# Sound energy density

Not to be confused with Sound pressure.
Sound measurements
Characteristic
Symbol
Sound pressure  p · SPL
Particle velocity  v · SVL
Particle displacement  ξ
Sound intensity  I · SIL
Sound power  P · SWL
Sound energy  W
Sound energy density  w
Sound exposure  E · SEL
Sound energy flux  Q
Acoustic impedance  Z
Speed of sound  c
Audio frequency  AF

Sound energy density or sound density (symbol w) describes the sound energy per volume unit at a given point in the medium.

## Mathematical definition

Sound energy density, denoted w and measured in J·m−3 or Pa, is given by:

$w = \frac{p v}{c}$

where:

The terms instantaneous energy density, maximum energy density, and peak energy density have meanings analogous to the related terms used for sound pressure. In speaking of average energy density, it is necessary to distinguish between the space average (at a given instant) and the time average (at a given point).

## Equations in terms of other measurements

Sound energy density can be related to other sound measurements:

$w = \frac{p^2}{c \mathfrak{R}(z)} = \frac{\mathfrak{R}(z) v^2}{c} = \frac{P}{c A} = \frac{I}{c}.$

For sine waves with angular frequency ω, the amplitude of the sound energy density can be related to those of the particle displacement and the particle acceleration:

$w_\mathrm{m}(\mathbf r) = \frac{\omega^2 z_\mathrm{m}(\mathbf r) \xi_\mathrm{m}(\mathbf r)^2}{c} = \frac{z_\mathrm{m}(\mathbf r) a_\mathrm{m}(\mathbf r)^2}{c\, \omega^2}.$
Symbol Unit Meaning
c m·s−1 speed of sound
v m·s−1 particle velocity
z Pa·m−1·s specific acoustic impedance
A m2 area
p Pa sound pressure
P W sound power
I W·m−2 sound intensity
w J·m−3 sound energy density