The authors study the asymptotic properties of the solution to the Helmoholtz equation with Neumann boundary conditions in a dumbbell-type domain in the regime when the "handle'' is thin and tightening to a curve. A mathematical analysis is done for the model problem posed in the half-plane with an infinite thin straight channel. It is proved that the solution of such a perturbed problem converges to the solution of the limiting problem for the same equation posed in the whole half-plane. Optimal estimates for the convergence rate are obtained in various norms. The authors also construct one more approximation for the solution of the perturbed problem which takes into account the presence of the channel. It is shown that this approximation is better in the sense that the estimates for the difference between this approximation and the "perturbed'' solution are smaller in order than similar estimates for the limiting solutions. The authors also conjecture that the last mentioned estimates are optimal; this conjecture is supported by a series of numerical results.