Asymptotics of solutions to the wave equation in a domain with a small hole
Dmitrii Vladimirovich Korikov · St Petersburg Mathematical Journal · 2015
In a bounded domain with a small hole (at all times $t\in \mathbb {R}$), the wave equation is considered with homogeneous Dirichlet condition on the boundary. The asymptotics of the solution as the diameter of the hole tends to 0 is deduced. To describe the behavior of long waves, the method of compound asymptotic expansions is used. The contribution of short waves (the wavelength is smaller than the diameter of the 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.