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

[nonequilibrium]

Homework Statement

"Solve for t > 0 the one-dimensional wave equation
$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$
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 $$\frac{\partial u}{\partial t}(x,0)$$ are piecewise continuous with the necessary properties to make Fourier transformation useful.

Homework Equations

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

The Attempt at a Solution

Well I basically applied the usual method and got the answer that $$u(x,t) = \frac{1}{2} \left( u(x+ct,0) + u(x-ct,0) \right) + \frac{1}{2} \int_{-t}^{t} \frac{\partial u}{\partial t}(x+cs,0) \mathrm d s$$
which is exactly what you'd get if $$x \in \mathbb R$$ (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?

 

[mighty2000]

Thank you for your post. It seems like you have made good progress in solving the one-dimensional wave equation using Fourier transformation. However, as you mentioned, you have not yet incorporated the boundary condition u(0,t) = 0 into your solution.

To incorporate the boundary condition, you can use the idea of extending the function f(x) to be an odd function in the interval [-L,L], where L is a large enough number such that the function becomes zero at the boundary x = 0. This can be achieved by defining f(x) = -f(-x) for x < 0 and f(x) = f(x) for x > 0. This extended function can then be used in the Fourier transformation, and the resulting solution will satisfy the boundary condition.

I hope this helps. Keep up the good work!