# 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 in the medium can be expressed as:

${\displaystyle P=P_{o}+p}$

where ${\displaystyle P_{o}}$ is the hydrostatic or ambient pressure and ${\displaystyle p}$ is the acoustic pressure or pressure disturbance.

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.

${\displaystyle \rho =\rho _{o}+\varrho }$

${\displaystyle T=T_{o}+dT}$

${\displaystyle \mathbf {U=u_{o}+u} }$

Notice that pressure, density, and temperature are scalar quantities and the particle velocity is a vectorial quantity.

Plane wave travelling inside a tube

Let's consider a plane wave travelling in the x-direction inside a tube filled with a fluid at rest with constant pressure, density and temperature. As the wave moves, 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, the plane wave inertial frame of reference is followed and a control volume is drawn around the wave.

Control volume around a travelling plane wave

Applying continuity and neglecting higher order terms

${\displaystyle \rho _{o}C_{o}A=(\rho _{o}+\varrho )(C_{o}-u)A}$

${\displaystyle \rho _{o}u=C_{o}\varrho }$

Applying conservation of momentum, neglecting higher order terms

${\displaystyle P_{o}+\rho _{o}C_{o}^{2}=P_{o}+p+(\rho _{o}+\varrho )(C_{o}-u)^{2}}$

${\displaystyle \rho _{o}C_{o}^{2}=p+(\rho _{o}+\varrho )(C_{o}^{2}-2C_{o}u)}$

${\displaystyle 0=p+-2C_{o}\rho _{o}u+\varrho C_{o}^{2}}$

${\displaystyle C_{o}^{2}={\frac {p}{\varrho }}}$

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.

${\displaystyle P(\rho _{o}+\varrho )=P_{o}+p=P(\rho _{o})+{\frac {\partial P(\rho _{o})}{\partial \rho }}\varrho +{\frac {1}{2}}{\frac {\partial ^{2}P(\rho _{o})}{\partial \rho ^{2}}}\varrho ^{2}+...}$

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

${\displaystyle C_{o}^{2}={\frac {p}{\varrho }}={\frac {\partial P(\rho _{o})}{\partial \rho }}}$

As a sound wave moves through a fluid, the fluid follows 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 following thermodynamic relation.

${\displaystyle P=C\rho ^{\gamma }}$

Taking the partial derivative and using the ideal gas law,

${\displaystyle {\frac {\partial P}{\partial \rho }}=C\gamma \rho ^{\gamma -1}={\frac {\gamma P}{\rho }}=\gamma R_{g}T}$

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

${\displaystyle C_{o}^{2}={\frac {\partial P(\rho _{o})}{\partial \rho }}={\frac {\gamma P_{o}}{\rho _{o}}}=\gamma R_{g}T_{o}}$

Using the definition of the Bulk modulus,the speed of sound can written in three different forms.

${\displaystyle B_{o}=\rho _{o}{\frac {\partial P(\rho _{o})}{\partial \rho }}=\gamma P_{o}}$

${\displaystyle C_{o}={\sqrt {\frac {\gamma P_{o}}{\rho _{o}}}}={\sqrt {\frac {B}{\rho _{o}}}}={\sqrt {\gamma R_{g}T_{o}}}}$

Original Document [1]

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