Gravity–capillary rings generated by water drops

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Abstract
A theory for water waves created by the impact of small objects such as raindrops on an initially quiescent body of water is established. Capillary and dissipative viscous effects are taken into account in addition to gravity. It is shown that the prevailing waves are in a mixed capillary–gravity regime around a wavenumber km which corresponds to the minimum value of the group velocity. The waves are described as function of time and distance by the linear superposition of two transient wave components, a ‘sub-km’ (k < km) component and a ‘super-km’ (k > km) component. The super-km components prevail at a short distance from the drop, whereas only the sub-km ones remain at a larger distance. The relative time history of the wavetrain is independent of the size of the drop, and its amplitude is proportional to the drop momentum when it hits the free surface. The wave pattern is composed of a multiplicity of rings of amplitude increasing towards the drop location and is terminated by a trailing wave with an exponential decay. The number of rings increases with time and distance.

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