Undergrad Bouncing sound pulse in cylinder, driving vibrations in shell

[Swamp Thing]

Consider a tube closed at both ends. An air compression pulse (e.g. Gaussian) bounces between the ends; let us assume nearly lossless propagation and reflection, and no group delay distortion.

At any instant, the elevated pressure around the pulse causes the cylinder to bulge slightly according to its own structural behavior. Assume that this bulging is too small to modify the behavior of the pulse itself. Over time the wall deflection evolves according to this driving pressure and its own dynamics.

Can we approximate the cylinder's shape, as a function of time, by considering a long series of time-bound pulses with, again, a temporal Gaussian growth and decay? We apply these spatio-temporal pulses sequentially, stepping along the cylinder, and sum the cylinder's spatio-temporal responses to each pulse?

In a nutshell, is this a problem where a convolution-like numerical solution would work? My intuition is somehow kind of conflicted between yes and no.

 

[.Scott]

Before I attack this, I'm going to add two more features of your tube.
1) The structure that you use to close the tube at each end has no effect on the "bulging". The reason I mention this is the normal physical world, it would.
2) Th tube is very long compared to the pulses and the wavelengths involved.

So, as a pulse moves down the pipe, for most of its journey it will have the same bulging effect on all the pipe that it crosses.
At each end, the air compression pulse will be reflected against an inelastic end-stop. The pressure at that end-stop will always be the statics pressure of the compressed air plus double the dynamic, time-dependent pressure created by the wave (a signed value). If we have a "bulge function" that operates on the histogram of the pressure history for a particular pipe location, then the histogram provided to this bulge function will be different for locations on the pipe close to the end-stops than for position along the bulk of the pipe.

There are different ways of dealing with those pulses. But since we are presuming that they are relatively short compared to the length of the pipe, then we describe the situation this way:
For pulse length S and pipe length I, then:
1) the bulk of the pipe from positions S to I-S will experience the same histogram profile.
2) For positions on the pipe P close to the end-stops (P<S), then the histogram profile will be from half as many pulses - but each pulse being the result of two overlapping pulses. When P=0, the overlapping pulses line up perfectly, hence a doubling.