Relation between string wave and sound wave

Question

What is the connection between an equation representing a string's movement, and the equation representing the sound wave produced by that string? For example: What can I learn about the sound wave produced by a string who's displacement equation is-

Answer 1

Effectively the sound is pretty much the same (in time). Keep in mind though that this sound is very faint. In order to amplify it, you need an acoustic chamber (say a guitar's body) which can have different resonance frequency (and hence different harmonics).

If the fundamental frequency of the string matches that of the acoustic chamber, it would be a good guess (or even a reasonable approximation) to assume they are the same. However one should study wave motion on a 2D membrane in order to capture the characteristic of the acoustic chamber a bit better. i.e. solve the wave equation:

$y_{tt} + c\nabla^2y=0$

for boundary $\bf{B}$ such that $\bf{B}$ is a closed loop (such as a circle): $y(t,\textbf{B})=0$

for initial condition $y(0,\textbf{x})=f(\textbf{x})$, $f(\textbf{x})=0$