There is really only one problem with your cancellation idea. To work in all
directions the sounds have to come form the same point in space. If they are
not the same, all you can do is local cancellation or an interference pattern
with some areas silent and some twice as loud.

Just take two points and start drawing circles around each. The intersections
will form lines of interference.

The PDE's are harder than pulse jets since the detonation wave is supersonic.
You exceed what you might call the ellasticity of the air and I don't know if
linear theory (like Fourier) will apply. There is probably a dramatic
discontinuity and Fourier requires at least piece wise continuity. But I have
not looked at shock physics in a long time. Sombody on RAM would know, like
Marry Shafer.

-- Charlie Springer