I am trying to represent a continuous function, $f(x)$, defined on an interval of length L by a superposition of standing wave functions $g(x,t)$.
Definitions for this problem are:
$f(x)$ is in $\mathbb{R}$, but can be looked at as $f(x,t)$ in $\mathbb{R}^2$, but constant with changes in $t$
$g(x,t)$ is in the same $\mathbb{R}^2$
$g(x,t) = \cos(wt + \phi)\cos\left(\dfrac{wx}{v} + \gamma\right)$
$\phi$ and $\gamma$ are phase variables
$w$ is a redial frequency variable
$t$ is a time variable
$x$ is a $1$-D spatial variable
$v$ is a velocity constant
I have tried to use what I know about Fourier theory, but that hasn't gotten me very far. Could someone point me in the right direction to solve this problem?
If I have made anything unclear, please let me know, and thanks in advance.