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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 

image or image

It can be expressed by Taylor series expansion as following

image

Since the acoustic fluctuation is small, the density variation is small also. The ρ2 and higher power terms can be neglected for acoustic fluctuation with moderate sound pressure. Implies:

image

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

image  

or

image

For a fixed of mass, the fractional change of density and specific volume per volume change are

image

or

image

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:

image , image and image

where

P is pressure of medium at initial state
ρo is density of medium at initial state
υo is specific volume of medium at initial state
Vo is volume 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 :

image

where

p is acoustic pressure variation
ρ is acoustic density variation
Rs is specific gas constant of medium
T is temperature of medium at initial state

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:

image

where

ρo is initial density of medium
Vo is initial volume of medium
ρ is density variation
V is volumetric variation

By neglecting the product of small quantities, then:

image

Therefore, by rearrangement, the acoustic pressure can also be expressed as a function of volumetric strain:

image

where

V is acoustic volumetric variation
ρ is acoustic density variation
ρo is initial density of medium
Rs is specific gas constant of medium
T is temperature of medium at initial state


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ID: 100900018 Last Updated: 9/11/2010 Revision: 0

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