  This article addresses a fundamental concern regarding the incompressible
approximation of fluid motions, one of the most widely used approximations in
fluid mechanics. Common belief is that its accuracy is $O(\epsilon)$ where
$\epsilon$ denotes the Mach number. In this article, however, we prove an
$O(\epsilon^2)$ accuracy for the incompressible approximation of the
isentropic, compressible Euler equations thanks to several decoupling
properties. At the initial time, the velocity field and its first time
derivative are of $O(1)$ size, but the boundary conditions can be as stringent
as the solid-wall type. The fast acoustic waves are still $O(\epsilon)$ in
magnitude, since the $O(\epsilon^2)$ error is measured in the sense of Leray
projection and more physically, in time-averages. We also show when a passive
scalar is transported by the flow, it is $O(\epsilon^2)$ accurate {\it
pointwise in time} to use incompressible approximation for the velocity field
in the transport equation.
