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Wave‐source expansions for water of infinite depth in the presence of surface tension
Author(s) -
Rhodes-Robinson P. F.
Publication year - 1991
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/s0025579300006458
Subject(s) - mathematics , surface wave , surface (topology) , surface tension , free surface , mathematical analysis , asymptotic expansion , series (stratigraphy) , velocity potential , geometry , mechanics , optics , physics , geology , boundary value problem , paleontology , quantum mechanics
In this paper two expansions are obtained by contour integration methods for the velocity potential describing two‐dimensional time‐harmonic surface waves due to a free‐surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as K r ®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short‐wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.