The continuity and the (potential flow) momentum equations for unsteady flows are given by
The first integral of the (potential flow) momentum equation is given by 
where  is an arbitrary function. Without loss of generality, we can set  since  is not uniquely defined. Combining these equations, we obtain
Substituting here  results in
Nearly parallel flows: As in before, for nearly parallel flows, we can write (after introudcing a recaled time )
provided the constant Mach number  is not close to unity. When  is small (transonic flow), we have the following nonlinear equation
Sound waves: In sound waves, the velocity magntiude  (or the Mach number) is very small, although the unsteady term is now comparable to the other leading terms in the equation. Thus neglecting all quadratic and higher-order terms and noting that in the same approximation,  is a constant (for example, in polytropic gas ), we have
which is a linear wave equation for the velocity potential . Again the oscillatory part of the velocity vector  is related to the velocity potential by , while as before  is the Laplace operator, and  is the average speed of sound in the homogeneous medium. Note that also the oscillatory parts of the pressure  and density  each individually satisfy the wave equation, in this approximation.