Speed of Sound - Basic Formula

Basic Formula

In general, the speed of sound c is given by the Newton-Laplace equation:

c = \sqrt{\frac{K}{\rho}}\,

where

K is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gases),
is the density

Thus the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density. For ideal gases the bulk modulus P is simply the gas pressure multiplied by the adiabatic index.

For general equations of state, if classical mechanics is used, the speed of sound is given by

c^2=\frac{\partial p}{\partial\rho}

where the derivative is taken with respect to adiabatic change.

where is the pressure and is the density

If relativistic effects are important, the speed of sound may be calculated from the relativistic Euler equations.

In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds, a mixture of oxygen and nitrogen constitutes a non-dispersive medium. But air does contain a small amount of CO2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic frequencies (> 28 kHz).

In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity, while the energy of the disturbance propagates at the group velocity. The same phenomenon occurs with light waves; see optical dispersion for a description.

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