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

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!