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On the resonant interaction between a surface wave and a weak surface current
Author(s) -
Stewartson K.
Publication year - 1977
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300008883
Subject(s) - current (fluid) , amplitude , wavelength , surface (topology) , surface wave , infinity , mathematics , resonance (particle physics) , stokes wave , phase velocity , free surface , wave equation , wave shoaling , classical mechanics , mechanical wave , mechanics , physics , wave propagation , mathematical analysis , breaking wave , longitudinal wave , optics , quantum mechanics , geometry , thermodynamics
An internal wave motion, below a layer of uniform fluid, induces a weak current on the free surface in the form of a long wave with phase velocity c I . A uniform progressive train of surface waves, whose wave‐length is much shorter than that of the current is incident on it from infinity and undergoes modification. In particular, when the group velocity c g of the progressive wave is equal to c I , the resonance takes place and then, even though the amplitude of the current is small, the interaction builds up near a number of its wavelengths until the train of surface waves is significantly modified. The equations governing the modifications are derived, using the method of multiple scales, and the roles of the Döppler shift and the radiation stress in resonant situations are elucidated. Three‐dimensional interactions are discussed and an analogy is drawn between the fundamental equation describing the interactions and Schrödinger's equation.