  In this paper we present a numerical scheme for solving a system of
Boussinesq-type equations. This can correspond to longitudinal displacements in
a multi-layered elastic bar with delamination, with conditions on the interface
between the sections of the bar. The method is initially presented for two
coupled equations in each section and multiple sections in the bar, and later
extended to any number of layers. Previous works have presented a similar
method constructed using finite-difference methods, however these only solved
for two sections of the bar at a time which limited the scope of studies using
these methods. The new method presented here solves for all sections at a given
time step and therefore the transmitted and reflected waves in each section of
the bar can be studied. The new results are shown to be in excellent agreement
with previously obtained results and a further study is performed showing that
the delamination width can be inferred from the changes to the incident
soliton. The generalised form of this method, for any number of sections and
layers with coupling terms independent of time derivatives, can be used to
study the behaviour of longitudinal waves in more complicated waveguides in
future studies.
