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

In summary, the problem is nonhomogeneous and the solution is not simple. The equation is nothomogeneous and the solution is dependent on x.
  • #1
scorpion990
86
0
Hey there!
I'm faced with this problem:
http://img7.imageshack.us/img7/4381/25686658nz9.png

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

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!
 
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  • #2
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
It's not. It's not a homogeneous equation.
 
  • #4
Yes you are right the equation is nonhomogeneous. Silly me. :shy:

Let me try to integrate that delta function.
Make the substitution [tex]\theta=\varsigma-x[/tex].

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

Hence
[tex]\\\\\\u(x,t)=\frac{1}{\omega}(1-cos(\omega t))[/tex]. :smile:
 
  • #5
Hmm... But that function isn't dependent on x. That goes against my geometrical intuition of the problem...
 
  • #6
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
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
I asked the professor today, and he gave me the hints I needed to figure it out. Thanks anyway guys.
 

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

1. What is the 1D wave equation with Dirac delta function as an external force?

The 1D wave equation with Dirac delta function as an external force is a mathematical model used to describe the behavior of a wave in a one-dimensional medium that is subject to an external force represented by a Dirac delta function. This equation takes into account the effects of both the wave's natural propagation and the sudden impact of the external force.

2. What is the physical significance of a Dirac delta function in this equation?

A Dirac delta function is a mathematical function that represents an infinitely narrow pulse of infinite magnitude. In the context of the 1D wave equation, it represents a sudden and localized impact on the wave that is applied at a specific point in time and space. This external force can be interpreted as a physical event such as an explosion or an impulse applied to a string.

3. How does the presence of a Dirac delta function affect the solution to the 1D wave equation?

The presence of a Dirac delta function as an external force in the 1D wave equation causes a sudden change in the amplitude and velocity of the wave at the point of impact. This results in a discontinuity in the solution to the equation, with the wave exhibiting a jump or shock at the point where the Dirac delta function is applied. The behavior of the wave after the impact is determined by the properties of the medium and the nature of the external force.

4. Can the 1D wave equation with Dirac delta function be used to model real-world phenomena?

Yes, the 1D wave equation with Dirac delta function as an external force can be used to model real-world phenomena such as vibrations in a guitar string or the propagation of seismic waves during an earthquake. However, in most cases, the Dirac delta function is only an approximation of the actual external force, which may be more complex and spread out over a larger area.

5. Are there any limitations to using the 1D wave equation with Dirac delta function as an external force?

One limitation of using the 1D wave equation with Dirac delta function as an external force is that it assumes a one-dimensional medium, which may not always be the case in real-world scenarios. Additionally, the equation does not take into account the effects of damping or dissipation, which may be present in certain systems. Therefore, it is important to carefully consider the applicability of this equation to a specific situation before using it to model real-world phenomena.

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