Jökull - 01.12.1977, Page 81

An Equation for Gravity Waves on Deep Water
GUNNAR BODVARSSON
SCHOOL OF OCEANOGRAPHY AND DEPARTMENT OF MATHEMATICS
OREGON STATE UNIVERSITY, CORVALLIS, OREGON 97331
abstract
A single scalar equation for the amplitude of
gravity waves on deep water can in the lineariz-
ed infinitesimal amplitude approximation be
derived on the basis of elementary potential
theory. The spatial part of the wave operalor
involved acts as the square root of the two-
dimensional Laplacian operating on the domain
defined by the static free surface. The charac-
teristics of gravity waves such as dispersion re-
sult from the special properties of this operator.
INTRODUCTION
The theory of gravity waves on deep water
has interested mathematicians and physicists for
centuries. The subject has a long and impres-
sive history which can be traced back to Cauchy
and Poisson at the beginning of the nineteenth
century. Research in this field is still very active,
and a number of excellent books have appeared
(Stoker, 1957; Kinsman, 1965; Phillips, 1966)
describing both the classical theory and notable
recent results.
It is of sorne interest to note that in spite of
the very rich and well developed tlieory, no
single equation which can be referred to as the
equation of gravity waves of infinitesimal ampli-
tude, appears to liave been set forth in the
literature. Well defined wave equations, on the
other hand, provide the basis for the theory of
other oscillatory scalar fields such as in tlie
theory of sound where d’Alembert’s equation
plays a central role.
Although it is quite obvious that the theory
of gravity waves has been progressing very well
without the benefit of a basic wave equation,
there is little doubt that the availability of such
an equation would help to both simplify and
clarify some aspects of the theory. Tlie present
short note intends to demonstrate that a gravity
wave equation can be obtained by elementary
means witiiin the framework of classical line-
arized theory, viz., involving waves of an in-
finitesimal amplitude on the surface of a deep
layer of an ideal fluid. We will commence by
introducing our principal mathematical tool,
the surface operator which generates the deriva-
tive of a harmonic function across a given sur-
face on the basis of the values of tlie function
on the surface.
THE SURFACE OPERATOR
Consider a rectangular coordinate system and
a simply connected 2-dimensional domain 2 with
a piecewise smooth boundary y embedded in
the plane z = 0. Let B be the 3-dimensionaI
cylindrical product domain of 2 antl the inter-
val (0,d) on the z-axis. Moreover, let 2 + T
denote the boundary of B where F consists of
the side surface of B and the end face in the
z = d plane. The general field point in B is
P = (x,y,z).
Let cf, (P) be a function which is harmonic
in B, that is,
-V20 = 0, P in B, (1)
and which satisfies the Neumann type boundary
condition on T,
d(þ/dn = 0, P on T, (2)
where n is the outward normal to F- Since cþ (l’)
is harmonic in B and satisfies a prescribed condi-
tion on r, its values in B are uniquely deter-
mined by the boundary values cþ0 (S) on 2
where S = (x,y) is a field point on 2- The
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