Flute acoustics: measurement, modelling and design - School of ...

28 CHAPTER 2. THEORY AND LITERATURE REVIEW
where t e is the effective length of the hole (i.e. including end corrections). Wolfe & Smith (2003)
derive this formula for an infinite array of tone holes.
The cutoff frequency increases with hole size and is one of the main reasons for the difference
in tone between baroque, classical and modern flutes. Flutes with larger holes have a
higher cutoff frequency and so produce sounds with a wider spectral content. These instruments
sound ‘bright’ in comparison to smaller-holed instruments. The cutoff frequency also
contributes to the highest note playable on the instrument. For most woodwind instruments
the cutoff frequency is in the range 1.5–2.0 kHz. For a detailed discussion of the influence of the
cutoff frequency on various types of flutes, see Wolfe & Smith (2003). In Chapter 5 impedance
spectra of flutes and clarinets are compared with regard to their cutoff frequency.
2.2.12 Air-jet–resonator interactions
A flutist plays a flute by directing a jet of air across the embouchure hole. In the steady-state,
i.e. when a standing wave is already established in the flute, the sound field within the flute
imposes oscillations on the jet of air, alternately directing the air in and out of the flute. If the
travel time of oscillations on the jet from the player’s embouchure to the embouchure edge is
such that energy is added to the standing wave in phase, then the standing wave is sustained.
For this reason the jet length and speed are critical, and a player can play several notes with
a single fingering simply by changing the blowing pressure or the position of the lips. The jet
is driven by acoustic flow into and out of the embouchure hole—therefore flutes operate near
impedance minima, in contrast to reed instruments such as the clarinet and bassoon, which
operate near impedance maxima.
The jet drive mechanism in flutelike instruments is inherently a nonlinear process, due in
large part to the shape of the velocity profile of the jet. As discussed by Fletcher & Rossing
(1998, chapter 16) the velocity profile of a jet has a bell-shaped form in the direction parallel to
the axis of the embouchure hole. Due to this non-uniform velocity profile, different harmonic
components are generated depending on the offset of the jet relative to the sound-generating
edge. For example, a symmetric jet generates no even harmonics and the third harmonic can
be made to vanish by inclining the jet into the instrument (in this case the first and second
harmonics are still strong). The harmonic content of the jet drive also depends on the amplitude
of excitation, with (in general) higher harmonic content at higher amplitude. Given such a
nonlinear sound production mechanism, the playing frequencies of flutes are expected to vary
with the level of excitation, and with jet offset. The playing frequency of a note may depend on
the frequencies of more than one input impedance minima, and if these are not harmonically
related, then the relative weighting will vary with amplitude.
A detailed discussion of the physics of air jets is beyond the scope of this work. For an
overview see Fletcher & Rossing (1998) and for a review of physical models see Fabre & Hirschberg
(2000). Experimental investigations in the context of the flute and organ pipes have been
provided by Coltman (1968a,b), Fletcher & Thwaites (1979, 1983), Thwaites & Fletcher (1980,
1982) and Vergez et al. (2005).