Welcome to the final part of a three-part series. It was originally published as part of an article I wrote for the May 2013 Hearing Review. Part 1 of this blog series dealt with standing waves, which are as important for behind-the-ear hearing aids as they are for our vocal tract and musical instruments. Part 2 dealt with damping and impedance, and while not these topics are not quite as exciting as standing waves, they are important for understanding why musical instruments inherently sound louder than speech.
This part of the blog series deals with the flaring of a tube,which is important for behind-the-ear hearing aids (but not custom hearing aids); discusses our vocal tracts during the articulation of low vowels such as the [a] in ‘father’; and explain why trumpets and other brass instruments have flares. Unlike damping, flaring of a tube has an effect only above certain frequencies (given by the formula F=v/2L), but more is given below.
As with the other two parts of this blog series, while this post is about hearing aids, try to substitute the words “trumpet” or “vowel [a]” when you read through it. And also like part 2 of this series, the equation numbering does not start at #1, but follows the numbering in the original article.
The acoustic transformer effect is a fancy way of saying that, depending on the length of a tube, increasing its cross-sectional area enhances the amplitude of the higher frequency components. An even fancier way of saying this is that “the amplitude of all frequencies whose one-half the total length of the tubing are enhanced by having a flare or horn.” Now that’s a great line sure to impress your friends… or there is always the less impressive, “a flare increases the loudness for the higher frequencies”.
Equation 4 looks like the equation in Equation 1 (from part 1 of this blog), as it is made up of the speed of sound (v) and the length of a tube (L), but that is where the similarity stops.
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Equation 4:
F = v/2L
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F is the frequency at which a flare would begin to enhance the amplitude or the loudness of the sound. For a tube that is 75 mm in length, such as that of a BTE hearing aid for an adult, Equation 4 suggests that all frequencies above 2266 Hz (F = 340,000 mm/sec/(2 x 75 mm)) would be enhanced by having a flared tubing. For young children (with a total length of only 60 mm), the high-frequency enhancement begins at 2833 Hz. That is, the shorter the tube, the higher frequency and the less the effect of a flare would begin at.
Libby horns are one example of tubing that gradually flares, but this also explains why opening one’s mouth wider during the articulation of some sounds generates an increased relative high-frequency output. It also explains why trumpets and many other musical instruments use a flared tube.
Flared tubes as amplifiers:
Finally, our trip through the land of the acoustics of tubes finishes with the amplification factor. Equation 5 shows an example of a doubling of the inner diameter of the tubing.
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Equation 5:
Amplification factor =
10log (πr2 of wider portion/πr2 of narrower portion)
For a doubling of inner diameter:
10log (22) = 2 x 10log (2) = 6 dB
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If the internal diameter doubles, such as with the 4-mm Libby horn where the inner diameter increases from 2 mm to 4 mm, then the maximum increase in amplitude above the frequency, shown in Equation 5, is 6 dB. Of significance, however, is that this is true for all doublings of inner diameters; it doesn’t matter whether the doubling is from 2 mm to 4 mm or from 1 mm to 2 mm. A thin tube that flares to the size of #13 tubing also will generate the same 6 dB of high-frequency output.
One may ask if this is equivalent to increasing the electrical output in a hearing aid by 6 dB. Although the end result would indeed be similar, there are two differences. With an acoustical flare of the tubing, the extra output is after the hearing aid receiver, so this would not deleteriously affect battery life. Additionally, the difference between the gain and the output in the high-frequency region is not altered. Acoustical high-frequency modifications maintain the necessary headroom.
Summary:
There are many laws of acoustics that have been forgotten since the advent of digital hearing aids. While it is true that most of the modifications can be accomplished digitally now instead of acoustically, there are some subtle differences. The five basic equations provided here remind us why many things are the way they are today in the realm of hearing aids.
