Standing waves and ¼ wavelength resonators. Part 1 of 5:
I have always found it interesting that the various acoustic principles that are at play with hearing aids, are the same principles that rear their sometimes ugly heads in the realm of speech acoustics and in musical acoustics. For those readers whose background is audiology or speech sciences, you won’t learn anything new – merely a restatement and a reorientation of what we already know, but applied to a different area of study. After all, a trumpet is merely a vocal tract that is 134 cm long (if it’s a C tuned trumpet) and not 17 cm for the vocal tract of an adult, or 7.5 cm for the tubing of a conventional behind the ear hearing aid. The principles are the same – it’s just a matter of scale.
So, what are the principles that explain to us why a trumpet is a trumpet and a violin is a violin? These can be summarized as:
- Standing waves and quarter wavelength resonators
- Standing waves and half wavelength resonators
- Impedance and damping
- Amplification and flaring of a tube
- Pinna effect and stage setting at a venue
And it should come as no surprise that these five principles constitute the five parts of this musical acoustics blog.
So, onwards with Part 1 – Standing waves and quarter wavelength resonators:
Standing waves are everywhere – in a tube, in a chamber, in a room, but not in a tree… sounds like the Dr. Seuss book – Green Eggs and Ham. Whenever there is a surface or edge of a tube that obstructs or reflects some of the sound energy, we will get interference in the form of a standing wave. A standing wave is the result of an incident wave going in one direction, with a reflected wave going in the opposite direction. There are parts like two water waves going in opposite directions where they add up constructively and become very large waves, and parts where they interact destructively and cancel each other out. For those who like physics, we call the first type constructive interference and the latter type destructive interference. Unfortunately, for those who don’t like physics, it’s still called constructive and destructive interference.
So why am I talking about interference? Well, when signals (such as the incident signal and its reflection) interact, they set up sound levels that can be very high (when they add up constructively) and sound levels that are almost silent (when they add up destructively). These are called nodes (destructive interference) and anti-nodes or loops (constructive interference). Next time you are in a theatre, and this is true of almost all theatres, when listening to the music, tilt your head and see if the sound changes. Now do this experiment – during the performance, jump over the seat and push the person in front of you out of the way – they won’t mind – just tell them this is for “science”. See if the sound has changed. The odds are that the sound will change and you are moving from a nodal to an anti-nodal position or the other way around. If sound waves were visible, we would not be able to see anything except for wavy lines.
There are three locations where we would not see standing waves – sky diving, being in an anechoic chamber, and custom hearing aids. In sky diving there is simply nothing to reflect your screams so there cannot be any constructive or destructive interference. (Well, that’s not exactly true – in the brief moment before you hit the ground, there will be some reflection of your screams, but only for a moment). In an anechoic chamber, as the name suggests, there is no echo – no reflection. Actually there is, but depending on the size and density of the absorptive wedges in the walls, floor, and ceiling, the standing waves may only exist for the very low frequency sounds. And what about custom hearing aids? Well there are standing waves but typically above the frequency range of interest for modern hearing aids and this has to do with the short length of the hearing aid tubing from the hearing aid receiver to the edge of the custom hearing aid shell.
I have actually made a small jump from standing waves caused by incident and reflected sound to frequency. The standing wave in a custom hearing aid is so short because the receiver tubing is so short that the standing waves are above 8000 Hz, and many modern hearing aids only transduce up to 6000-7000 Hz. There are formulae for converting standing waves and the associated length of the tube that creates them and this is where frequency comes in, but the choice of the formula depends on which flavor of standing waves were are dealing with.
There are two commonly found “resonators” in musical acoustics – quarter wavelength resonators and half wavelength resonators. Scientists would say that there is a resonance, or series of resonant frequencies, associated with the standing waves.
Examples of quarter wavelength resonators are all brass instruments, the lower register of a clarinet, and our vocal tracts when uttering many sonorants such as the vowel [a] as in ‘father’. Typically a quarter wavelength resonance occurs if the “tube” is open at one end and closed at the other end. This structure helps to define whether a reflected sound will add up constructively or destructively with the incident sound- it all has to do with phase- if its in-phase, the sound adds up constructively like two water waves adding up to create a gigantic wave; if it’s out of phase, the sound adds up destructively and there is minimal energy (or node).
