The Acoustics of Hearing Aids: Standing Waves- Part 1

On May 6, 2013, an article of mine was published in the Hearing Review about the acoustics of hearing aids. This was divided into three categories- 1. standing waves, 2. damping and impedance, and 3. flared or belled tubes. While this article was about hearing aids, the acoustic principles underlie speech acoustics and also the acoustics of musical instruments- something that has been discussed previously on this blog.

One of the exciting things about the field of acoustics is that there can be so many different applications of acoustics. And, unlike quantum mechanics, acoustics does rear its head(s) frequently in our field. While this blog post (presented in three parts, one for each of the categories listed above) is about hearing aids, it would be instructive to read it as if the phrase “hearing aid” and also the word “trumpet” were mentioned. This will help you in seeing the relationships.

In the case of standing waves, these are found in: all behind-the-ear hearing aids (but not in custom hearing aids); in all brass, woodwind, and stringed musical instruments; and in our vocal tracts, during speech production.

Part 1 of this series of posts deals with standing waves only; we will leave damping and impedance to part 2 and flared tubing to part 3).

Standing waves come in two flavors in our field: quarter- and half-wavelength resonators. Behind-the-ear hearing aids function as quarter-wavelength resonators (closed at the receiver end and open at the end of the earmold tubing) and generate odd-numbered multiples of the fundamental. This is the case for all brass instruments and the lower register of the clarinet.

Oh, and I guess that I should mention that our ear canals also function as a quarter-wavelength resonator (closed at the eardrum end and open at the lateral part of the ear canal). Half-wavelength resonators (closed at both ends) are not typically found in hearing aids, but are found in all stringed instruments and in our vocal tracts- these have integer multiples of the fundamental.

So, on to standing waves, and let’s see how many musical instruments you can identify?

Back to the 1980’s

How much do you remember about all of the neat acoustics that were clinically necessary for working with hearing aids back in the 1980s? Some of what we, as a field, may have forgotten is increasingly resurfacing in the form of basic problems related to open and occluded fittings. For example, can you explain the various elements of the real-ear coupler difference transfer functions when it comes to deep-canal hearing aid fittings? What about the benefits of flaring out a slim tube in a non-occluding behind-the-ear (BTE) style hearing aid? Is an acoustic solution any different from merely turning up the high-frequency gain electrically? And why don’t we use acoustic resistance anymore while fitting modern slim-tube hearing aids? Is there a reason, or is it simply more efficient to equalize out an unwanted resonant peak by using the hearing aid fitting software?

Initially, this article was called “Forgotten Acoustics” because it focuses on the acoustical laws of physics that every clinician needed to know in the 1980s and earlier. Since the advent of digital hearing aids in the 1990s, much of this knowledge has been relegated as “redundant.” Some may find this article to be a glimpse into how the last generation of audiologists may have functioned, and some, like myself, may just find the explanations about why things are the way they are to be interesting.

Standing Waves—They Are Everywhere… Well, Almost!

Whenever there are reflections of a wave, you can get sound energy flowing in two (or more) different directions: the incident wave and the reflected wave. As they pass in the night (or in a tube), they add up constructively and also destructively.

There are points in a room where a reflected wave may cancel the incident wave, and if you are unlucky enough to be in such a location, sound is reduced. These nulls or nodes occur in musical theatres and other performance venue halls—the more of them there are, the poorer the hall. Imagine a violinist tilting his head slightly and receiving a different sound than when sitting upright. This can happen. Welcome to architectural acoustics!

These same standing waves, brought about by an interplay between constructive interference and destructive interference, also can occur in tubes. In fact, the only place they don’t occur is in a free field, such as when skydiving or when seated in an anechoic chamber. And, like standing waves, there are tubes everywhere: in our vocal tract when we utter vowels and nasal sounds, in musical instruments like the trumpet, and even in BTE hearing aids. There are no standing waves in custom hearing aids simply because the tubing length is so short (actually there are, but the first resonant peak associated with standing waves in a canal or CIC hearing aid occurs above 8000 Hz or above the range that frequencies are typically amplified).

So now you know the answer to the question, “What do skydiving and CIC hearing aids have in common?” No standing waves.

The quarter-wavelength resonator. The most important tube in the hearing field is one that is open at one end and closed at the other end—a tube called a quarter wavelength resonator. BTE hearing aids are “closed” at the receiver end and “open” at the tip of the earmold tubing. This is also true of our vocal tracts, which are “closed” at the vocal chords and “open” at the lips during the articulation of [a] as in “father.” And it is also true of a trumpet: “closed” at the lips and “open” at the end of the bell.

