150
4 After the Lips: Acoustic Resonances and Radiation
representing a sinusoidal wave with angular frequency ω propagating with a speed
c. For cylindrical or conical bores, U = 0 everywhere in the tube, and c = c 0 . In a
flaring bell, on the other hand, U increases with the rate of flare of the tube; Eq. 4.62
shows that this results in an increase in the local speed of sound propagation c. If
the speed of propagation increases at constant frequency, the local wavelength must
also increase: this means that nodes and antinodes in a standing wave are further
apart in the flaring bell of a trombone than they are in the slide section.
If the maximum value U max of the horn function is greater than ω 2 /c 0 , there will
be a region of the tube in which the solution to Eq. 4.62 is imaginary. To understand
what happens in this region, it is helpful to rewrite Eq. 4.63 in terms of a local wave
number:
k = ω/c =
k
2
0 − U
1/2 ,
(4.64)
where k 0 = ω/c 0 . The solution to Eq. 4.61 then takes the form
ψ = Ae
j (kz−ωt)
=
Ae
jkz
e
jωt .
(4.65)
When both k and ω are real numbers, Eq. 4.65 represents a wave propagating in
the positive z direction. When U > k 2
0 , the local wave number k is imaginary, and
the term in brackets on the right-hand side of Eq. 4.65 can be written as A exp(−kz).
This solution represents an oscillating pressure change which is not propagating, but
decaying exponentially; such a wave is described as evanescent.
It is now possible to understand why the lower modes in a flaring tube are
higher in frequency than the corresponding modes of a cylindrical tube of the same
length. At a frequency for which U k 2
0 , the region of the flaring tube in which
a propagating wave can exist is much shorter than its sounding length, and the
standing wave pattern is effectively squeezed into this shorter length. It should also
be borne in mind that in the region of rapid flare, the distance between nodes in the
standing wave pattern expands; the pressure distribution in a realistic instrument can
be calculated numerically if the bore profile is known.
4.3.8 A Theoretical Example: The Bessel Horn
There is a family of theoretical horn profiles for which analytic solutions to the
Webster equation can be found; because the solutions are expressed in terms of
Bessel functions, these profiles are known as Bessel horns. The general expression
for a Bessel horn bore profile is
r =
B
(x 0 − x) α ,
(4.66)
4 After the Lips: Acoustic Resonances and Radiation
representing a sinusoidal wave with angular frequency ω propagating with a speed
c. For cylindrical or conical bores, U = 0 everywhere in the tube, and c = c 0 . In a
flaring bell, on the other hand, U increases with the rate of flare of the tube; Eq. 4.62
shows that this results in an increase in the local speed of sound propagation c. If
the speed of propagation increases at constant frequency, the local wavelength must
also increase: this means that nodes and antinodes in a standing wave are further
apart in the flaring bell of a trombone than they are in the slide section.
If the maximum value U max of the horn function is greater than ω 2 /c 0 , there will
be a region of the tube in which the solution to Eq. 4.62 is imaginary. To understand
what happens in this region, it is helpful to rewrite Eq. 4.63 in terms of a local wave
number:
k = ω/c =
k
2
0 − U
1/2 ,
(4.64)
where k 0 = ω/c 0 . The solution to Eq. 4.61 then takes the form
ψ = Ae
j (kz−ωt)
=
Ae
jkz
e
jωt .
(4.65)
When both k and ω are real numbers, Eq. 4.65 represents a wave propagating in
the positive z direction. When U > k 2
0 , the local wave number k is imaginary, and
the term in brackets on the right-hand side of Eq. 4.65 can be written as A exp(−kz).
This solution represents an oscillating pressure change which is not propagating, but
decaying exponentially; such a wave is described as evanescent.
It is now possible to understand why the lower modes in a flaring tube are
higher in frequency than the corresponding modes of a cylindrical tube of the same
length. At a frequency for which U k 2
0 , the region of the flaring tube in which
a propagating wave can exist is much shorter than its sounding length, and the
standing wave pattern is effectively squeezed into this shorter length. It should also
be borne in mind that in the region of rapid flare, the distance between nodes in the
standing wave pattern expands; the pressure distribution in a realistic instrument can
be calculated numerically if the bore profile is known.
4.3.8 A Theoretical Example: The Bessel Horn
There is a family of theoretical horn profiles for which analytic solutions to the
Webster equation can be found; because the solutions are expressed in terms of
Bessel functions, these profiles are known as Bessel horns. The general expression
for a Bessel horn bore profile is
r =
B
(x 0 − x) α ,
(4.66)