Introduction to Acoustics

1054 Part H Engineering Acoustics
Part H 25.1
advantages of this measurement technique is that sound
intensity measurements make it possible to determine
the sound power of sources in situ without the use of
costly special facilities such as anechoic and reverberation
rooms, and sound intensity measurements are now
routinely used in the determination of the sound power
of machinery and other sources of noise in situ. Other
25.1 Conservation of Sound Energy
In the absence of mean flow the instantaneous sound
intensity is the product of the sound pressure p(t) and
the particle velocity u(t)
I(t) = p(t)u(t) . (25.1)
By combining the fundamental equations that govern
a sound field, the equation of conservation of mass, the
adiabatic relation between changes in the sound pressure
and in the density, and Euler’s equation of motion,
one can show that the divergence of the instantaneous
intensity equals the (negative) rate of change of the sum
of the potential and kinetic energy density w(t),
∇·I(t) =− ∂w(t)
. (25.2)
∂t
This is the equation of conservation of sound energy
[25.1, 2], expressing the simple fact that if there
is net flow of energy away from a point in a sound
field then the sound energy density at that point is reduced
at a corresponding rate. Integrating this equation
over a volume V enclosed by the surface S gives, with
Gauss’s divergence theorem,
�
I(t) · dS =− ∂
⎛ ⎞
�
⎝ w(t)dV⎠
∂t
S
V
=− ∂E(t)
, (25.3)
∂t
in which E(t) is the total sound energy in the volume
as a function of time. The left-hand term is the total
net outflow of sound energy through the surface, and
the right-hand term is the rate of change of the total
sound energy in the volume. In other words, the net
flow of sound energy out of a closed surface equals
the (negative) rate of change of the sound energy in
the volume enclosed by the surface because energy is
conserved.
In practice we are often concerned with stationary
sound fields and the time-averaged sound intensity rather
important applications of sound intensity include the
identification and rank ordering of partial noise sources,
the determination of the transmission losses of partitions,
and the determination of the radiation efficiencies
of vibrating surfaces. Because the intensity is a vector
it is also more suitable for visualization of sound fields
than, for instance, the sound pressure.
than the instantaneous intensity, that is,
〈I(t)〉 t = 〈p(t)u(t)〉 t . (25.4)
For simplicity the symbol I is used for this quantity
in what follows rather than the more precise notation
〈I(t)〉 t. If the sound field is harmonic with angular
frequency ω = 2π f the complex representation of the
sound pressure and the particle velocity can be used,
which leads to the expression
I = 1/2Re(pu ∗ ) , (25.5)
where u∗ denotes the complex conjugate of u.
A consequence of (25.3) is that the integral of the
normal component of the time-averaged sound intensity
over a closed surface is zero,
�
I · dS = 0 , (25.6)
S
when there is neither generation nor dissipation of sound
energy in the volume enclosed by the surface, irrespective
of the presence of steady sources outside the surface.
If the surface encloses a steady source then the surface
integral of the time-averaged intensity equals the sound
power emitted by the source Pa,thatis,
�
I · dS = Pa, (25.7)
S
irrespective of the presence of other steady sources outside
the surface. This important equation is the basis for
sound power determination using sound intensity.
In a plane wave propagating in the r-direction the
sound pressure p and the particle velocity ur are in phase
and related by the characteristic impedance of air ρc,
where ρ is the density of air and c is the speed of sound,
ur(t) = p(t)
. (25.8)
ρc