Forced nonlinear oscillations of elastic membranes

An initial-boundary-value problem is considered for the spatially two-dimensional damped Boussinesq equation with a forcing term. The problem in question is argued to serve as a model for describing small nonlinear oscillations of a circular membrane under the influence of acoustic pressure. Eigenfunction expansion method is used for constructing the global-in-time solution of the problem in question. Existence and uniqueness follow from the construction. Long-time asymptotics is computed on the basis of the obtained series representation. Numerical simulations are conducted. The algorithm shows excellent convergence properties.