Even gases like air are assumed (for sufficiently low flow velocities) to have constant density. Is it only because the hydrodynamic equations of motions are easier to solve when incompressibility is assumed? Or can it be proven with Statistical mechanics why incompressibility is frequently assumed?
In case of Sound waves, small deviations in Density are respected and kinetic Energy of many sound waves are a lot lower than the air flow around a car at 100mph. Will compressibility come into account also at low speeds and if yes, why?
That is a hystorical assumption in the fluid dynamics (and more often in aerodynamics) field. This assumption is not on the physical property of the fluid but rather on the assumption of the involved characteristic velocity compared to the sound velocity. Below a certain value of v/a=Mach number the flow (not the fluid) problem is assumed to be governed by the simplified NSE equations. Historically, other assumptions were associated such as steady flow and inviscid flow to get a simplified set of equations (see potential flows).
Actually, if the flow is assumed to be unsteady and viscous, the set of equations is somehow more mathematically complicated than the corresponding set for fully compressible flows. Modern methods try to solve the compressible low-Mach equations.
Of course, the incompressible flow is a mathematical model, therefore you cannot describe some physical property such as pressure wave at finite velocity.
I agree with the previous answers. I think statistcal mechanic is not the main reason. For most common engineering problems the motion equations for incompressible flow are easier to solve.
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