# What is the Sound of a Solution to the Wave Equation?

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

1. find a frequency spectrum (or magnitude/amplitude spectrum) or
2. 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 $t=0$ • all depends on your initial conditions. what $f(x)$ was at $t=0$. the period (in time) will be $\frac{2L}{c}$ and the pitch will be $log_2\left( \frac{c}{2L f_0} \right)$ measured in octaves from your reference pitch that has fundamental frequency $f_0$. Mar 27, 2015 at 0:20

Since you already have the solution to the wave equation, what you need to do now is define a "pick-up" point somewhere along the length of the string (some $x$) and obtain its trace in time, i.e. look at what that specific $x$ value is "doing" across all the $t$ instances that you have. In other words, draw a straight line down your vibrating string diagram and take the vertical displacement of your string at each time point as the amplitude of your sound signal at that time point.