Equation of State
Draft for Information Only Thermodynamic Equation of State
Since sound fluctuation can be treated as an adiabatic process, the
pressure is a function of density fluctuation only. That is or It can be expressed by Taylor series expansion as following Since the acoustic fluctuation is small, the density variation is small also. The ρ^{2} and higher power terms can be neglected for acoustic fluctuation with moderate sound pressure. Implies: The relation of acoustic pressure variation and acoustic density variation becomes linear. Similarly, as sound fluctuation is assumed an adiabatic process, the fractional change of pressure per displacement change in term of density and specific volume generally can be expressed as
or For a fixed of mass, the fractional change of density and specific volume per volume change are or As the equilibrium pressure is much greater than the acoustic pressure, the acoustic variation of density, volume or specific volume only cause a very small acoustic pressure variation in the equilibrium pressure. Therefore, taking the fractional change at the equilibrium state is accurate enough to relate the acoustic pressure and the acoustic variation of density, volume or specific volume. And can be expressed as: , and where P is pressure of medium at initial state Sub into the Tayor expansion and together with the ideal gas law, then the relationship between acoustic pressure and acoustic density variations are : where
p is acoustic pressure variation Therefore the relationship can be a function of absolute temperature of the medium. Considering a fixed mass with a small changes in volume and density, then: where
ρ_{o} is initial density of medium By neglecting the product of small quantities, then: Therefore, by rearrangement, the acoustic pressure can also be expressed as a function of volumetric strain: where
V is acoustic volumetric variation ©sideway ID: 100900018 Last Updated: 9/11/2010 Revision: 0 
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