Extending the length of the left and the right halves of the pipe by
50 cm each, we expect to affect the standing wave properties, but not
the reactive boundary conditions at the origin. These should
depend on the Helmholtz resonator only. With this longer model,
the three test point sound pressure levels are again interesting
functions of frequency.
The high levels that were experienced at 535 Hz are now lower than
480 Hz (150% of the wavelength would put this resonance at 357
Hz). The minimum for the origin is still at 565 Hz, and the
minimum for the output end is still at 585 Hz. However, now there
is a pronounced peak between them, at 575 Hz. Closer examination
shows
that at this frequency, the pressure level in the cavity is
extreme. Compare this maximum (147 dB) with the three frequencies
of the 2 m pipe, and only the 535 Hz plot comes close (at 137
dB). In effect, this is the next frequency that satisfies the
boundary condition for the standing wave in the left half of the pipe.
The addition of the Helmholtz resonator, does reduce noise
propagating down the pipe - at a particular frequency. However,
it also modifies the overall acoustic situation. It adds a
"branch" to the pipe that has its own inertia and compliance, and at
this branch point there will be reflection as well as
transmission. The reflection can create a standing wave in
the upstream part of the system that can actually increase levels, both
upstream and downstream of the resonator.
Also, a bit of a Reality Check: This model assumes a source
that provides a
constant pressure amplitude regardless of frequency. The
frequency-dependent standing wave phenomenon in the left half of the
pipe may also influence a real source, which is subject to a
frequency-dependent load. This could be seen, for example, if we
built a test apparatus using a loadspeaker driver at the input.
Copyright Daniel O. Ludwigsen, 2005
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