Consider a vibrating quarter-circular membrane, 0 < r < a, 0 < θ < π/2, with u = 0 on the entire boundary. [Hint: You may assume without derivation that λ > 0 and that product solutions u(r, θ, t) = φ(r, θ)h(t) = f(r)g(θ)h(t) satisfy ∇2φ + λφ = 0 dh dt = −λkh d2g dθ2 = −μg r d dr r df dr + (λr2 − μ)f = 0.]

Solve the initial value problem if u(r, θ, 0) = g(r, θ), ∂u ∂t (r, θ, 0) = 0.