Abstract :
In 1888, A. M. Basset ‘‘A Treatise on Hydrodynamics,’’ Deighton, Bell and Co.,
Cambridge, 1888. gave a formal argument to show that slow modes of decay do not
exist for the free surface and velocity field of a linearized, two dimensional, viscous,
finite-depth ocean with small viscosity. His dispersion relation for waves of the
form eik xqv t was v fO k2 .. His argument did not apply to very small wave-
lengths large k.which are important to consider since nonlinear effects can cause
small wavelengths to focus and form singularities. In Lamb’s book on hydrodynamics,
there is a formal argument which shows that slow modes of decay exist for
small wavelengths on the free surface of a linearized, two dimensional, viscous
ocean of infinite depth if surface tension is neglected. There, Lamb showed that
vfO 1rk.. The lack of boundary conditions on a fixed bottom makes the analysis
somewhat easier than for the finite depth case. Lamb’s analysis, as well as Basset’s,
relies on physical intuition rather than careful mathematical analysis}the smallness
of certain quantities is assumed before the dispersion relation is derived.
More importantly, both Basset and Lamb rely on a particular decomposition we
will be specific below. of the stream function which becomes degenerate in the
small wavelength limit. They obtain dispersion relations by setting to zero a
determinant, but the zeros of that determinant do not all correspond to physically
nontrivial pressures or velocities. Some zeros merely reflect the degeneracy of the
decomposition they used. The purpose of this article is to warn others of this
pitfall, while pointing out a simple direct method of calculation which avoids it.
Our approach applies equally well to the cases of finite or infinite depth, with or
without surface tension. In addition, our method yields a complete asymptotic
expansion for the dispersion relation for small wavelengths. In order to obtain a
complete expansion from other methods, one would have to decide at which point
in the expansion small terms that were dropped would need to be reinserted, etc.
