Problème de Helmholtz intérieur : formulation modale couplée à une représentation intégrale

Let Ω be an opened domain of R3 with a boundary Γ. The problem numbered (1) in the text has a unique solution in H1(Ω), if k∈R, Re(1ζ)>0 on a part of Γ the area S of which is different from zero and g(k,y,ɛ)∈H12(Γ). ɛ is the damping of an elastic structure and ζ is the normalised acoustic impedance of the internal wall of the cavity. ɛ and 1ζ are small parameters. Thanks to a proper modal expansion and a mean over a narrow band of wave number k, an integral relation between the trace of u on Γ,ζ,ɛ and g is built to the first order ϑ(1ζ,ɛ) in 1ζ and ɛ, using the residues theorem. It is not an equivalent equation to the problem, but just a step towards its resolution, which will be published in future papers. To cite this article: D. Brenot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).