Undercompressive shock wave

An undercompressive shock wave is a shock wave that does not fulfill the Peter Lax conditions.

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Details

Ordinary shock waves are compressive, that is, they fulfill the Peter Lax conditions: the characteristic speed behind the shock is greater than the speed of the shock, which is greater than the characteristic speed in front of the shock. The characteristic speed is the speed of small, traveling perturbations. The Lax conditions seem to be necessary for a shock wave to come to existence; if the top of a wave goes faster than its bottom, then the wave front becomes sharper and sharper and eventually becomes a shock wave (a "discontinuous" wave, a sharp wave front which remains sharp when it travels).

A shock wave is undercompressive if and only if the Lax conditions are not fulfilled. Undercompressive shock waves are astonishing: how can a wave front remain sharp if little perturbations can escape from it? At first sight, it seems that such a wave should not exist. But it exists. It has been observed that a sharp wave front remained sharp in its traveling and that little perturbations behind the front traveled slower than it.

The experiment can be made with traveling liquid steps : a thick film is spreading on a thin one. The liquid steps remain sharp when they travel because the spreading is enhanced by the Marangoni effect. Making little perturbations with the tip of a hair, one can see whether shock waves are compressive or undercompressive.

Notes & references

Non-linear waves and the classical theory of shock waves

The mathematical theory of undercompressive shock waves

Experiments with liquid films

Experimental undercompressive shock waves