Helmholtz interior problem: coupling a modal expansion and an integral representation. (Q557118)

The author deals with the following Helmholtz interior problem \[ \Delta u+ k^2 u= 0\quad\text{in }\Omega,\tag{1} \] \[ \Biggl({\partial u\over\partial n}- {i_k\over\zeta}\Biggr)|_\Gamma= g\quad\text{in }\Gamma= \partial\Omega,\tag{2} \] where \(\Omega\) is an open domain in \(\mathbb{R}^3\). The problem (1) and (2) possesses a unique solution in \(H^1(\Omega)\), if \(k\in\mathbb{R}\), \(\text{Re}({1\over\zeta})> 0\) on a part of \(\Gamma\) the area of which is different from zero and \(g(k,y,\varepsilon)\in H^{1/2}(\Gamma)\). Here \(\varepsilon\) is the damping of an elastic structure and \(\zeta\) is the normalized acoustic impedance of the internal wall of the cavity. 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 \(\Gamma\), \(\zeta\), \(\varepsilon\) and \(g\) is built to the first order \(\theta({1\over\zeta}, \varepsilon)\) in \({1\over\zeta}\) and \(\varepsilon\), using residues theorem.