asymptotic Convergence of Collocation Methods With Spline Functions of Even Degree

“…r Au = ß, f = z(r) on I\ have been used more recently in acoustics and electromagnetic fields and corresponding numerical treatments, [16], [17], [18], [35]. Employing the Cauchy-Riemann equations and integration by parts, (2.6) can be rewritten as whose principal part is given bỹ…”

“…For this equation preliminary convergence results can be found in [1], [2], [5], [14] and [33]. Our strongly elliptic systems with convolutional principal part contain, in addition, systems of integro-differential equations [3] (see [7]) with constant coefficients, certain singular integral equations, in particular, those of plane elasticity [7,Appendix], [24], [25], [26], [34], Fredholm integral equations of the second kind [6], [8], [11], [12], [13], [17], [27], [35], and also the integro-differential operator of Prandtl's wing theory [16], [17], [18], [24], [35].…”

The Neumann problem for the 2‐D Helmholtz equation in a domain, bounded by closed and open curves

“…Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [1][2], [5][6][7][8] and problems in the exterior of open arcs [5], [9][10][11] were treated separately, because different methods were used in their analysis.…”

“…Previously the Neumann problem in the exterior of an open arc was reduced to the hypersingular integral equation [9][10] or to the infinite algebraic system of equations [11], while the Neumann problem in domains bounded by closed curves was reduced to the Fredholm equation of the second kind [1], [6][7][8] [3][4], where the problems in the exterior of open curves were reduced to the Fredholm integral equations using the angular potential.…”

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

“…Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [2], [13]- [17] and problems in the exterior of cuts (cracks) [14,16], [18]-[20] were treated separately, because different methods were used in their analysis. Previously the Neumann problem in the exterior of a cut was reduced to a hypersingular integral equation [14,16,18,19] or …”