Two-Dimensional Wave Equation
Response to the following questions:
1. Find the solution to the two-dimensional wave equation
∂2u/∂t2=∂2u/∂x2+∂2u/∂y2,0
initial conditions u(x,y,0)=sin2(πx)sin (πy) and ∂u/∂t(x,y,0)=0 and
boundary condition u(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=0
2. Solve the two-dimensional wave equation for a quarter-circular membrane,
0
initial conditions u(r,Θ,0)=a(r,Θ), and ∂u/∂t(r,Θ,0)=0
The boundary condition is such that u=0 on the entire boundary.
3. Consider Laplace's equation ∂2u/∂t2=c2(∂2u/∂x2+∂2u/∂y2)-k(∂u/∂t) with k>0.
a. Give a brief physical interpretation of this equation.
b. Suppose that u(x,y,t)=f(x)g(y)h(t)
What ordinary differential equations are satisfied by f, g, and h?