  Long weakly-nonlinear waves in a layered waveguide with an imperfect
interface (soft bonding between the layers) can be modelled using coupled
Boussinesq equations. Previously, we considered the case when the materials of
the layers have close mechanical properties, and the system supports radiating
solitary waves. Here we are concerned with a more challenging case, when the
mechanical properties of the materials of the layers are significantly
different, and the system supports wave packet solutions. We construct a
weakly-nonlinear solution of the Cauchy problem for this system, considering
the problem in the class of periodic functions on an interval of finite length.
The solution is constructed for the deviation from the evolving mean value
using a novel multiple-scales procedure involving fast characteristic variables
and two slow time variables. By construction, the Ostrovsky equations emerging
within the scope of this procedure are solved for initial conditions with zero
mean while initial conditions for the original system may have non-zero mean
values. Asymptotic validity of the solution is carefully examined numerically.
We also discuss the application of the solution to the study of co-propagating
waves generated by the solitary or cnoidal wave initial conditions, as well as
the case of counter-propagating waves and the resulting wave interactions. One
local and two nonlocal conservation laws ae obtained and used to control the
accuracy of the numerical simulations.
