Compressor stall.
 

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Can someone explain why the pulses are so rhythmic in nature?

This is pretty simplistic, but it goes something like this;
A compressor stalls because the compressor blades can't maintain the pressure gradient across the individual compressor stages - in other words, the airflow across the compressor blades separates (not unlike what happens to a wing at an excessive angle of attack), this breakdown in flow allows the higher-pressure air ahead of the compressor stage to reverse direction and come back out the front.
When that happens, the pressure difference across the compressor stage(s) goes away - and which 'unstalls' the compressor blades and the compressor starts working again - until the pressure difference becomes too great again and the blade stalls, causing the process to repeat.

Clear?