Question

7.7.3. 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 drrdf dr+(λr2 −μ)f =0 .]

*(a) Determine an expression for the frequencies of vibration.

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

(c) Solve the wave equation inside a quarter-circle, subject to the conditions
∂u ∂r
(a,θ,t)=0,u (r,0,t)=0 ur, π 2,t=0,u (r,θ,0) = 0 ∂u ∂t (r,θ,0) = β(r,θ)

Answer 1