Boussinesq equations and other
for small-amplitude long waves in nonlinear dispersive media. Part
derivation and the linear theory
Considered herein are a number of
variants of the classical Boussinesq system and their
higher-order generalizations. Such equations were first
derived by Boussinesq to describe the two-way propagation
of small-amplitude, long
wavelength, gravity waves on the surface of water in a canal.
These systems arise also when modeling the propagation of long-crested
waves propagating on large lakes, the ocean and in other contexts.
Depending on the modeling of dispersion, the resulting system may or
may not have a linearization about the rest state which is well posed.
Even when well posed, the linearized system may exhibit a lack of
conservation of energy that is at odds with its status as an
approximation to the Euler equations. In the present script, we
derive a four-parameter family of Boussinesq systems from the
equation for free-surface flow and
isolate those members of the class that are
linearly well posed and energy conserving.
Min Chen (email@example.com)