Solve u t = k1 2u x2 + k2 2u y2 on a rectangle (0 < x < L, 0 <y subject to the initial

Chapter 7, Problem 7.3.4

(choose chapter or problem)

Consider the wave equation for a vibrating rectangular membrane \((0<x<L, 0<y<H)\)

\(\frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)\)

subject to the initial conditions

\(u(x, y, 0)=0 \text { and } \frac{\partial u}{\partial t}(x, y, 0)=\alpha(x, y)\).

[Hint: You many assume without derivation that the product solutions \(u(x, y, t)=\phi(x, y) h(t)\) satisfy \(\frac{d^{2} h}{d t^{2}}=-\lambda c^{2} h\) and the two-dimensional eigenvalue problem \(\nabla^{2} \phi+\lambda \phi=0\), and you may use results of the two-dimensional eigenvalue problem.]

Solve the initial value problem if

(a) \(u(0, y, t)=0, \quad u(L, y, t)=0, \quad \frac{\partial u}{\partial y}(x, 0, t)=0, \quad \frac{\partial u}{\partial y}(x, H, t)=0\)

(b) \(\frac{\partial u}{\partial x}(0, y, t)=0, \quad \frac{\partial u}{\partial x}(L, y, t)=0, \quad \frac{\partial u}{\partial y}(x, 0, t)=0, \quad \frac{\partial u}{\partial y}(x, H, t)=0\)

(c) \(u(0, y, t)=0, \quad u(L, y, t)=0, \quad u(x, 0, t)=0, \quad u(x, H, t)=0\)