Asymptotics of solutions to wave equation in domain with a small hole

In a cylinder Q(ε) = {(x:, t) : x ∈ Ώ(ε), t ∈ ℝ} (whose section Ω(ε) is a domain in R with a small hole) we consider the wave equation U_tt - ΔU = F under the condition U = 0 on ∂Q(ε). We derive the asymptotics of a solution as the diameter of the hole tends to 0. To describe the behavior of long waves, we use the method of compound asymptotic expansions. The contribution of short waves (the wavelength is smaller than the diameter of hole) to the energy of the solution is negligible due to the smoothness of the right-hand side of the wave equation with respect to time.