Uniqueness and existence of an outgoing solution of Helmholtz problem using Green's formula

Layouni Amamou Manel

Layouni Amamou Manel Faculty of Science Mathematics Physics and Natural of Tunis, University Tunis-El Manar, Tunisia

Keywords: Helmholtz problem; Green formula; outgoing solution; acoustic wave; sound-soft obstacle

Abstract

In this article, first we present a new approach based on Green's formula, to describe the uniqueness and existence of the solution of the Helmholtz equation. By imposing at infinity the outgoing wave condition or also called Sommerfeld radiation condition, we show how it is possible to define in a natural way an outgoing solution of the Helmholtz equation based on physical arguments. Then, we resolve the exterior problem, given by the scattering of time-harmonic acoustic wave by sound-soft obstacle, which leads to find a radiating solution to the Helmholtz equation.

Downloads

Download data is not yet available.

References

S. Amini and D. T. Wilton, An investigation of boundary element methogs for the exterior acoustic problem, Comput. Methods Appl. Mech. Engrg., 54 (1986), pp.49-65.

K. E. Atkinson and G. Chandler, The collocation method for solving the radiosity equation for unoccluded surfaces, J. Integral Equations App., 10 (1998), pp. 253-290.

K. E. Atkinson and D. Chien, Piecewise polynomial collocation for boundary integral equations, SIAM J. Sci. Comput., 16 (1995), pp. 651-681.

H. Barucq, J. Diaz and S. Tordeaux, Introduction a l’analyse mathematique de l’equation de Helmholtz, Projet magique 3D, INRIA Bordeaux sud- oust (2012).

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, (1983).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer-Verlag, New York, (1998).

B. Gieseke, Zum Dirichletschen prinzip fur selbstadjungierte elliptische differentialoperatoren. Math. Z. 68, 54-62 (1964).

P. Hahner, scattering by media. In Scattering (Pike and Sabatier, eds). Academic Press, new York, 74-94 (2002).

L. M. Levine, A uniqueness theorem for the reduced wave equation. Comm. Pure Appl. Math. 17, 147-176 (1964)

P.A. Martin, Multiple Scattering, Interaction of Time-Harmonic Waves with N obstacles, Encyclopedia of Mathematics and its Appliction 107, Cambridge, (2006).

F. Rellich, Uber das asymptotische verhalten der Losungen van delta u+lambda u=0, in unendlichen Gebieten. Jber. Deut sch. Math. Verein. 53, 57-65(1943).