Acoustic impedance for the lowest note on
the flute
The lowest note on a flute is the easiest to explain, because
for this note the instrument has all tone holes closed, and
so most closely approximates a cylinder. These notes should
be read while consulting the impedance spectrum for the note
C4 (Open a new
window for C4). For an introduction to acoustic impedance,
see What is
acoustic impedance?
The impedance spectrum for C4 on a C foot flute
For this case, the flute most closely approximates a simple
cylindrical pipe, open at the far end, which is a standard
case considered in textbooks about acoustics. (There are differences,
of course: the closed tone holes each have a little volume
added to the side of the bore; the head joint of the flute
is not cylindrical but tapered to a smaller diameter near
the embouchure hole and we are driving it at the embouchure
hole which is, in effect, a little side tube some distance
from the end of the pipe.)
The impedance spectrum shows a regular series of maxima
and minima. These correspond to the resonances of the flute
(or pipe) in this configuration. The minima correspond to
resonances for the condition when the embouchure end is open
(like a flute) while the maxima are resonances for the condition
when the embouchure end is closed (like a clarinet - see Flutes
vs Clarinets).
With this fingering, the flute will play a series of notes
at frequencies near to the minima in this spectrum. Keeping
all the keys closed, and just by blowing harder and adjusting
the embouchure one can play at least the first 7 or 8 notes
in the harmonic series, as shown below (Note the half sharp
- the seventh note in the series is between A6 and A#6, slightly
closer to the latter.)
As is the case for a simple cylindrical pipe, the amplitude
of the maxima and minima decreases with frequency - you will
have noticed that the notes get harder and harder to play
as you go up. This is due to viscous losses. The air flow
is greatest in the centre of the flute, and zero at the walls.
Some energy is lost in overcoming the viscous drag of the
air, and this limits the sharpness (lowers the Q factor) of
the resonances. The effect is greater for high frequencies.
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