On the poles of an acoustic resonator (Q1337907)

We prove that under some conditions there exists a unique resonator pole, its order is equal to one, and a unique eigenfunction is associated with it. The existence of exactly two vanishing first-order poles is shown, and asymptotic expansions of these poles into power series with respect to a small parameter (the channel cross-section diameter) are constructed. We give a representation of the solution to the boundary value problem which allows us to justify asymptotic expansions of the poles and the solution.