Introduction to Acoustics

94 Part A Propagation of Sound
Part A 3.16
consists of a cavity of volume V with a neck of length ℓ
and cross-sectional area A.
In the limit where the acoustic frequency is sufficiently
low that the wavelength is much larger than any
dimension of the resonator, the compressible fluid in the
resonator acts as a spring with spring constant
ksp = ρc2 A2 , (3.599)
V
and the fluid in the neck behaves as a lumped mass of
magnitude
m = ρAℓ ′ . (3.600)
Here ℓ ′ is ℓ plus the end corrections for the two ends of
the neck. If ℓ is somewhat larger than the neck radius a,
and if both ends are terminated by a flange, then the two
end corrections are each 0.82a. (The determination of
end corrections has an extensive history; a discussion can
be found in the text by the author [3.30].) The resonance
frequency ωr, in radians per second, of the resonator is
given by
where
ωr = 2π fr = (ksp/m) 1/2 = (MACA) −1/2 , (3.601)
MA = ρℓ ′ /S (3.602)
gives the acoustic inertance of the neck and
CA = V/ρc 2
(3.603)
gives the acoustic compliance of the cavity. The ratio of
the complex pressure amplitude just outside the mouth
of the neck to the complex volume velocity amplitude
of flow into the neck is the acoustic impedance ZHR of
the Helmholtz resonator and given, with the neglect of
damping, by
ZHR =−iωMA +
1
, (3.604)
−iωCA
which vanishes at the resonance frequency.
If a Helmholtz resonator is inserted as a side branch
into the wall of a duct, it acts as a reactive muffler that
has a high insertion loss near the resonance frequency
of the resonator. The analysis assumes that the acoustic
3.16 Ray Acoustics
When the medium is slowly varying over distances comparable
to a wavelength and if the propagation distances
U(0 – )
UHR
p(0 – ) = p(0 + )
U(0 + )
Fig. 3.50 Helmholtz resonator as a side-branch in a duct
pressures in the duct just before and just after the resonator
are the same as the pressure at the mouth of the
resonator. Also, the acoustical analog of Kirchhoff’s circuit
law for currents applies, so that the volume velocity
flowing in the duct ahead of the resonator equals the sum
of the volume velocities flowing into the resonator and
through the duct just after the resonator. These relations,
with (3.604), allow one to work out expressions for the
amplitude reflection and transmission coefficients at the
resonator, the latter being given by
2ZHR
Ì =
. (3.605)
2ZHR + ρc/AD
The fraction of incident power that is transmitted is
consequently given by
�
1
τ = 1 +
4β2 ( f/ fr − fr/ f ) 2
�−1 , (3.606)
where β is determined by
β 2 = (MA/CA)(AD/ρc) 3 . (3.607)
The fraction transmitted is formally zero at the resonance
frequency, but if β is large compared with unity, the
bandwidth over which the drop in transmitted power is
small is narrow compared with fr.
are substantially greater than a wavelength, it is often
convenient to regard acoustic fields as being carried
x