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## Homework Statement

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

[tex]u(x,0)=\Phi(x)[/tex] and [tex]u_t(x,0)=\Psi(x)[/tex] for [tex]-\infty\leq x \leq \infty[/tex]

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

[tex]u(x,t)=G(x+ct)[/tex]

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

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 [tex]u(x,t)=G(x+ct)[/tex]

So with the initial conditions...

[tex]

u(x,0)=\phi(x)=G(ct)

[/tex]

[tex]

u_t(x,0)=\psi(x)=cG'(ct)

[/tex]

After this I got stuck.

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