The answers to these questions can be understood through an
understanding of what the situation looks like at the origin. The
following series of plots show the sound pressure level in the pipe and
resonator at the three key frequencies.
These images allow us to see why our virtual sound level meter
probes highlighted these three frequencies. It appears that the
535 Hz wave fits the left half of the pipe quite well, with its
particular
boundary conditions. The cavity of
the resonator is subject to the greatest pressure levels in the
model. We also see the nodal planes shown
in blue start to "wrap around" near the origin as a result of the
oscillation of the air in the Helmholtz resonator. As
an aside, the wavelengths at these frequencies
are much longer than the dimensions of the resonator cavity or neck -
therefore the treatment as lumped mass (inertia) and spring
(compliance) elements is justified. For example, we expect the
air in the neck to move as a unit, and the cavity to have fairly
uniform pressure. To visualize the lumped mass motion, the
following vector plot illustrates the particle velocity at the
neck. The arrows show the magnitude of the oscillatory
motion. The moving plug of air obviously extends past the end of
the neck, motivating the use of "end corrections" to take into account
additional inertia.
The development of the interaction at that junction as we go to
higher frequencies forces the nodal plane (in blue) to deform
further. At 565 Hz, it passes through the origin, explaining our
dip in the sound meter results at that frequency and location.
Finally, by 585 Hz, the Helmholtz resonator interaction causes that
nodal plane to carry on through the right half of the pipe.
Copyright Daniel O. Ludwigsen, 2005
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