Visual physics and mathematics/Shock waves

A shock wave generally obeys the Lax conditions: the speed of small perturbations behind the wave front is greater than the speed of the wave front which is itself greater than the speed of small perturbations in front of the wave front:

Such shock waves are said to be compressive.

All shock waves are expected to be compressive. Peter D. Lax proposed his conditions as universal conditions for a mathematical theory of all shock waves. If the Lax conditions are not verified, the wave front is expected to gradually spread out and therefore a shock wave can never form. However, we can observe, in exceptional conditions, shock waves which do not obey the Lax conditions:

Such shock waves are said to be undercompressive.