4.3 Bore Profiles of Brass Instruments
149
When discussing spherical waves in conical tubes (Sect. 4.3.3), we found that the
wave equation was simplified by changing the variable from p to ψ = pr. A similar
substitution is helpful in the more general case of flaring tubes:
ψ = pS
1
2 .
(4.58)
Assuming that the wave has a sinusoidal time dependence with angular frequency
ω, the Webster equation becomes
∂ 2 ψ
∂z 2 +
ω 2
c 2
0
− U
ψ = 0,
(4.59)
where
U =
1
a
∂ 2 a
∂z 2
(4.60)
is known as the horn function. In the plane wave approximation, a = r and z = x;
the horn function can then be calculated numerically from a known bore profile
r(x) using Eq. 4.60. In a more realistic treatment using curved wavefronts, it is first
necessary to establish the functional relationships between (x, r) and (z, a) from the
geometry of the assumed wavefront shape.
It has been pointed out by several authors that the form of the Webster
equation 4.59 is formally identical to the time-independent Schrödinger Equation in
quantum mechanics, with the horn function playing the role of the potential energy.
It is a striking example of the power of the fundamental concepts of physics that
the emission of light from a quantum well laser and the emission of sound from a
trombone are governed by the same basic equation.
The constant c 0 which appears in Eqs. 4.56 and 4.59 was defined as the speed of
sound in free air. The musically useful ‘harmonicity-correcting’ property of a flaring
bell arises from the fact that the speed of sound is not constant everywhere in the
tube. This is evident if we rewrite Eq. 4.59 as
∂ 2 ψ
∂z 2 +
ω 2
c 2
ψ = 0,
(4.61)
with
c = ω
ω 2
c 2
0
− U
−1/2
.
(4.62)
Equation 4.61 has a solution which can be written as
ψ = Ae
jω(z/c−t) ,
(4.63)
149
When discussing spherical waves in conical tubes (Sect. 4.3.3), we found that the
wave equation was simplified by changing the variable from p to ψ = pr. A similar
substitution is helpful in the more general case of flaring tubes:
ψ = pS
1
2 .
(4.58)
Assuming that the wave has a sinusoidal time dependence with angular frequency
ω, the Webster equation becomes
∂ 2 ψ
∂z 2 +
ω 2
c 2
0
− U
ψ = 0,
(4.59)
where
U =
1
a
∂ 2 a
∂z 2
(4.60)
is known as the horn function. In the plane wave approximation, a = r and z = x;
the horn function can then be calculated numerically from a known bore profile
r(x) using Eq. 4.60. In a more realistic treatment using curved wavefronts, it is first
necessary to establish the functional relationships between (x, r) and (z, a) from the
geometry of the assumed wavefront shape.
It has been pointed out by several authors that the form of the Webster
equation 4.59 is formally identical to the time-independent Schrödinger Equation in
quantum mechanics, with the horn function playing the role of the potential energy.
It is a striking example of the power of the fundamental concepts of physics that
the emission of light from a quantum well laser and the emission of sound from a
trombone are governed by the same basic equation.
The constant c 0 which appears in Eqs. 4.56 and 4.59 was defined as the speed of
sound in free air. The musically useful ‘harmonicity-correcting’ property of a flaring
bell arises from the fact that the speed of sound is not constant everywhere in the
tube. This is evident if we rewrite Eq. 4.59 as
∂ 2 ψ
∂z 2 +
ω 2
c 2
ψ = 0,
(4.61)
with
c = ω
ω 2
c 2
0
− U
−1/2
.
(4.62)
Equation 4.61 has a solution which can be written as
ψ = Ae
jω(z/c−t) ,
(4.63)