Consider a two dimensional flexible membrane whose equilibrium position occupies a region in the horizontal x y-plane. Suppose this membrane vibrates up and down with u(x, y,t) denoting the vertical displacement of the point (x, y) of the membrane at time t. The vertical displacement function u(x, y,t) satisfies the two dimensional wave equation
∂^2u/ ∂t^2 +c^2 ∇^2 u
where c^2 =T/ρ for T and ρ the membranes tension and density respectively. Suppose that the rectangular membrane 0 ≤ x ≤ a, 0 ≤ y ≤ b is released from rest with given initial displacement u(x, y, 0) = f (x, y). Further, suppose that we have the following boundary conditions u(0, y,t) = u(a, y,t) = u(x, 0,t) = u(x,b,t) = 0, and ut(x, y, 0) = 0. Find an expression for the vertical displacement of the membrane.