AcademySound
Academy
Speed of Sound
Level 1 - Physics topic page in Sound.
Principle
Sound speed is set by the medium's stiffness divided by its inertia.
Notation
\(v\)
sound speed
\(\mathrm{m\,s^{-1}}\)
\(B\)
bulk modulus
\(\mathrm{Pa}\)
\(\rho\)
medium density
\(\mathrm{kg\,m^{-3}}\)
\(\gamma\)
adiabatic index for a gas
1
\(p\)
gas pressure
\(\mathrm{Pa}\)
\(T\)
absolute temperature
\(\mathrm{K}\)
Method
A sound wave compresses and expands the medium. The restoring effect is the bulk modulus, while density measures the inertia being accelerated.
Elastic medium
\[v=\sqrt{\frac{B}{\rho}}\]
Adiabatic gas stiffness
\[B=\gamma p\]
Rapid sound compressions exchange little heat with surroundings.
Gas speed
\[v=\sqrt{\frac{\gamma p}{\rho}}\]
Ideal-gas form
\[v=\sqrt{\frac{\gamma RT}{M}}\]
For an ideal gas, pressure and density combine into temperature and molar mass.
In a given gas, warmer temperature usually means a larger sound speed.
Rules
These are the compact sound-speed models.
Elastic speed
\[v=\sqrt{\frac{B}{\rho}}\]
Gas speed
\[v=\sqrt{\frac{\gamma p}{\rho}}\]
Ideal gas speed
\[v=\sqrt{\frac{\gamma RT}{M}}\]
Examples
Question
Air has
\[\gamma=1.40\]
pressure \[1.01\times10^5\,\mathrm{Pa}\]
and density \[1.20\,\mathrm{kg\,m^{-3}}\]
Estimate the sound speed.Answer
\[v=\sqrt{\frac{\gamma p}{\rho}}=\sqrt{\frac{(1.40)(1.01\times10^5)}{1.20}}=343\,\mathrm{m\,s^{-1}}\]
Checks
- A stiffer medium raises sound speed if density is unchanged.
- A larger density lowers sound speed if stiffness is unchanged.
- Use kelvin, not degrees Celsius, in the ideal-gas form.
- Sound speed is a property of the medium, not the loudness of the sound.