1. Find the solution to the two-dimensional wave equation
ð2u/ðt2= ð2u/ðx2+ð2u/ðy2, 0
boundary conditions 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
The boundary condition is such that u=0 on the entire boundary.
3. Consider Laplace's equation
ð2u/ðt2 = c2(ð2u/ðx2+ð2u/ðy2)-kðu/ðt 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?