Solving the Wave Equation PDE: A General Solution Approach

In summary, the conversation is about a problem with a PDE and finding the general solution. The person initially thought the solution was u(x,t)=0, but it was pointed out that this is only the solution for the homogeneous wave equation without a source term. The conversation then moves on to finding the general solution for a non-homogeneous equation by adding the general solution for the related homogeneous equation to a specific solution. The conversation ends with the person sharing their solution for part (a) and (b) of the problem.
  • #1
sarahisme
64
0
Hi everyone,

I'm having a bit of trouble with this pde problem:
http://img243.imageshack.us/img243/9313/picture3ui3.png

i get the answer to be u(x,t)=0 but i am guessing that's not right.

is the general solution to this problem: u(x,t) = f(x+ct) + g(x-ct) ??

thanks

sarah :)
 
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  • #2
No, it's not. That would be the general solution to the homogeneous wave equation, uxx= c2utt, without a "source" term. Likewise, u(x,t)= 0 is the solution to the homogeneous equation with those conditions.

Remember that with a linear equation, you can construct a general solution to a non-homogeneous equation by adding the general solution to the related homogeneous equation to a single specific solution to the entire equation.
 
  • #3
hmmm ok, i tried again and this is what i get...

for (a) http://img291.imageshack.us/img291/6120/picture4td7.png

for (b) http://img301.imageshack.us/img301/5572/picture5ii4.png

what do you think?
 
Last edited by a moderator:
  • #4
lol, well i think its right anyway, thanks for the help HallsofIvy! your a legend! :D
 

1. What is the wave equation PDE problem?

The wave equation PDE problem is a partial differential equation that describes the behavior of waves in a physical system. It is commonly used in physics, engineering, and other scientific fields to model various wave phenomena such as sound, light, and water waves.

2. How is the wave equation PDE problem solved?

The wave equation PDE problem is typically solved using mathematical techniques such as separation of variables, Fourier series, or Laplace transforms. Depending on the specific problem, numerical methods such as finite difference or finite element methods may also be used.

3. What are the applications of the wave equation PDE problem?

The wave equation PDE problem has a wide range of applications in fields such as acoustics, electromagnetics, seismology, and fluid dynamics. It is used to model and study various phenomena such as sound propagation, electromagnetic radiation, earthquake waves, and ocean waves.

4. What are the boundary conditions for the wave equation PDE problem?

The boundary conditions for the wave equation PDE problem depend on the specific physical system being modeled. In general, there are two types of boundary conditions: initial conditions, which specify the initial state of the system, and boundary conditions, which describe the behavior of the system at the boundaries of the domain.

5. Are there any real-world limitations to the wave equation PDE problem?

While the wave equation PDE problem is a powerful tool for modeling wave phenomena, it does have some limitations. It assumes ideal conditions, such as a perfect medium, and does not account for factors such as damping or non-linear behavior. Additionally, it is not always applicable to complex systems or systems with discontinuities.

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