  We prove that the divergence-free component of the compressible Euler
equations with solid-wall boundary condition converges strongly towards the
incompressible Euler equations at the same order as the Mach number. General
initial data are considered and are not necessarily close to the
divergence-free state. Thus, large amplitude of fast oscillations persist and
interact through nonlinear coupling without any dissipative or dispersive
mechanism. It is then shown, however, that the contribution from fast
oscillations to the slow dynamics through nonlinear coupling is of the same
order as the Mach number when averaged in time. The structural condition of a
vorticity equation plays a key role in our argument.
