# Deriving the longitudinal sound wave from the transverse string vibration

I am trying to derive the longitudinal sound wave (i.e. either the pressure $\psi_P(x,t)$ or the displacement $\psi(x,t)$, where both $\psi$s are in the direction of $x$) produced by the vibrations of a finite string.

For example, if a string looks like this:

Then the air\gas that was, at time $t$, between the curves, has moved in the $y$ direction (approx.). This should produce a sound wave, who's properties are determined by the string (i.e. the transverse string displacement and its dimensions1).

So, I tried to find an expression for the pressure wave, by using compressibility: $\kappa=-\frac{1}{V}*\frac{\partial V}{\partial P}$ and expressing the difference in the volume of the gas with $y(x,t)$ (at some specific $x$, along the axis between the string's fundamental nodes).

But I can't seem to get to a second derivative of the longitudinal coordinate (which is in the direction of $y$, but it's confusing to call it that).

Am I missing something? Perhaps some assumption or approximation that makes it simple.

Note:

I got into all this from this section of a Wikipedia page. Curious about how that is derived. Searched a lot online as well as here and I didn't find an actual mathematical derivation.

1I think it's ok to assume that the string is more or less a cylinder, and it's radius is the dimensions I think are relevant:

• Are you attempting a 2D problem?
– Deep
Dec 22, 2017 at 5:17
• @Deep Not exactly. Eventually I am trying to find the frequency of the produced sound wave, at a distance from the string (i.e. assuming that the longitudinal pressure wave doesn't depend on x (the horizontal axis in the graph in the question) Dec 22, 2017 at 11:57
• Frequency of the sound wave will be equal to the forcing frequency i.e. the oscillation frequency of the string. Only its amplitude will decrease with distance from the string.
– Deep
Dec 23, 2017 at 11:20
• @Deep But why are they equal? I am looking for the mathematical derivation that shows that (and finding the pressure wave should show that...) Dec 23, 2017 at 11:22