Irrotational Water Waves Research Articles

Conservation laws of free boundary problems and the classification of conservation laws for water waves

The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis.

On the existence theory for irrotational water waves

AbstractThis paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, andwhere g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

Temperature fluctuations at an air-water interface caused by surface waves

This brief report predicts the variation of surface temperature induced by a plane progressive irrotational linear water wave when a thermal boundary layer exists. The following conclusions are drawn. (1) When the wave period is sufficiently small (≲3 sec), the amplitude of the wave-induced temperature fluctuation is proportional to the wave amplitude and to the square root of the wave period. This result is in perfect agreement with experiments that, however, are by no means definitive. (2) Surface temperature maximums lead wave crests by one-eighth of a period when the average surface temperature is less than the bulk water temperature; surface temperature minimums lead wave crests by one-eighth of a period when the average surface temperature exceeds that of the water below.

On the decrease of velocity with depth in an irrotational water wave

ABSTRACTThe following theorems are proved for irrotational surface waves of finite amplitude in a uniform, incompressible fluid:(a) In any space-periodic motion (progressive or otherwise) in uniform depth, the mean square of the velocity is a decreasing function of the mean depth z below the surface. Hence the fluctuations in the mean pressure increase with z.(b) In any space-periodic motion in infinite depth, the particle motion tends to zero exponentially as z tends to infinity. The pressure fluctuations at great depths are therefore simultaneous, but they do not in general tend to zero.(c) In a progressive periodic wave in uniform depth the mass-transport velocity is a decreasing function of the mean depth of a particle below the free surface, and the tangent to the velocity profile is vertical at the bottom. This result conflicts with observations in wave tanks, and shows that the waves cannot be wholly irrotational.(d) Analogous results are proved for the solitary wave.