t EndFraction equals minus StartFraction partial-differential left-parenthesis rho u Subscript x Superscript 2 Baseline right-parenthesis Over partial-differential x EndFraction minus StartFraction partial-differential upper P Over partial-differential x EndFraction plus f Subscript x"/> (2.5)
Here, fx=Fx/(Adx) is the volume force density (force per volume). Using the chain law the partial derivatives of the first and second term lead to
The term in brackets is the homogeneous continuity Equation (2.1), and Equation (2.6) simplifies to
As with the conservation of mass, this can be extended to three dimensions:
This equation is the non-linear, inviscid momentum equation called the Euler equation.
2.2.3 Equation of State
The above equations relate pressure, velocity and density. For further reducing this set we need a third equation. The easiest way would be to introduce the . Here we start with the first law of thermodynamics in order to show the difference between isotropic (or adiabatic) equation of state and other relationships.
With the following specific quantities per unit mass
With the specific entropy ds=dq+drT we get:
The relation dv=d(1/ρ) comes from the fact that v is a mass specific value and therefore the reciprocal of the density ρ=1/v. For an ideal gas we have
cp and cv are the specific thermal heat capacities for constant pressure and volume, respectively. That is the ratio of temperature change ∂T per increase of heat ∂q. From the total differential
we can derive
Using all above relations the change in density dρ is:
with κ=cv/cp. In most acoustic cases the process is isotropic: i.e. time scales are too short for heat exchange in a free gas; thus ds=0, and the change of pressure per density is
In case of constant temperature (isothermal) dT=0 we get with (2.12) and the ideal gas law (2.11):
As we will later see, c0 is the . Newton calculated the wrong speed of sound based on the assumption of constant temperature that was later corrected by Laplace by the conclusion that the process is adiabatic. For fluids and liquids like water a different quantity is used because there is no such expression as the ideal gas law. The bulk modulus is defined as:
