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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:

string example

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:
piece of string

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  • $\begingroup$ Are you attempting a 2D problem? $\endgroup$
    – Deep
    Dec 22, 2017 at 5:17
  • $\begingroup$ @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) $\endgroup$
    – ItamarG3
    Dec 22, 2017 at 11:57
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    $\begingroup$ 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. $\endgroup$
    – Deep
    Dec 23, 2017 at 11:20
  • $\begingroup$ @Deep But why are they equal? I am looking for the mathematical derivation that shows that (and finding the pressure wave should show that...) $\endgroup$
    – ItamarG3
    Dec 23, 2017 at 11:22

1 Answer 1

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The sound waves are produced by a moving surface that is the minimum surface of revolution formed by the string orbit. Your graph has the correct curve but the orbit of the string is that curve rotated about the x-axis. Now if you check, you will see the curve you have drawn is a cosh x function and not sin x. It has constant curvature even at rest. It is easy to understand the sound comes from the string manifold like a bell.

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  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Nov 5, 2023 at 16:15
  • $\begingroup$ There is no reference to this in the literature that I can find. $\endgroup$ Nov 5, 2023 at 18:02

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