Introduction to Acoustics

where the overbars represent time averages. When p1
and p2 are uncorrelated, the cross-term in (23.5) is
zero, and it is the mean square pressures that add. This
is the case in most noise control situations where the
noise comes from one or more different sources. When
p1 = p2, it is the pressures that add. When p1 =−p2,
generally the objective in active noise control, the resulting
mean-square pressure is zero. For sinusoidal signals,
the mean-square pressure depends on the amplitudes of
the waves and the phase difference between them.
Other quantities of importance are the intensity I of
the sound wave, a vector in the direction of propagation
that is the sound energy per unit of area perpendicular to
an element of surface area dS, Chap. 6. For a source of
sound, the total sound power W radiated by the source
is given by
�
W = I · dS , (23.6)
S
where the integral is over a surface that surrounds
the source. Sound power is a widely used measure of
the noise emission of the source. Its determination using
sound intensity is advantageous in the presence of
background noise because, in the absence of soundabsorptive
materials inside the surface, any sound energy
that enters the surface from the outside also leaves the
surface from the inside, and thus does not affect the radiated
power. In practice, the sound intensity on the surface
is determined by scanning or by measurements at a finite
number of measurement positions, and only an approximation
to the ideal situation is obtained Chap. 6. Other
techniques for the determination of the sound power of
sources depend on an approximation of the intensity in
terms of mean-square pressure, and are discussed later
in this chapter.
The Decibel as a Unit of Level
As will be shown below, the magnitude of the quantities
associated with a sound wave described above vary
over a very wide range, and it is convenient to use logarithms
to compress the scale. Logarithms are also useful
in many noise transmission problems because quantities
that are multiplicative in terms of the quantities themselves
are additive using logarithms. The decibel is a unit
of level, and is defined as
Lx = 10 log X
, (23.7)
X0
where X is a quantity related to energy, X0 is a reference
quantity, and log is the logarithm to the base 10.
In the case of sound pressure, X = p 2 , the mean square
pressure and X0 = p 2 0 . The reference pressure p0 is standardized
at 20 µPa (2 × 10 −5 N/m 2 ). The sound pressure
level can then be written as
Lp = 10 log
p2 (2 × 10−5 dB . (23.8)
) 2
For sound power and sound intensity, the reference levels
are 10 −12 W and 10 −12 W/m 2 , respectively, and the
corresponding sound power and sound intensity levels
are
LW = 10 log W
dB , (23.9)
10−12 I
LI = 10 log dB . (23.10)
10−12 Note that all three quantities above are expressed in decibels;
the decibel is a unitoflevel,andmaybeusedto
express a variety of quantities related to energy relative
to a reference level. This fact often causes confusion because
the quantity being expressed in decibels is often
not stated (“The level is 75 dB.”), and the meaning of
the statement is not clear. The wide use of sound power
level as a measure of noise emission can easily cause
confusion between sound power level and sound pressure
level. In this situation, it is convenient, and common
in the information technology industry, to omit the 10
before the logarithm in (23.9), and express the sound
power level in bels, where one bel = 10 dB.
Relative Magnitudes
As mentioned above, the quantities associated with
a sound wave are small. For example, a sound pressure
level of 90 dB is relatively high and corresponds to
an RMS sound pressure of
�
��4×10
−10 � x1090/10� = 0.63 Pa . (23.11)
pA =
Since atmospheric pressure pat is nominally 1.01 × 10 5 Pa,
the ratio p0/pat is very small: 0.62 × 10 −5 .
The particle velocity can be determined from (23.3)
assuming plane wave propagation of a sinusoidal wave
having a radian frequency ω (ω = 2π f ). In this case,
(23.3) reduces to p = ρcu where ρ is the density of air,
nominally 1.18 kg/m 2 . The particle velocity is then
u = 0.63/(1.18x344) = 1.6×10 −3 m/s (23.12)
and the ratio u/c is 4.7×10 −6 . The particle velocity
is very small compared with the speed of sound, and
the model of a wave traveling with speed c and having
a particle velocity u is justified. When this is not the
case, the compressional portion of the wave travels with
Noise 963
Part G 23