• Support PF! Buy your school textbooks, materials and every day products Here!

Can you have fourier transform + boundary condition? (solving wave equation)

  • #1
1,434
2

Homework Statement


"Solve for t > 0 the one-dimensional wave equation
[tex]\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}[/tex]
with x > 0, with the use of Fourier transformation.
The boundary condition in x = 0 is u(0,t) = 0.
Assume that the initial values u(x,0) and [tex]\frac{\partial u}{\partial t}(x,0)[/tex] are piecewise continuous with the necessary properties to make Fourier transformation useful.

Homework Equations


Suggested form of Fourier transformations:
[tex]F(k) = \int_{-\infty}^{\infty}f(x)\exp(-ikx)\mathrm d x[/tex]
[tex]f(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty}F(k)\exp(ikx)\mathrm d k[/tex]


The Attempt at a Solution


Well I basically applied the usual method and got the answer that [tex]u(x,t) = \frac{1}{2} \left( u(x+ct,0) + u(x-ct,0) \right) + \frac{1}{2} \int_{-t}^{t} \frac{\partial u}{\partial t}(x+cs,0) \mathrm d s[/tex]
which is exactly what you'd get if [tex]x \in \mathbb R [/tex] (instead of specifically x > 0), and indeed I never used the boundary condition u(0,t) = 0, simply because there was nowhere to use it... How do I put in a boundary condition whilst using Fourier transform?
 

Answers and Replies

Related Threads for: Can you have fourier transform + boundary condition? (solving wave equation)

Replies
0
Views
2K
Replies
8
Views
8K
Replies
5
Views
3K
Replies
3
Views
3K
Replies
5
Views
2K
Replies
1
Views
608
Replies
2
Views
5K
Top