Engineering Acoustics/Speed of sound

When sound waves propagate in a medium, they cause fluctuations in the pressure, density, temperature and particle velocity in the medium. The total pressure, $$P$$, in the medium can be expressed as:

$$       P=P_o+p' $$

where $$P_o$$ is the hydrostatic or ambient pressure and $$p'$$ is the acoustic pressure or pressure fluctuation.

The hydrostatic pressure can be thought as the mean pressure while the acoustic pressure represents fluctuations around the mean pressure. Similarly, the density, temperature, and particle velocity are separated into mean and fluctuating components.

$$\rho=\rho_o+\varrho ' $$

$$       T=T_o + T' $$

$$               \vec{U} =\vec{u}_o+\vec{u}'

$$

where $$       \rho_o

$$ is the ambient density; $$       \varrho '

$$, the density fluctuation; $$       T_o

$$, the ambient temperature; $$       T'

$$, the temperature fluctuation; $$       \vec{u}_o

$$, the mean velocity; and $$       \vec{u}'

$$, the particle velocity. Notice that pressure, density, and temperature are scalar quantities and the particle velocity is a vectorial quantity. Let's consider a planar wave traveling in the x-direction inside a tube filled with a fluid at rest ($$       \vec{u}_o=0

$$) with constant pressure, density and temperature ($$P_o$$, $$\rho_o$$, and $$T_o$$, respectively). As the wave moves through the fluid at a speed, $$       c_o

$$,  it creates infinitesimally small fluctuations in the initially stagnant fluid in front. All four quantities describing the fluid vary around their mean value, increasing or decreasing depending on whether the fluid is being compressed or expanded. To obtain a relation for the propagating speed, $$       c_o

$$, an inertial frame of reference is used and a control volume is drawn around the wave. Applying continuity,

$$       \rho_o c_o A = (\rho_o+\varrho')(c_o-u') A $$

and neglecting higher order terms,

$$           \rho_o u'= c_o \varrho' $$

Applying conservation of momentum,

$$       P_o+ \rho_o c_o^2 = P_o + p' + (\rho_o+\varrho')(c_o-u')^2 $$

neglecting higher order terms,

$$    \rho_o c^2_o = p' + (\rho_o+\varrho')(c_o^2-2 c_o u') $$

$$    0 = p' + -2c_o \rho_o u' +\varrho' c_o^2 $$,

and using the continuity equation,

$$       c_o^2=\frac{p'}{\varrho'} $$

The speed of sound is related to the ratio of pressure fluctuation (acoustic pressure) to density fluctuation. Given that the speed of sound is always a positive quantity, an increase in the fluid pressure implies an increase in the fluid density and vice versa. The total pressure is expressed as a Taylor series expansion about the ambient density to relate its infinitesimally small fluctuations to the total pressure and density.

$$       P(\rho_o+\varrho')= P_o + p' = P(\rho_o)+ \frac{\partial P(\rho_o)}{\partial \rho} \varrho' + \frac{1}{2} \frac{\partial^2 P(\rho_o)}{\partial \rho^2} \varrho'^2 + ... $$

Neglecting second order terms and higher, the speed of sound can be related to the total pressure and density.

$$       c_o^2=\frac{p'}{\varrho'}= \frac{\partial P(\rho_o)}{\partial \rho} $$

As a sound wave moves through a fluid, the fluid is usually assumed to follow an adiabatic and reversible thermodynamic path. So, the heat transfer between fluid particles is negligible and the changes caused by the sound wave onto the fluid can be reverse to their original state without changing the entropy of the system. For an isentropic process, the total pressure and density are related by the thermodynamic relation,

$$P=C \rho^\gamma$$.

Taking the partial derivative and using the ideal gas law, $$P=\rho R_g T $$,

$$\frac{\partial P}{\partial \rho} = C \gamma \rho ^{\gamma -1 } = \frac{\gamma P}{\rho} = \gamma R_g T$$.

The speed of sound can be expressed in terms of the ambient pressure, density and temperature using,

$$ c_o^2= \frac{\partial P(\rho_o)}{\partial \rho} = \frac{\gamma P_o}{\rho_o} = \gamma R_g T_o$$.

Using the definition of the Bulk modulus,

$$B_o=\rho_o \frac{\partial P(\rho_o)}{\partial \rho}= \gamma P_o$$

the speed of sound can written as,

$$       c_o = \sqrt{\frac{\gamma P_o}{\rho_o}} = \sqrt{\frac{B_o}{\rho_o}} = \sqrt{ \gamma R_g T_o} $$.