What is the sound of a solution $y(x,t)$ to the Wave Equation
$\frac{\partial^2y}{\partial t^2}=c^2\,\frac{\partial^2y}{\partial x^2}\enspace$?
To be clear, I don't look for a solution of the Wave Equation, I have a concrete solution $y(x,t)$. I only want to describe the sound of this concrete solution.
In my case I want to describe the sound of a vibrating string, so the ends of the string are fixed, i.e. $y(0,0)=y(L,0)=0$ where $L$ is the length.
With a solution $y(x,t)$, I get perfectly the shape of the string between $x=0$ and $x=L$ for every time $t$.
I can show that $y(x,t)$ must be also $2L$-periodic according to $x$, i.e. $y(\lambda+2L,t)=y(\lambda,t)$, and $(2L/c)$-periodic according to $t$, i.e. $y(x,\lambda+2L/c)=y(x,\lambda)$.
To describe the sound, I would like to
- find a frequency spectrum (or magnitude/amplitude spectrum) or
- produce a sound file with a series of samples $x(i)$ to hear the vibrating string.
For 1. I don't know how to do that. Should I develop the Fourier-Series according to $x$ or to $t$ (a Fourier-Series exists, because I've showed the periodicity) or do complety other things? For a Fourier-Series according to $x$, the Fourier-Coefficients $a_k,b_k$ depends on $t$ and for a Fourier-Series according to $t$, the Fourier-Coefficients depends on $x$.
Any idea?
Edit:
Here is an example for a string that get plucked at $x_p$ with a displacement of $h$. Its shape is a triangle as follows:
$y_0(x)= \begin{cases} x(h/x_p)&\text{if}\enspace 0\leq x < x_p\\[1ex] h\left(1-\frac{x-x_p}{L-x_p}\right)&\text{if}\enspace x_p \leq x \leq L \end{cases}$
The initial velocity is 0 so finally the string is described as: $y(x,t)=y(x,t)=\frac{1}{2}\left(y_0(x+ct)+y_0(x-ct)\right)$
In the following picture, you can see the process of the string starting at $t=0$