Shock waves are tough.
Shock waves are formed when the speed of the upstream flow is greater than the speed of the downstream flow. This causes increased pressure on the wave front, causing it to increase in density and accelerate.
Shock waves propagate faster than the speed of sound, and decrease speed as they propagate. When the speed of the shock wave slows to the speed of sound, the shock wave dissipates into a sound wave.
The wave speed tends to slow at an inverse square of distance covered. This is the same rate of energy dissipation as the decrease in volume over distance as a sound wave. However, it is in fact the minimum rate of energy dissipation of a shock wave, since a shock wave creates heat and turbulence in the flow after it passes.
So that's a start.
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It gets complicated because a shock wave in air is different from a lot of practical examples.
If a faucet is running in an empty, smooth-bottomed sink, after a moment, there will be a ring where the depth of the water increases. This zone where the depth of the water changes is a shock wave. This shock wave is "moving" at a speed equal and opposite to the speed of the incoming flow, and appears stationary.
If you had a thin layer of honey on a plate, then you let a stream of honey trickle onto the plate, there will be a shock wave. That shock wave is visible on the surface of the honey as an expanding circle which is thicker than the underlying layer. The leading edge from this expanding circle is a shock wave, characterized by dramatic increase in density and speed over a small length.
When ocean waves break on a beach, the top of the wave, which breaks, does so because it is a shock wave. The top of the wave is moving faster than the bottom of the wave, and it spills forward, exchanging forward energy into heat and turbulence.
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It gets even more complicated because of the fact that waves of different wavelengths travel at slightly different speeds. This is called dispersion.
"Complicated" waves are superpositions (additive combinations) of "simple" sine waves. That means that the complicated wave's shape is composed of some number of simple sine waves added together. Those sine waves may propagate at different "speeds of sound" within the medium. So the shape of the wave will change as it propagates.
(Water waves ripple and spread and combine because they are a superposition of simple waves which move at different speeds.)
This is pertinent, because some shock waves "leak" information in the direction of flow. E.g. in some viscous fluids, you can indent a mark on the leading edge of the shock wave. That mark will flow forward, ahead of the shock wave.
This all brings on a discussion of phase speed vs. group speed of the wave's propagation.
Each frequency of sine wave that composes the main wave moves at its own phase speed. The overall wave moves at the group speed.
The "center" of the wave moves as the group speed, but the width of the wave may spread or contract based on the dispersion characteristics of the medium. If the medium is non-dispersive, then the wave will move without changing shape.
So you can have a wave, which has portions of it's composition waves acting in a shock wave, while other portions are free to pass through the shock wave. This is true, even though the shock waves move faster than the speed of sound, because the speed of sound for the frequencies in the shock wave is different than the speed of sound for frequencies beyond that range.
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