| Solutions of free, barotropic waves over variable topography are derived. In particular, we
examine two cases: waves around axisymmetric seamounts and waves along a sloping
bottom. Even though these types of oscillations have been studied before, we revisit the
problem because of two main reasons: (i) The linear, barotropic, shallow-water equations
with a rigid lid are now solved with no further approximations, in contrast with previous
studies. (ii) The solutions are applied to a wide family of seamounts and bottom slopes with
profiles proportional to exp(rs) and ys, respectively, where r is the radial distance from the
centre of the mountain, y is the direction perpendicular to the slope, and s is an arbitrary
positive real number. Most of previous works on seamounts are restricted to the special
case s = 2. By varying the shape parameter one can study trapped waves around
flat-topped seamounts or guyots (s > 2) or sharp, cone-shaped topographies (s < 2).
Similarly, most of previous studies on sloping bottoms report cases with s = 1
(linear slopes), whilst the present results are applied to more general bottom profiles.
The resulting dispersion relation in both cases possess a remarkable simplicity that
reveals a number of wave characteristics as a function of the topography shape. |