Reference no: EM13899158

1. Find the solution to the two-dimensional wave equation

ð²u/ðt²= ð²u/ðx²+ð²u/ðy², 0<x<1, 0<y<1 with initial conditions u(x,y,0)=sin2(Πx)sin(Πy) and ðu/ðt(x,y,0)=0 and

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<r<1, ,0<Θ<Π/2 with the initial condition u(r,Θ,0) = a(r,Θ) and ðu/ðt(r,Θ,0)= 0

The boundary condition is such that u=0 on the entire boundary.

3. Consider Laplace's equation

ð²u/ðt^{2 = c2(ð2u/ðx2+ð2u/ðy2)-k}ðu/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?