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...

<br /> u(x,0)=\phi(x)=G(ct)<br />
<br /> u_t(x,0)=\psi(x)=cG&#039;(ct)<br />

After this I got stuck.

 

[hunt_mat]

Assume that the general solution is:

<br /> u(t,x)=f(x-t)+g(x+t)<br />

and use the two initial condition.