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 can 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.
G.B. Whitham. Linear and non-linear waves Wiley-Interscience 1974
Peter D. Lax. Hyperbolic systems of conservation laws and the mathematical theory of shock waves Society for industrial and applied mathematics Philadelphia, Pennsylvania 1973, Hyperbolic systems of conservation laws II Comm. Pure Appl. Math., 10 :537-566, 1957
Andrea L Bertozzi, A. Munch, M. Shearer, Undercompressive Shocks in Thin Film Flow, Physica D, 134(4), 431-464, 1999
A. Munch. Shock transition in Marangoni and gravitation driven thin film flow 1999
A. Munch, A. L. Bertozzi, Rarefaction-Undercompressive Fronts in Driven Films, Physics of Fluids (Letters) 11(10), pp. 2812-2814, 1999
Herbert E. Huppert. Flow and instability of a viscous current down a slope Nature Vol. 300, 427-429, 1982
A.M. Cazabat, F. Heslot, S.M. Troian, P. Carles. Fingering instability of thin spreading films driven by temperature gradients Nature Vol. 346, 824-826 1990
A.L. Bertozzi, A. Münch, X. Fanton, A.M. Cazabat, Contact Line Stability and "Undercompressive Shocks" in Driven Thin Film Flow, Physical Review Letters, Volume 81, Number 23, 7 December 1998, pp. 5169-5172
T. Dugnolle, Des chocs non-classiques lors de l'étalement forcé d'un liquide, Mémoire de DEA (Paris 6, Physique des Liquides), LPMC, Collège de France 1999