Water Waves by James Johnston Stoker

Author:James Johnston Stoker
Language: eng
Format: epub, pdf
Publisher: Dover Publications, Inc.
Published: 2019-04-27T16:00:00+00:00

and g is a potential function in η < 0 which satisfies

on η = 0. The formula

(obtained from the well-known analogous representation for 1/r) in which the Bessel function J0 can be expressed as

allows us to write

for η = 0 and y < 0. It is now easy to see that

is a potential function in η < 0 which satisfies the boundary condition. An interchange of the order of integration gives

where denotes the real part. If we think of p as a complex variable, the path from 0 to ∞ in the last result can be replaced by any equivalent path L, to be chosen later:

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