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

k = ω c and pressure given by p(x, t) = [Ae −ikx + Be ikx ]e iωt , (2.10)
16 CHAPTER 2. THEORY AND LITERATURE REVIEW
decomposed into the various modes, but often the plane wave or spherical wave approximations
provide a sufficiently accurate description of the behaviour of an instrument, provided
length or impedance corrections are applied at points where the plane wave approximation
breaks down.
2.2.4 Acoustic impedance
The acoustic impedance Z at any point in an acoustic field is defined as the ratio of pressure to
volume flow. For a plane wave propagating in an infinite cylindrical pipe, the acoustic impedance
is independent of frequency, having the value
Z 0 = ρc
S , (2.8)
where S is the cross-sectional area of the pipe. Z 0 is known as the characteristic impedance of
the pipe. For finite length pipes and cones, the plane-wave acoustic impedance is a function of
frequency; formulae for various situations are given below. The acoustic impedance is related
to the reflection coefficient R (the amplitude of the reflected wave relative to that of the incident
wave) according to the relation
1 + R
Z = Z 0
1 − R . (2.9)
A review of techniques for impedance measurement is given in Chapter 3 and the acoustic
input impedance of flutes and clarinets is discussed at length in Chapter 5.
2.2.5 Plane waves in cylindrical pipes
The plane wave mode in a lossless cylindrical pipe will always propagate with wavenumber
where A and B are the amplitudes of the forward- and backward-going waves respectively. The
acoustic volume flow is given by
( ) S
U(x, t) = [Ae −ikx − Be ikx ]e iωt . (2.11)
ρc
For real pipes with non-negligible wall losses k is complex, and the amplitude of the wave decreases
exponentially as it propagates (see §2.2.7).
The transfer matrix T for a cylindrical pipe of length L relates the pressure and volume flow
at the input to those at the output (Figure 2.2). Using (2.10) and (2.11), it can be shown that
[ ] [ ]
p1 p2
= T
(2.12a)
U 1 U 2
where
[ ]
cosh(ikL) Z0 sinh(ikL)
T =
. (2.12b)
1
Z 0
sinh(ikL) cosh(ikL)
The input impedance of a cylindrical pipe section terminated by a load impedance Z L may
be derived from (2.12) by multiplication after setting p 2 = Z L U 2 . This gives
[ ]
ZL cosh(ikL) + Z 0 sinh(ikL)
Z IN = Z 0 . (2.13)
Z L sinh(ikL) + Z 0 cosh(ikL)