Wave equation and fourier transformation

[prehisto]

Homework Statement

u_tt=a²u_xx
Initial conditions:
1)When t=0,u=H,1<x<2 and u=0,x[itex]\notin[/itex](1<x<2)
2)When t=0,u_t=H,3<x<3 and u=0,x[itex]\notin[/itex](3<x<4)

The Attempt at a Solution

So I transformed the first initial condition
[itex]\hat{u}[/itex]=1/[itex]\sqrt{2*\pi}[/itex] [itex]\int[/itex] Exp[-i*[itex]\lambda[/itex]*x)*H dx=
Hi/[itex]\sqrt{2*\pi}[/itex][itex]\lambda[/itex])[Exp(-i*[itex]\lambda[/itex]2)-Exp(-i*[itex]\lambda[/itex])]

integration boundaries are from x=1 to x=2

This condiotions is clear.

Now i have to deal with the 2nd.
Thats the problematic one.

My thought is:
du/dt=[itex]\hat{u}[/itex],only with proper boundaries.
Then maybe i can find the solution to this DE,and it would be my transformed boundary condition?

 

[strangerep]

OK, you really need to put more effort into writing your post. I can only guess that you're looking for a linear combination of solutions of the wave eqn that satisfies the given boundary conditions.

If that is the question, then...

1) Write out your general solution of the wave equation (carefully).

2) Show in more detail how you're trying to restrict it using the boundary conditions. I don't see you got where you did with the 1st boundary condition. (Since you didn't show enough of your work, it's difficult for me to guess where your mistakes begin.)

 

[dauto]

If [tex]\hat u(k,t) = \frac{1}{\sqrt{2\pi}} \int u(x,t) e^{-ikx} dx,[/tex] than [tex]\hat u_t(k,t) = \frac{1}{\sqrt{2\pi}} \int u_t(x,t) e^{-ikx} dx.[/tex]