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Wave Equation traveling to the left

  • Thread starter roldy
  • Start date
  • #1
232
1

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.
 
Last edited:

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,739
18
Assume that the general solution is:

[tex]
u(t,x)=f(x-t)+g(x+t)
[/tex]

and use the two initial condition.
 

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