The time‐averaged radial sound intensity in the far field of a dipole is given by [4]
3.7 Sound Power of Sources
3.7.1 Sound Power of Idealized Sound Sources
The sound power W of a sound source is given by integrating the intensity over any imaginary closed surface S surrounding the source (see Figure 3.7):
Figure 3.7 Imaginary surface area S for integration.
The normal component of the intensity In must be measured in a direction perpendicular to the elemental area dS. If a spherical surface, whose center coincides with the source, is chosen, then the sound power of an omnidirectional (monopole) source is
(3.42)
and from Eq. (3.35) the sound power of a monopole is [4, 13]
It is apparent from Eq. (3.44) that the sound power of an idealized (monopole) source is independent of the distance r from the origin, at which the power is calculated. This is the result required by conservation of energy and also to be expected for all sound sources.
Equation (3.43) shows that for an omnidirectional source (in the absence of reflections) the sound power can be determined from measurements of the mean square sound pressure made with a single microphone. Of course, for real sources, in environments where reflections occur, measurements should really be made very close to the source, where reflections are presumably less important.
The sound power of a dipole source is obtained by integrating the intensity given by Eq. (3.40) over a sphere around the source. The result for the sound power is
The dipole is obviously a much less efficient radiator than a monopole, particularly at low frequency.
Example 3.7
Two monopoles of equal sound power W = 0.1 watt at 150 Hz, but pulsating with a phase difference of 180° are spaced λ/12 apart. Determine the sound power of this dipole at 150 Hz.
Solution
We know that l = λ/12, λ = 343/150 = 2.29 m, and Wm = 0.1 watt. If we compare the sound power of a dipole Wd with that of a monopole Wm (Eqs. (3.44) and (3.45)) we find that
Therefore, the sound power radiated by the dipole is 9 mW.
In practical situations with real directional sound sources and where background noise and reflections are important, use of Eq. (3.43) becomes difficult and less accurate, and then the sound power is more conveniently determined from Eq. (3.41) with a sound intensity measurement system. See Ref. [22] in this book and chapter 106 in the Handbook of Acoustics [1].
We note that since p/ur = ρc (where ρ = mean air density kg/m3 and c = speed of sound 343 m/s) for a plane wave or sufficiently far from any source, that
where Eq. (3.46) is true for random noise as well as for a single‐frequency sound, known as a pure tone.
Note that for such cases we only need to measure the mean‐square sound pressure with a simple sound level meter (or at least a simple measurement system) to obtain the sound intensity from Eq. (3.46) and then from that the sound power W watts from Eq. (3.41) is
for an omnidirectional source (monopole) with no reflections and no background noise. This result is true for noise signals and pure tones that are produced by omnidirectional sources and in the so‐called far acoustic field.
For the special case of a pure‐tone (single‐frequency) source of sound pressure amplitude,
For measurements on a hemisphere, W = 2πr2 p2rms /ρc and for a pure‐tone source
Note that in the general case, the source is not omnidirectional, or more importantly,
