# Thread: Determine whether the given function satisfies the wave equation

1. ## Determine whether the given function satisfies the wave equation

$\displaystyle w(x,t)=e^{x-ct}$

Thank you for the help on the last problem, but for this one I'm still stumped as to what I need to do, even after reading a bit on wikipedia. Any help would be much appreciated.

Wikipedia has:

$\displaystyle \frac{\partial u^{2}}{\partial t^{2}}=c^{2}\nabla^{2}u$

But, I don't know what u and $\displaystyle \nabla$ are.

2. Originally Posted by downthesun01
$\displaystyle w(x,t)=e^{x-ct}$

Thank you for the help on the last problem, but for this one I'm still stumped as to what I need to do, even after reading a bit on wikipedia. Any help would be much appreciated.

Wikipedia has:

$\displaystyle \frac{\partial u^{2}}{\partial t^{2}}=c^{2}\nabla^{2}u$

But, I don't know what u and $\displaystyle \nabla$ are.
u is the function that satisfies the wave equation. In your case, you're asked to show that u = w(x, t) is a solution.

$\displaystyle \nabla^2$ is the Laplace operator.

3. In Cartesian coordinates in three dimensions, $\displaystyle \nabla^2 f= \frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}$.

Here, you don't have any dependence on y or z so your equation reduces to
$\displaystyle \frac{\partial^2 u}{\partial t^2}= c^2\frac{\partial^2 u}{\partial x^2}$.

"u" is function. You are asked to replace u by $\displaystyle e^{x- ct}$ in that equation and see if it is true.

But what course did you see this in? I cannot imagine you being asked to do a problem like this if you were not already expected to know what $\displaystyle \nabla^2$ was!