Introduction to Acoustics

Under such conditions the intensity is
Ir =〈p(t)ur(t)〉t =〈p 2 �
(t)〉t ρc
= p 2 � |p|
rms ρc = 2
, (25.9)
2ρc
where p2 rms is the mean square pressure (the square of
the root-mean-square pressure) and p in the rightmost
expression is the complex amplitude of the pressure in
a harmonic plane wave. In the particular case of a plane
propagating wave the sound intensity is seen to be simply
related to the rms sound pressure, which can be measured
with a single microphone. Under free-field conditions
the sound field generated by any source of finite extent
is locally plane sufficiently far from the source. This
is the basis for the free-field method of sound power
determination (which requires an anechoic room); the
sound pressure is measured at a number of points on
a sphere that encloses the source [25.3].
25.2 Active and Reactive Sound Fields
Some typical sound field characteristics can be identified.
For example, the sound field far from a source
under free-field conditions has certain well-known properties
(dealt with above); the sound field near a source
has other characteristics, and some characteristics are
typical of a reverberant sound field. One of the characteristics
of the sound field near a source is that the sound
pressure and the particle velocity are partly out of phase.
To describe such a phenomenon one may introduce the
concepts of active and reactive sound fields.
It takes four second-order quantities to describe the
distributions and fluxes of sound energy in a stationary
sound field completely [25.4–6]: potential energy
density, kinetic energy density, active intensity (the
quantity given by (25.4) and(25.5), usually simply referred
to as the intensity), and the reactive intensity. The
last of these quantities represents the non-propagating,
oscillatory sound energy flux that is characteristic
of a sound field in which the sound pressure and
the particle velocity are in quadrature (90◦ out of
phase), as for instance in the near field of a small
source. The reactive intensity is a vector defined as
the imaginary part of the product of the complex
pressure and the complex conjugate of the particle
velocity,
J = 1/2Im � pu ∗�
(25.12)
Sound Intensity 25.2 Active and Reactive Sound Fields 1055
A practical consequence of (25.9) is the following
extremely simple relation between the sound intensity
level (Iref = 1pW/m2 ) and the sound pressure level
(pref = 20 µPa),
L I � Lp . (25.10)
This is due to the fortuitous fact that
ρc � p2 ref
Iref
= 400 kg/m 2 s (25.11)
in air under normal ambient conditions. At a static pressure
of 101.3 kPa and a temperature of 23 ◦ C the error
of (25.10) is about 0.1dB.
However, it should be emphasized that in the general
case there is no simple relation between the sound intensity
and the sound pressure, and both the sound pressure
and the particle velocity must be measured simultaneously
and their instantaneous product time-averaged as
indicated by (25.4). This requires the use of a more
complicated device than a single microphone.
with the eiωt sign convention [cf. (25.5)]. Units:
power per unit area (W/m2 ). More-general timedomain
formulations based on the Hilbert transform
are also available [25.7]. Unlike the usual active intensity,
the reactive intensity remains a somewhat
controversial issue although the quantity was introduced
more than half a century ago [25.8], perhaps
because the vector J has no obvious physical meaning
[25.9], or perhaps because describing an oscillatory
flux by a time-averaged vector seems peculiar to
some. However, even though the reactive intensity
is of no obvious direct practical use it is nevertheless
quite convenient that we have a quantity that
makes it possible to describe and quantify the particular
sound field conditions in the near field of sources
in a precise manner. This will become apparent in
Sect. 25.3.2.
It can be shown from (25.5) that the active intensity
is proportional to the gradient of the phase of the sound
pressure [25.10],
I =− |p|2 ∇ϕ
, (25.13)
2ρc k
where k = ω/ c is the wavenumber. Thus the active intensity
is orthogonal to surfaces of equal phase, that is,
the wavefronts [25.5]. Likewise it can be shown from
(25.12) that the reactive intensity is proportional to the
Part H 25.2