1D wave equation with dirac delta function as an external force.

1. Feb 13, 2009

scorpion990

Hey there!
I'm faced with this problem:
http://img7.imageshack.us/img7/4381/25686658nz9.png [Broken]

It's a 1D nonhomogeneous wave equation with a "right hand side" equaling to the dirac delta function in x * a sinusoidal function in t. I have to find its general solution with the constraints:
http://img177.imageshack.us/img177/8083/38983002rq3.png [Broken]

I know that the solution, by D'Alembert's theorem, is equal to a double integral over the external function. I showed this in the original problem.

I don't have a lot of experience with the dirac delta function. I know that integrals over [a,b] of the diract delta function = 1 if 0 is an element of [a,b]. The integral is 0 otherwise.

I tried switching the order of integration. Didn't help much. I don't think that integration by parts helps, either. Can somebody point me in the right direction?

Thanks!

Last edited by a moderator: May 4, 2017
2. Feb 17, 2009

matematikawan

Just curious with your initial conditions: u(x,0)= ut(x,0) = 0.
Since the initial position and velocity are zero I presume the solution must be
u(x,t) = 0.

Can't be that simple!

3. Feb 17, 2009

scorpion990

It's not. It's not a homogeneous equation.

4. Feb 17, 2009

matematikawan

Yes you are right the equation is nonhomogeneous. Silly me. :shy:

Let me try to integrate that delta function.
Make the substitution $$\theta=\varsigma-x$$.

$$Integral = \int_{-c(t-\tau)}^{c(t-\tau)} \delta (\theta +x) d\theta = 1 \\\ if \\\\ |x| < c(t-\tau)$$.

Hence
$$\\\\\\u(x,t)=\frac{1}{\omega}(1-cos(\omega t))$$.

5. Feb 17, 2009

scorpion990

Hmm.... But that function isn't dependent on x. That goes against my geometrical intuition of the problem...

6. Feb 18, 2009

matematikawan

So the problem is not simple ha! I give up. I thought it is only an integration problem and can be solve quite easily. I'm wrong. Really sorry for the expectation.

Good that you have physical interpretation of a solution. I should have checked my solution first.

Any possibility solving the original wave equation using the Fourier or Laplace transforms? Is your x from -inf to +inf ?

7. Feb 18, 2009

scorpion990

Yes. x can range from -inf to +inf. But the professor specifically mentioned that no question in the entire course will require Fourier transforms.

8. Feb 19, 2009

scorpion990

I asked the professor today, and he gave me the hints I needed to figure it out. Thanks anyway guys.