2.3Equation of State

The third equation needed for deriving the acoustic wave equation is called the equation of state which is the most difficult equation to derive and comprehend among the three equations. This is because it involves both the kinetic theory of gases and the principle of conservation of energy. Despite this difficulty, the physics involved in the equation of state is both interesting and important for understanding acoustics.

The equation of state describes the relationship between air pressure change and mass density change as shown in the figure below:

: Absolute density

: Averaged density

: Absolute pressure

The equation of state can be expressed as:

where:

Po is the average air pressure. P is the instantaneous pressure.

ρo is the averaged air mass density. ρ is the instantaneous mass density.

γis the ratio of specific heats and is defined as:

γ3α þ 2 3α

where α is a ratio between the total energy (rotational energy + translational energy) and the translational energy of molecules as:

12 Iaθ2ca ¼ ðα 1Þ 12 mav2ca

where α is equal to or greater than 1. When the particle does not have rotational energy, α¼1; when the particle has rotational energy, α >1:

=

̇

Using this linear relationship between rotational energy and translational energy, the total energy of a particle is given as follows:

Ka ¼ 12 mav2ca þ 12 Iaθ2ca ¼ 12 mav2ca þ ðα 1Þ 12 mav2ca ¼ α2 mav2ca

According to the conservation of energy, in a closed system, the increase of energy is equal to the work done to the system:

The total energy Ka includes both translational energy and rotational energy and is equal to α2 mav2ca , as shown before. And the work done by a constant external pressure Po due to a system of decreased volume dVa is PodVa; therefore:

Ka ¼ W

!α2 d mav2ca ¼ PodVa

2.3.2Pressure due to Colliding of Gases

According to the kinetic theory, the pressure is directly related to the translational energy and is independent of the rotational energy. The pressure due to the particle collision is:

Po ¼ 1 ma v2

3 Va ca

The derivation of the collision pressure is based on the fundamentals of kinetic energy, as shown in the figure below:

Colliding Pressure (Kinetic theory of gases)

1 = 3

4 = 3

According to Newton’s law of motion, the force is equal to the change of momentum when the particle collides with the wall of the sphere as:

The pressure on the surface of the sphere is the force divided by the total surface area of the sphere as:

2

Note that the pressure is directly related to the translational energy only (not include the rotational energy), and rewrite the relation as: