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.