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This page contains optional extra work (mostly aimed at IA Nat. Sci. maths), interesting links, and assorted other things.

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Additional comments on the Lecture Notes

Derivation of Sound Wave Equation

In the IB Physics A Osciallations and Waves course, there is a derivation of the wave equation for sound waves in a gas. (The lecture handout is on the course web page, which also contains a good section of interesting links.) A number of groups have asked me about the approximation used in deriving equation 47. I think that the main confusion here comes from the use of p and \Psi_p, and I try to explain what's going on below.

In the run-up to equation 44, \Psi_p is introduced as the excess pressure due to the wave. Therefore the actual pressure in the gas, as a function of x and t, is p + \Psi_p, where p is the constant ambient pressure. This notation is used carefully and consistently to derive equations 44 and 45. In equation 46 and what follows, it's a bit sloppier. The thing that's being called p is really p+\Psi_p, the actual pressure. There is also an approximation going on: the small but finite changes in pressure and volume as the wave goes past, which are \Psi_p and \Delta V respectively, are being assumed to be in the same ratio as an infinitesimal dp and dV. This is a very good approximation, but it's important to be aware that an approximation is being made.

Thus what equation 46 is really trying to tell us is that \Psi_p = - \gamma (p + \Psi_p) times partial da/dx.

Once we come to take a derivative of equation 46, we need to use the product rule, because the thing written p is really the constant p plus the spatially-varying \Psi_p. Then the first p (after the \gamma) in the formula for the derivative ought to be p + \Psi_p, but we can legitimately ignore that \Psi_p, because it's then a small part of the equation that we've got. (The relies on \Psi_p's being a small correction to p, which is equivalent to the previous approximation.) In the next term, the partial dp/dx is really partial d(p + \Psi_p)/dx, which is partial d(\Psi_p)/dx. So we've now got a term with that derivative on both sides of the equation. In principle we should take them both to the LHS and divide by the 1 + \gamma (partial da/dx). This would make a vile equation. But here we can make a further approximation: if a is much smaller than \lambda, which it usually is, then partial da/dx is much smaller than 1.* As \gamma is of order 1, that means that the whole term can be neglected and the derivation proceeds as given.

* Since a has some sort of wave-like solution, partial da/dx is going to be something like ka, which is proportional to a/(\lambda). This seems to be what the lecturer is thinking, anyway. But really we have already decided that partial da/dx is small, because equation 46 tells use that it's a number of order 1 times \Psi_p/p, and the assumption that \Psi_p is small compared to p is one that we've been making since beginning to derive equation 46 itself.