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

2.2. ACOUSTICS OF WOODWIND INSTRUMENTS 15
infinitesimal width dx and area S is E(x)dx where the energy function E(x) is given by
E(x) = 1 2
2.2.3 Wave propagation in pipes and horns
( p 2 S
ρc 2 + ρU 2 )
. (2.2)
S
The acoustic pressure wave inside wind instruments with rigid walls can be determined generally
by solving the wave equation
∇ 2 p = 1 ∂ 2 p
c 2 ∂t 2 (2.3)
subject to the boundary condition that n·∇p = 0 at the instrument walls, where n is a unit vector
normal to the wall. Simple solutions to this equation exist for several geometries, including
cylindrical tubes and conical horns (Fletcher & Rossing 1998). Approximate solutions can be
found for other geometries, such as the exponential horn. The acoustic flow velocity u is related
to the pressure according to Newton’s law
u =
i ∇p (2.4)
ρω
where ω = 2πf is the angular frequency.
The solution to (2.3) in cylindrical polar coordinates for a cylindrical pipe of radius a is proportional
to exp[i(−k mn x + ωt)], where the proportionality depends on r and φ and describes
how the pressure wave is distributed over the (r,φ) plane. The square of the wave vector k mn
for the mode (m,n) is given by
) 2 ( πqmn
) 2
− , (2.5)
( ω
kmn 2 = c a
where m and n are mode numbers corresponding to the numbers of nodal diameters and nodal
circles in the (r,φ) plane and q mn is defined by the boundary condition that there can be no flow
in the radial direction at the cylinder walls.
The (m,n) mode propagates (extends infinitely in the x direction, neglecting any losses) if
k mn is real; otherwise the acoustic pressure decays exponentially with x and the mode is termed
evanescent. The plane wave mode given by m = n = 0 will always propagate since q 00 = 0;
higher modes will only propagate at frequencies above the cutoff frequency for the mode given
by
ω c = πq mnc
. (2.6)
a
The higher mode with the lowest cutoff frequency is the (1, 0) mode, which has a single nodal
plane. For this mode
ω c = 1.83c
a , (2.7)
giving a cutoff frequency of approx. 10 kHz for a typical flute bore radius of 10 mm.
Similar equations may be derived for conical horns although the cutoff frequency for each
mode varies with radius along the length of the horn, and the wavefronts for the always-propagating
mode are spherical rather than planar.
Fortunately, in the bores of most wind instruments all higher modes are non-propagating at
frequencies of musical interest. Higher modes are evoked in the vicinity of bore disturbances,
such as discontinuities or tone holes, but decay quickly. The fields around tone holes may be