Introduction to Acoustics

nant mode ∆ω resulting from small distributed changes
∆S(x) in bore area S(x)isgivenby
∆ωn
=−
ωn
1
� �
c0
2 ωn
�L
� � �
∂ ∆S(x) ∂pn
pn
∂x S(x) ∂x dx
� �
0
L
�
0
�
S(x)p 2 n dx
�
. (15.115)
An alternative equivalent derivation uses Rayleigh’s
harmonic balance argument and equates the peak kinetic
energy to the peak potential energy. To first order,
the perturbation is assumed to leave the shape of the
modal wavefunction unchanged. The kinetic and potential
energy stored in a particular resonant mode can
be expressed in terms of the local kinetic 1 2 ρω2 n ξ2 n and
strain 1 2 γ P0(∂ξn/∂x) 2 energy densities. For simplicity,
we consider the perturbation of the nth resonant mode
of a cylindrical air column open at one end, with particle
displacement ξn ≈ sin(nπx/L)cos(ωnt), where n is an
odd integer. Equating the peak kinetic and potential energy
over the perturbed bore of the cylinder, we can then
write
ω ′ 2
n
�
0
L
= γ P0k 2 n
ρ [S + ∆S(x)] sin 2 (kx)dx
�L
0
[S + ∆S(x)] cos 2 (kx) dx , (15.116)
where ω ′ n is the perturbed frequency. This can be rewritten
as
�L
= [S + ∆S(x)] cos 2 �
(kx) dx
ω ′ 2
n
ω 2 n
0
L
�
0
[S + ∆S(x)] sin 2 (kx)dx . (15.117)
Because the perturbations are assumed to be small, we
can rearrange (15.117) to give the fractional change in
frequency
∆ωn
ωn
= 1
L
�L
0
∆S(x)
S
�
cos 2 kx − sin 2 �
kx dx .
(15.118)
Musical Acoustics 15.3 Wind Instruments 613
Hence, if the tube is increased in area close to
a displacement antinode, where the particle flow is
large (low pressure), the modal frequency will increase,
whereas the frequency will decrease, if constricted close
to a nodal position (large pressure) (Benade [15.133],
Sect. 22.3). This result can be generalised to a tube
of any shape. Hence, by changing the radius over an
extended region close to a node or antinode, the frequencies
of a particular mode can be either raised or
lowered, but at the expense of similar perturbations to
other modes. Considerable art and experience is therefore
needed to correct for the inharmonicity of several
modes simultaneously.
Electric Circuit Analogues
It is often instructive to consider acoustical systems in
terms of equivalent electric circuit analogues, where
voltage V and electrical current I can represent the
acoustic pressure p and flow along a pipe U.Forexample,
a volume of air with flow velocity U in a pipe of area
S and length l has a pressure drop (ρl/S)∂U/∂t across its
length, which is equivalent to the voltage L∂I/∂t across
an inductor in an electrical circuit. Likewise, the rate
of pressure rise, ∂p/∂t = γ P0U/V, as gas flows into
a volume V, is equivalent to the rate of voltage rise,
∂V/∂t = I/C across a capacitance C ≡ V/γ P0.
As a simple example, we re-derive the Helmholtz
resonance frequency, previously considered in relation
to the principal air resonance of the air inside a violin or
guitar body (Sect. 15.2.4), but equally important, as we
will show later, in describing the resonance of air within
the mouthpiece of brass instruments.
In its simplest form, the Helmholtz resonator consists
of a closed volume V with an attached cylindrical
pipe of length l and area S attached, through which the
air vibrates in and out of the volume. All dimensions are
assumed small compared to the acoustic wavelength, so
that the pressure p in the volume and the flow in the pipe
U can be assumed to be spatially uniform. The volume
acts as an acoustic capacitance C = V/γ P0, whichresonates
with the acoustic inductance L = ρl/S of the air
in the neck. The resonant frequency is therefore given
by
ωHelmholtz = 1
�
�
S γ P0 S
√ = = c0
LC ρl V lV ,
(15.119)
as derived earlier.
Any enclosed air volume with holes in its containing
walls acts as a Helmholtz resonator, with an
Part E 15.3