γ 3α þ 2 3α

ð3αþ2Þ γ

Replacing α with the ratio of specific heats yields:

3

The ratio, γ, of specific heats is related to the ratio, α, between total energy and translation energy. The ratio, γ, for air can be measured (α ¼ 53) and is approximately equal to:

γ ¼ 1:4

Note that γ ¼ 1.4 is approximately equivalent to α ¼ 53 where α relates to the rotational energy and translational energy as mentioned earlier.

The ratio of specific heats, γ, is also referred to as the heat capacity ratio, the adiabatic index, adiabatic exponent, or Laplace’s coefficient. The ratio of specific heats, γ, can be formulated as the heat capacity, Cp, at a constant pressure condition divided by the heat capacity, Cv, at a constant volume condition. Further discussion is beyond the scope of this course.

2.4Derivation of Acoustic Wave Equation

The acoustic wave equation is based on the three physics principles, as introduced in the previous section.

The sound pressure can be related to the air density using the first principle:

The air density can be related to the flow velocity using the second principle:

Finally, the air density can be related back to the sound pressure using the third principle:

40 2 Derivation of Acoustic Wave Equation

III. Equation of State

(Based on Kinetic theory and Conservation of Energy)

III. =

From Euler’s force equation, we have the relationship between pressure difference and flow velocity.

The first step is to replace the flow velocity with the air mass density so that the relationship between pressure difference and mass density can be formed. This is accomplished by combining (I) Euler’s force equation and (II) the equation of continuity and eliminating the flow velocity, as shown in the figure below. Now we have a direct relationship between pressure difference and the air mass density.

The second step is to replace the air mass density with the pressure difference so that the acoustic wave equation has only one variable – the pressure difference P P0. This is accomplished by combining (III) equation of state and the relationship between the pressure difference and the air mass density from the previous step.

The third step is to replace the pressure difference with sound pressure, defined as p ¼ P Po. This yields:

This is the one-dimensional acoustic wave equation in Cartesian coordinates:

= − = =

By introducing the speed of sound c as:

c2 ¼ γPo

ρo

Finally, we can summarize that the acoustic wave can be formatted in the general wave equation as:

∂2 p ¼ 1 ∂2 p

∂x2 c2 ∂t2

where c is the speed of the sound:

r

c ¼ γPo

ρo

where γ is the ratio of specific heats and the approximate value of γ for air is:

3α þ 2 γ 3α ¼ 1:4

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

The approximated value of α for air molecules is:

α ¼ 53