Quarter-wavelength resonators have odd-numbered multiples of the first mode of resonance. This may not sound exciting until one realizes that this explains the resonant pattern of the vowel [a] as in “father,” the spectrum of a trumpet, and the frequency response of BTE hearing aids. In the case of a BTE, the “tubing” length from the receiver, through the ear hook, and through the hearing aid tubing is about 75 mm long for an adult.
Using the equation shown in Equation 1 below (F=v/4L), assuming the speed of sound (v) at 340,000 mm/sec, and setting L = 75 mm, F equals 1000 Hz. That is, the first mode of resonance is at 1000 Hz, and then there are odd-numbered multiples of 1000 Hz, namely 3000 Hz, 5000 Hz, and 7000 Hz.
_____________________
Equation 1:
F = v/4L where…
F is Frequency (Hz)
v is speed of sound (34,000 cm/sec or
340,000 mm/sec)
L is the length of the tube (in cm or mm)
_____________________

Turning our attention to custom hearing aids, such as a CIC, let us assume that the tubing length is now only 10 mm from the receiver to the end of the shell. Again, using the quarter-wavelength resonator formula with L = 10 mm, we find that F = 8500 Hz. Since this is typically above the frequency range of most modern hearing aids, we can safely say that there are no standing wavelength resonances for custom hearing aids; any resonances found in the frequency response of a CIC (or any other custom hearing aid) are due to the mechanical attributes of the receiver, not the acoustical pathway.

Another occurrence of a quarter-wavelength resonator is in our ear canals, which are “closed” at the eardrum and “open” at the lateral portion of the meatal opening. Figure 1 shows the unobstructed ear canal with the first mode of a standing wave, which is associated with the 2700-Hz real-ear unaided response (REUR). Obstruction of the outer portion of the ear canal (gray area in Figure 2) with either wax or a pathological collapse of the cartilage (e.g., polychondritis) will significantly reduce the level of the associated resonance at 2700 Hz.

Figure 1: An unobstructed ear canal showing a standing wave that corresponds with a 2700-Hz resonance. There is a large volume velocity at position A and a minimal one at position B. (See also text for Figure 2.)

Figure 2. Obstruction of the outer portion of the ear canal (at position A) with either wax or a pathological collapse of the cartilage (e.g, polychondritis) will serve to significantly reduce the level of the associated resonance at 2700 Hz. If the obstruction were at position B, there would be minimal effect.

In contrast, if the obstruction is near the tympanic membrane where there isn’t a lot of particle movement in the standing wave pattern (ie, near a node at position B), there would not be a significant change in the amplitude of the ear canal resonance. This is why ear canal occlusion with earwax may result in a significant high frequency conductive hearing loss (with its greatest effect around 2700 Hz).

Also think about this in terms of how French horn players place their right hand inside the bell of the instrument to alter the sound!

This last example demonstrates another property of a wavelength-associated resonance; depending on where the obstruction is, there will be different effects on the amplitude of the resonant peak. This is a well-known phenomenon when acoustic dampers are used in hearing aid tubing to smooth the frequency response. If the damper is at a maximum of a standing wave (position A), there will be a greater destruction of the resonance than if the damper is at a nodal position (B) in the standing wave.

Returning to Figure 1, let us assume that the average adult ear canal length is 28 mm. Using the quarter wavelength formula from Equation 1, then…
F = 340,000 mm/sec/4 x 28 mm = 3035 Hz

Effective length.

Why does this formula suggest that the resonance should be at 3035 Hz, but as measured using real-ear measurement techniques, the resonance is typically around 2700 Hz? This question brings us to a discussion of length versus “effective length.”

It turns out that the measured length of a tube is only a first approximation of how sound is transmitted through the air. There are two additional factors that affect the real or effective length: the nature of the “closed” end and the nature of the “open” end. In the case of the human ear canal, the “closed” end is the tympanic membrane and this adds the equivalent of a few mm of acoustic length because of its compliance.

This is not surprising; if one were to measure the length of the Zwislocki coupler used in KEMAR, the measured length is 21.5 mm, yet it functions acoustically as if it were much longer. The REUR as measured in KEMAR is 2700 Hz, and the explanation is that the microphone that replaces the tympanic membrane in KEMAR adds most of the missing acoustic length—the diaphragm of the microphone is quite compliant, which adds 7-8 mm of “length.”

Turning our attention to the lateral end of the ear canal, this opening contains a mass of air called an inertance. The inertance oscillates as one unit and provides an additional several millimeters of acoustic length. It also can generate its own lower frequency resonance, and this is what occurs in earmold vents (also known as a vent-associated resonance).

So the frequency of quarter-wavelength resonances, such as the “tubing resonance” in BTE hearing aids, or the real-ear unaided response (REUR), is governed by the formula in Equation 1. These resonances generate odd-numbered multiples of its mode or lowest frequency, and are altered by the physics of both the “open” end and the “closed” end. Of importance is that we have not discussed any effect on the amplitude of the associated resonances.