The formula for a quarter wavelength resonator is F = (2k-1)v/4L. Don’t panic! The frequency of the resonance (F) is equal to the speed of sound (v) divided by 4 times the length (L) of the tube. So, in the adult human vocal tract, the length L is 17 cm. If we assume that the speed of sound (v) is 34,000 cm/sec, then F is 34,000/4 x 17 which is 500 Hz. And oh yes, the (2k-1) part is simply a fancy way of saying odd numbered multiples. If k is 1, then (2k-1) is 1; if k is 2, then (2k-1) is 3 …,then 5, 7, 9, and so on. For the vowel [a], the resonances are 500 Hz, 1500 Hz, 2500 Hz, and so on. In the vocal tract we call these resonances formants and frequently you will see speech scientists referring to F1 = 500 Hz, F2 = 1500 Hz, and so on, meaning the first formant is at 500 Hz, the second formant is at 1500 Hz.
The trumpet and its brass cousins are all “closed” at the mouth piece end and “open” at the end of the bell. It is a tube that is 137 cm long so its associated resonances from the quarter wavelength formula mentioned above are: 34,000/4 x 137 = 62 Hz. The trumpet, being a quarter wavelength resonator also has resonances at 3 x 62 Hz, 5 x 62 Hz, and so on. Quarter wavelength resonators have resonant peaks at odd numbered multiples of 62 Hz.
In brass instruments, the “length” L can be changed by the judicious use of valves (e.g., trumpets, tuba, French horn) or by actually increasing the length with additional tubing such as with the trombone.
In the case of woodwinds, the “length” is the distance between the reed or mouthpiece and the first open hole. If there is an opening somewhere in the tube, it behaves as if it is much shorter, with an associated higher fundamental frequency.
When I play my clarinet, I have a “register key” and not an “octave key”- we’ll talk about octave keys next week in part 2 of this blog when we talk about half wavelength resonators. My register key multiplies the playing frequency by a factor of 3, or one and a half octaves. When I play a C at 262 Hz, and hit the register key, using the very same fingering, I am now playing a G which is three times the frequency of 262 Hz – namely 786 Hz. This is a hallmark of quarter wavelength resonators. Instruments with “octave keys’ that double (not triple) the playing frequency are a different animal- see part 2!
A behind-the-ear hearing aid is also a quarter wavelength resonator. The earhook/tubing combination for an adult is roughly 75 mm (= 7.5 cm). Using the quarter wavelength resonator formula, the first resonance is at 34,000/4 x 7.5 = 1133 Hz. This is sometimes referred to as the “1000 Hz resonance” and is the large first resonance found in behind the ear hearing aids. The second resonance is at 3 x 1133 Hz (3400 Hz) and the third resonance is at 5 x 1133 Hz (5665 Hz). These resonances are evident is all behind the ear hearing aids. However there are more of them- typically five resonances in behind the ear hearing aids- two additional ones at about 3000 Hz and 6000 Hz. These are mechanical and Helmholtz resonances associated with the hearing aid receiver. Modern “class D” receivers have a resonance of about 3000 Hz- hearing aids of the 1970s and 1980s had a class A or B output stage and had receiver related resonances that were closer to 2000 Hz and 4000 Hz.
Before we finish up, let’s do a quick calculation of the first (quarter) wavelength resonance in a custom canal hearing aid. Let’s assume that the length of tubing between the receiver and the end of the shell is 1 cm or 10 mm (L). Using the formula F = (2k-1)v/4L we get 34,000/4 x 1 = 8500 Hz. For all practical purposes then a custom hearing aid has no wavelength associated resonances (assuming that hearing aids do not generate sufficient gain above 8000 Hz). Any resonances seen in a custom product are related to the mechanical characteristics of the receiver.
Had enough of quarter wavelength resonators? What about the wonderful world of half wavelength resonators? Well that would be part 2 of this blog. Stay tuned!