Last week in part 1 of this blog series, we reviewed standing waves found in hearing aids, but drew a parallel between behind-the-ear hearing aids, our vocal tract, and musical instruments (not to mention our ear canals… well, I just mentioned it, so I guess that I should not have said, “not to mention”…).
This three-part series is based on an article I wrote in the Hearing Review in May 2013 and we continue the parallel between hearing aids and the exciting world of impedance and damping. As we will see, unlike standing waves and also flaring of the tubes, damping is independent of frequency and has to do with the loss of energy in hearing aids, our vocal tracts, and musical instruments (though obviously there is less overall damping in musical instruments than our vocal tracts, which is one reason why music is more intense than speech).
Since this is part 2 of this blog, we have continued the original numbering of the equations from part 1 (and as they appeared in the original article).
Impedance and Damping
The amplitude of the associated resonances found in a BTE hearing aid or musical instrument is well defined by simple acoustic principles. Equation 2 reflects the physics behind our acoustic immittance testing used routinely in a clinical setting. The equation shows that impedance (Z) is a function of both reactance and resistance. An interesting acoustic result is that, at resonance, reactance is 0. Using this fact, one can say that, at a resonant peak, the impedance Z is pure resistance.
Unlike reactance—which is frequency dependent—resistance is independent of frequency. The implication is that the amplitudes of the lower frequency resonances and the higher frequency resonances are similar and are more governed by the general properties of the tube; a trumpet has less resistance (damping) than the human vocal tract, but the higher frequency resonances in both cases are of similar amplitude as the respective lower frequency amplitudes.
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Equation 2:
Z=(reactance 2+resistance 2) 1/2
at resonance, reactance = 0
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The hearing aid industry has used acoustic resistors (aka “dampers”) that can reduce the amplitude of the frequency response at resonant frequencies. However, depending on where the acoustic resistor is placed—at a node of a standing wave or at an anti-node of a standing wave—there are differing effects on the frequency response.
In the case of wax occlusion of the ear canal, since the wax is in the outer (lateral) portion of the ear canal where there is an anti-node (see position A in Figures 1 and 2 from part 1 of this blog), there is a great resistive affect. An obstruction in a more medial location (nearer a node such as position B) results in very little alteration of the amplitude of the resonant peak.
This brings us to the wonderful world of specific impedance. Equation 3 shows the specific impedance of a tube, as well as an example of how to determine the specific impedance for #13 hearing aid tubing.
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Equation 3:
Z = ρv/area (cgs) where…
Z is the specific impedance
ρ is the density of air
v is the speed of sound
area is the cross-sectional area of tube (cgs)
For example, for #13 tubing:
Z = 0.0012 gr/cm3 x 34,000 cm/sec/0.0314 cm2
Z=1300 Ω
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If we use the cross-sectional area of standard #13 tubing commonly found with many BTE hearing aids, the specific impedance works out to be 1300 Ω (ohms). This means that #13 hearing aid tubing requires about 1300 Ω of resistance to fully reduce the amplitude of the resonances. The industry has used 1500 Ω of resistance in the past, but this is close enough.
Thinner-diameter tubes, such as those found with slim-tube BTE fittings, require a significantly higher specific impedance to rid the frequency response of annoying resonances. And, for the human vocal tract with a relatively large cross-sectional area, very little acoustic resistance is required to alter the resonant pattern.
And in the part 3 post of this blog we will delve into the exciting world of flares and horns- something shared by musical instruments, hearing aids, and speech acoustics (at least for the low back vowel [a] as in ‘father’).
