Introduction to Acoustics

so the time-averaged incident energy is totally reflected.
Acoustic energy is present in the second fluid, but its
time average over a wave period stays constant once the
steady state is reached.
3.10.3 Theory of the Impedance Tube
Impedance tubes are commonly used in the measurement
of specific acoustic impedances; the underlying
theory [3.66, 67] is based for the most part on (3.295),
(3.296), and (3.297) above. The incident and the reflected
waves propagate along the axis of a cylindrical
tube with the sample surface at one end. A loudspeaker
at the other end creates a sinusoidal pressure disturbance
that propagates down the tube. Reflections from the
end covered with the test material create an incomplete
standing-wave pattern inside the tube.
The wavelength of the sound emitted by the source
can be adjusted, but it should be kept substantially
larger than the pipe diameter, so that the plane wave
assumption holds. With kx identified as being 0, the
complex amplitude that corresponds to the sum of the
incident and reflected waves has an absolute magnitude
given by
| ˆp|=| f ˆ||1
+ Êe 2iky | , (3.309)
where y is now the distance in front of the sample. The
second factor varies with y and repeats at intervals of
a half-wavelength, and varies from a minimum value of
1 −|Ê| to a maximum value of 1 +|Ê|. Consequently,
y
y
ymax, 2
Incident
Reflected
ymin, 1
Sample
ymax, 1
(p 2 )av
(p 2 )av, max
(p 2 )av, min
Fig. 3.20 Incident and reflected waves inside an impedance
tube. The time-averaged pressure within the tube has minimum
and maximum values whose ratio depends on the
impedance of the sample at the end. Another measured parameter
is the distance back from the sample at which the
first maximum occurs
Basic Linear Acoustics 3.10 Reflection and Transmission 63
the ratio of the peak acoustic pressure amplitude | ˆp|max
(which occurs at one y-position) to the minimum acoustic
pressure amplitude | ˆp|min (which occurs at a position
a quarter-wavelength away) determines the magnitude
of the reflection coefficient via the relation
| ˆp|min
=
| ˆp|max
1 −|Ê|
. (3.310)
1 +|Ê|
The phase δ of the reflection coefficient can be
determined with use of the observation that the peak
amplitudes occur at y-values where δ + 2ky is an integer
multiple of 2π, while the minimum amplitudes occur
where it is π plus an integer multiple of 2π. Once the
magnitude and phase of the reflection coefficient are determined,
the specific acoustic impedance can be found
from (3.296)and(3.297).
3.10.4 Transmission through Walls
and Slabs
The analysis of transmission of sound through a wall or
a partition [3.68] is often based on the idealization that
the wall is of unlimited extent.
If the incoming plane wave has an angle of incidence
θI and if the fluid on the opposite side of the wall has the
same sound speed, then the trace velocity matching principle
requires that the transmitted wave be propagating
in the same direction.
A common assumption when the fluid is air is that
the compression in the wall is negligible, so the wall is
treated as a slab that has a uniform velocity vsl throughout
its thickness. The slab moves under the influence
of the incident, reflected, and transmitted sound fields
according to the relation (corresponding to Newton’s
θI
nI
pfront
vfront
vback
pback
Fig. 3.21 Transmission of an incident plane wave through
a thin flexible slab
Part A 3.10