In order to understand what happens when a classic Helmholtz
Resonator is applied to control sound propagating through a duct, it
helps to visualize the situation. Through a finite element model
(created in Femlab 3.1), the effect of this noise control solution can
be seen easily. The basic problem involves noise propagating down
a pipe or tube. Most standard acoustics texts provide background
on the theory behind the Helmholtz resonator, for example, Fundamentals of Acoustics, 4th ed.,
by Kinsler, Frey, Coppens, and Sanders, has a discussion in section
10.8.
The finite element model geometry consists of a square cross section
pipe two meters long, and ten centimeters in width. The end at z = -1 m is driven at a single
frequency with a given pressure amplitude (10 Pa, or, equivalently, 111
dB). The end at z = 1 m has a
"radiation" boundary condition, a feature of Femlab that models an
infinite pipe. Thus there is no reflection from the end at z = 1 m. The Helmholtz
resonator is a small cube connected to the pipe with a square
neck. The dimensions of the cube and neck were chosen to provide
a resonance frequency of 500 Hz. The speed of sound is set at 344
m/s, and the density of the gas is 1.2 kg/m3.
The combination of the long pipe with the Helmholtz resonator
located at the middle (z = 0)
is shown above.
The pipe alone is a fine channel for propagating noise. With
one end of the pipe set to drive the system at a given pressure
amplitude and the other to radiate into a virtual infinite pipe, the
pressure wave looks like this:
If we were to insert a sound level meter at various places along the
pipe, and read unweighted sound pressure level (in dB re: 20 microPa),
the finite element model predicts essentially no variation along the z axis. All frequencies
tested give roughly the same undulation, which is attributed to the
size of the mesh in this model:
While the pressure at a particular point will certainly oscillate in
time, the sound pressure level will not, as long as we have a wave
traveling down the pipe. The rest of this discussion will use
sound pressure level instead of a snapshot of pressure at a single
time. The sound pressure level is effectively an average over
time, and represents what we would hear, or measure with a sound level
meter.
When the Helmholtz resonator is added at the center of the length of
the pipe, the entire situation changes. We might expect the first
meter to act as a pipe element on its own, driven at one end and
terminated with a reactive (inertia and compliance) boundary condition
at the other. Therefore, the Helmholtz resonator creates a
standing wave in half of the pipe. By solving the model at a
series of frequencies (from 480 Hz to 600 Hz in steps of 5 Hz), we can
see that particular frequencies fit these boundary conditions
well. The following plot shows pressure at the center of the pipe
(in dB re: 20 microPa) as a function of location along the z axis. This is significantly
different than the previous constant propagation through the pipe.
Obviously, the noise injected at z
= -1 m is partially reflected back to the source (creating a standing
wave), and partially transmitted through beyond the Helmholtz
resonator. The sound pressure levels downstream from the
resonator depend strongly on frequency - ranging from over 120 dB to
just above 70 dB. To get to the bottom of this, we could take a
closer look at the sound pressure levels at three key points: right at
the driven end (0, 0, -1 m), exactly over the resonator (0, 0, 0 m),
and at the output end of the tube (0, 0, 1 m). These are plotted
as functions of frequency:
It is no surprise that the driven end is at 111 dB regardless of the driving frequency; this is part of the applied boundary condition. However, three important frequencies emerge on this plot. Near 535 Hz, both the origin and the output test points yield maximum levels. Near 565 Hz, there is a frequency which yields a minimum level just over the resonator at the origin. Finally, near 585 Hz, the level at the output is a mimimum.
Copyright Daniel O. Ludwigsen, 2005
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