Helmholtz Resonator Model - Daniel Ludwigsen

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Interaction between the Helmholtz resonator and the standing wave

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.
Pressure at 535 Hz 
Pressure at 565 Hz 
Pressure at 585 Hz 

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.
Velocity vector field

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.

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Copyright Daniel O. Ludwigsen, 2005