Introduction to Acoustics

1066 Part H Engineering Acoustics
Part H 25.3
position in a one-dimensional interference field with
a standing-wave ratio of 24 dB. It is apparent that the
pressure-intensity index varies strongly with the position
in such a sound field. Accordingly, the influence
of a given phase error depends on the position, as
shown in the calculations presented in Fig. 25.16b. Figure
25.17 shows an example of measurements with an
intensity probe drawn through a standing-wave tube.
The standing-wave test will also reveal other sources
of error than phase mismatch, for example, the influence
of an unacceptably high vent sensitivity of the
microphones [25.51].
25.3.2 The p–u Measurement Principle
A p–u sound intensity measurement system combines
two fundamentally different transducers, a pressure microphone
and a particle velocity transducer. The sound
intensity is simply the time average of the instantaneous
product of the pressure and particle velocity signal,
Ir = 〈p(t)ur(t)〉 t = 1/2Re � pu ∗� r , (25.31)
where the latter expression is based on the complex
exponential representation. In the frequency domain the
expression takes the form
Ir(ω) = Spu(ω). (25.32)
Equation (25.32) gives the same result as (25.31)
when the intensity spectrum is integrated over the frequency
band of concern.
A p–u sound intensity probe that combined a pressure
microphone with a transducer based on the
convection of an ultrasonic beam by the particle velocity
flow was produced by Norwegian Electronics for some
years [25.2, 52], but the device was somewhat bulky,
very sensitive to airflow, and difficult to calibrate, and
production was stopped in the middle of the 1990s. More
recently, a micromachined transducer called the Microflown
has become available for measurement of the
particle velocity [25.15], and an intensity probe based
on this device in combination with a small pressure microphone
is now in commercial production [25.16]; see
Fig. 25.18. The Microflown particle velocity transducer
consists of two short, thin, closely spaced wires, heated
to about 300 ◦ C [25.15]. Their resistance depends on the
temperature. A particle velocity signal in the perpendicular
direction changes the temperature distribution
instantaneously, because one of the wires will be cooled
more than the other by the airflow. The frequency response
of this device is relatively flat up to a corner
Fig. 25.18 A p–u sound intensity probe (by Microflown
Technologies, The Netherlands)
frequency of the order of 1 kHz caused by diffusion
effects related to the distance between the two wires.
A second corner frequency at about 10 kHz is caused by
the thermal heat capacity of the wires. Between 1 and
10 kHz there is a roll-off of 6 dB per octave. The particle
velocity transducer is combined with a small electret
condenser microphone in the 1/2 inch sound intensity
probe shown in Fig. 25.18. The velocity transducer is
mounted on a small, solid cylinder, and the condenser
microphone is mounted inside another, hollow cylinder.
The geometry of this arrangement increases the
sensitivity of the velocity transducer. Unlike Norwegian
Electronics’ intensity probe the Microflown probe
is very small – in fact much smaller than a standard p–p
probe. Thus it is possible to measure very close to a vibrating
surface with this device. At the time of writing
there is still relatively little experimental evidence of the
practical utility of the Microflown intensity probe, but it
seems to be promising [25.17]. Figure 25.19 shows the
results of sound power measurements with a Microflown
intensity probe in comparison with similar results made
with a Brüel & Kjær p–p sound intensity probe.
A variant of the p–u method, the surface intensity
method, combines a pressure microphone with
a transducer that measures the vibrational displacement,
velocity or acceleration of a solid surface, for example
with an accelerometer or a laser vibrometer [25.53–55];
see Fig. 25.20. Obviously, this method can only give the
normal component of the sound intensity near vibrating
surfaces. A disadvantage of the surface intensity method
is that sound fields near complex sources are often very