Helmholtz interior problem: coupling a modal expansion and an integral representation Final version

Overview & Synopsis

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( 1/ζ ) > 0 on a part of Γ the area S of which is different from zero and g(k, y, ε) ∈ H 1/2 (Γ): ε is the damping of an elastic structure and ζ is the normalised acoustic impedance of the internal wall of the cavity. Parameters ε et 1/ζ are small and real. Thanks to a proper modal expansion, an integral relation between u, ζ, ε and g is built to the first order ϑ( 1/ζ , ε) in 1/ζ and ε, using the residues theorem. The result is the solution to problem (1). A less elaborate version is included in the publication [6] relating to the transmission of sound inside a cavity.