Wave Equation traveling to the left

[roldy]

Homework Statement

Derive the general nontrivial relation between $$\phi$$ and $$\psi$$ which will produce a solution to $$u_{tt}-u_{xx}=0$$ in the xt-plane satisfying

$$u(x,0)=\Phi(x)$$ and $$u_t(x,0)=\Psi(x)$$ for $$-\infty\leq x \leq \infty$$

and such that u consists solely of a wave traveling to the left along the x-axis.

Homework Equations

d'Alemberts Formula

The Attempt at a Solution

So for a wave traveling to the left, the equation takes the form of

$$u(x,t)=G(x+ct)$$

Is this correct? It was my assumption that wave equations have two parts to them, i.e $$F(x-ct)$$ and $$G(x+ct)$$.

I think I know the general procedure of doing this, although I am probably wrong as I am just guessing here. We were never shown in class how to derive relations between the functions of the initial conditions.

I think that I would probably work in reverse from $$u(x,t)=G(x+ct)$$

So with the initial conditions...

$$u(x,0)=\phi(x)=G(ct)$$
$$u_t(x,0)=\psi(x)=cG'(ct)$$

After this I got stuck.

 

[hunt_mat]

Assume that the general solution is:

$$u(t,x)=f(x-t)+g(x+t)$$

and use the two initial condition.