How would a layer of hot air affect the normal frequencies in a pipe?

Question

Imagine that you have a pipe of length $L$ with one open end and one closed end. If the sound speed inside the pipe is $v_s$, then the fundamental frequency is:

$$f_1=\frac{v_s}{4L}$$

and the overtone frequencies are: $3f_1$, $5f_1$, etc.

I am trying to gain insight into how those normal frequencies would change if a thin layer of very hot air is placed somewhere inside the pipe.

By thin I mean compared to the total length $L$ of the pipe. By very hot air I mean such that the sound speed $v_s$ increases significantly/abruptly as it goes through the layer. I am also thinking the normal vector of the layer pointing parallel to the propagation of the waves inside the pipe.

Would you expect any change on the normal frequencies?

Can perturbation theory be used to find the putative new frequencies?

Any suggestions / ideas are welcome!

Answer 1

Let's take the case where the layer of heated air is very, very hot and let's assume it is fixed in position, at least for a moment so we can perform our thought experiment.

We then send a sound wave down the length of the pipe. where the hot layer is, the density of the air in the pipe is discontinuous, and its characteristic impedance is too. When the sound wave hits it, the impedance mismatch causes part of the sound wave to be transmitted through the hot layer and part of it bounces off and is reflected backwards down the pipe- as it would if it had instead hit the open end of the pipe.

If the thickness of the hot layer is small then we won't worry about a second reflection/transmission split as the transmitted sound wave emerges from the hot layer and resumes its normal speed down the rest of the pipe, where it will establish just the same resonance it would have had without the hot layer getting in the way. So whatever the effect of the hot layer will be, it will appear as a superposition on the response of the pipe without the hot layer.

The reflected wave will eventually come back to the point of origin of the sound wave and will establish a second resonance in that length of pipe between the beginning of the pipe and the position of the hot layer.