A bit on the acoustics of musical instruments: Part B flat

Marshall Chasin
May 18, 2011

This is a continuation of last week’s blog on the acoustics of musical instruments.  Last week we talked about quarter wavelength resonators and these are typically the clarinet and the brass instruments.  This latest blog is about the exciting half wavelength resonators- vocalists, strings, sax, flute, oboe and the bassoon.

Our vocal chords are tightly held at both ends of our larynx.  The guitar and violin strings are held tightly at either end of the instrument…  Whenever the same condition (physicists call this a “boundary condition”) is found at both ends of a vibrating tube or string, it typically functions as a half wavelength resonator.  A feature of all half wavelength resonators is that there is are harmonics at integer multiples of the fundamental- 100 Hz, 200 Hz, 300 Hz, 400 Hz, …  This is in contrast to quarter wavelength resonators where there are harmonics at odd numbered (or every other) multiples of the fundamental.  Therefore a half wavelength resonator instrument has twice the number of harmonics as a quarter wavelength resonator.

For those who like equations, and who doesn’t, the formula is given as f=kv/2L…. or in English, the frequencies of the fundamental and the harmonics are at integer multiples (k) of the speed of sound (v) divided by twice the length of the tube or string.  My fundamental frequency of my own voice is 125 Hz (exactly an octave below middle C).  If I wasn’t such a macho guy, my fundamental frequency may be higher such as 135 Hz.  This means that my vocal chords generates energy at 125 Hz, 250 Hz, 375 Hz, 500 Hz, and so on and this gives me my rich sexy male voice.

All stringed instruments also behave this way- the first harmonic is exactly an octave above the fundamental (250 Hz is an octave above 125 Hz).

Some woodwind instruments also function as half wavelength resonators as well- the sax, oboe, flute, and bassoon (but not a clarinet).  They all have an “octave key” which allows the musician to play at the second mode of resonance (or k = 2 if you like equations).  These instruments are odd (actually they are even) because they look somewhat like a clarinet (mouth piece on one end and open bore or bell on the other) but because they have a conical flare in their tubing, things change.  The reasons are complex, but suffice it to say that if we were able to construct a rubber clarinet where we could deform the straight tube and gradually flare it out like a cone, the clarinet would gradually change from a quarter wavelength resonator to a half wavelength one.

Until we have a rubber clarinet though, clarinets will have “register keys” which triple the fundamental (1, 3, 5,..) and saxophones and oboes will have “octave keys”.

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