What are the Boundary Conditions for Solving a Wave Equation?

[blalien]

Homework Statement

The problem is to solve
\phi_{yy}-c^2 \phi_{xx} = 0
\phi_y (x,0) = f'(x), x>0
\phi_x (0,y) = \phi(0,y) = 0, y>0 or y<0

Homework Equations

The solution, before applying boundary conditions is obviously
\phi(x,y)=F(x+c y)+G(x-cy)

The Attempt at a Solution

I start with the general solution
\phi(x,y)=F(x+c y)+G(x-cy)
and apply the two vanishing boundary conditions
\phi(0,y)=F(c y)+G(-cy)=0 or
1) F(\omega)+G(-\omega)=0
\phi_x(0,y)=F'(c y)+G'(-cy)=0 or
2) F'(\omega)+G'(-\omega)=0
Take the derivative of equation 1:
F'(\omega)-G'(-\omega)=0

So F' and G' both vanish. Then how do we apply the first boundary condition?

 

[fzero]

The first boundary condition is (in your notation)

F'(\omega) - G'(\omega) = f'(\omega).

As long as G is neither odd nor even, there's no inconsistency.

 

[blalien]

The problem is that the two boundary conditions
F(\omega)+G(-\omega)=0 and
F'(\omega)+G'(-\omega)=0
imply that F(\omega) and G(\omega) are constant. This does create an inconsistency.

 

[blalien]

I missed a detail that the solution is symmetric with y. That solves the problem. Thanks anyway.