Deriving Non-linear acoustic wave models, equilibrium state assumption

[binbagsss]

The standard derivation in obtaining a single wave equation involves making use of the heat equation with a Taylor expansion of the equation of state, then differentiating this equation and the continuity equation with respect to time, and combining with the divergence of the NS equation.

From the literature I have found, it tends to be assumed an equilibrium state of:
- #u_0=0# (vanishing equilibrium velocity)
-#\partial_t \rho_0 =0# (no time dependence on the equilibrium density)
(which together, by the continuity => vanishing spatial dependence on the equilibrium velocity as well).Probably a stupid question but, if assuming a non-vanishing equilibrium velocity, (and of course under the same assumptions corresponding to used to derive the specific model), do we expect to obtain the same wave equation as we would if vanishing equilibrium velocity was assumed?

Many thanks

 

[Valerie Park]

in advance!No, assuming a non-vanishing equilibrium velocity will not result in obtaining the same wave equation as if vanishing equilibrium velocity was assumed. The assumption of a non-vanishing equilibrium velocity will affect the terms which are included in the Taylor expansion of the equation of state and the resulting wave equation. As such, the wave equation obtained when assuming a non-vanishing equilibrium velocity will be different from the wave equation obtained when assuming a vanishing equilibrium velocity.