  Most of the asymptotically derived Boussinesq systems of water wave theory
for long waves of small amplitude fail to satisfy exact mechanical conservation
laws for mass, momentum and energy. It is thus only fair to consider
approximate conservation laws that hold in the context of these systems.
Although such approximate mass, momentum and energy conservation laws can be
derived, the question of a rigorous mathematical justification still remains
unanswered. The aim of this paper is to justify the formally derived mechanical
balance laws for weakly nonlinear and weakly dispersive water wave Boussinesq
systems. In particular, two asymptotic expansions used for the formal and
rigorous derivation of the Boussinesq systems and the same are employed for the
derivation and rigorous justification of the balance laws. Numerical validation
of the asymptotic orders of approximation is also presented.
