On scattering by bodies with narrow channels

Boundary value problems are considered for the Helmholtz equation outside a domain having a finite cavity of small 'radius' $\varepsilon\ll1$. It is shown that in the case of Neumann boundary conditions the analytic continuation of the solution of the boundary value problem has poles with small imaginary part. The method of matching asymptotic expansions is used to construct asymptotic expressions for these poles with respect to the small parameter $\varepsilon$, and it is shown that they have a resonance character.