Introduction to Acoustics

592 Part E Music, Speech, Electroacoustics
Part E 15.2
where L ′ is the effective length of the open hole. For
a circular hole of radius a, Rayleigh ([15.3] Vol.2,
Sect. 306) showed that L ′ = π 2 a, while for an ellipse
L ′ ∼ π 2 (ab)1/2 , provided the eccentricity is not too large.
Noting that the effective length depends largely on area,
Cremer ([15.29], Fig. 10.6) modelled the f-hole as an ellipse
having the same width and area as the f-hole. The
two f-holes act in parallel to give an air resonance for the
violin at ≈ 270 Hz, at an interval of just over a fifth above
the lowest open string. For the acoustic guitar, the circular
rose-hole produces an air resonance around 100 Hz,
which, like for the violin, is close to the frequency of the
second-lowest open string on the instrument.
Any induced motion of the top and bottom plates
that involves a net change in volume results in coupling
to the Helmholtz mode. Such coupling will perturb the
Helmholtz and body-mode frequencies, in just the same
way that string resonances are perturbed by coupling
to the body resonances (see Cremer [15.29], Sect. 10.3
for a detailed discussion of such coupling). Since the
acoustically important coupled modes are at consider-
a) F(t)
b)
dB
0
–20
–40
mp
Kp
mb
Kb
mh
V
Mp Cp Rp
F(t)
100 200 300 400 Hz
Ap
Up Mb
Ub
Cv
Cb
Rv
Uv
Rb Uh
Fig. 15.60a,b The mechanical (a) and equivalent electrical
(b) circuit for a three-mass model describing the vibrations
of the front and back plates of a stringed instruments coupled
via a Helmholtz resonance (after Fletcher and Rossing
[15.5]). The modulus of the admittance at the top plate has
been evaluated for identical front and back plates with uncoupled
frequencies of 300 Hz, coupled via a Helmholtz
air resonator at 250 Hz in the absence of coupling. The
frequencies of the uncoupled air and body resonances are
indicated by the vertical lines
Mh
Rh
ably higher frequencies than the Helmholtz resonance,
the mutual perturbation is not very large. Because of
such coupling, purists often object to describing this
resonance as a Helmholtz resonance. Similar objections
could apply equally well to string resonances, since
they too are perturbed by their coupling to body modes.
But, as already discussed, in many situations the normal
modes largely retain the character of the individually
coupled modes other then when their frequencies are
close together and, even then, when the damping of
either of the coupled modes is large compared to the
splitting in frequencies induced by the coupling in the
absence of damping (Fig. 15.46).
Well below the Helmholtz resonance, any change
in volume of the shell of the violin or guitar induced
by the vibrating strings will be matched by an identical
volume of air flowing out through the rose- or f-holes,
with no net volume flow from the instrument as a whole.
Since at low frequencies almost all the radiated sound is
monopole radiation associated with the net flow of air
across the whole surface of an instrument, little sound
will be radiated. However, above the air resonance, the
response of the air resonance will lag in phase by 180 ◦ ,
so that the flow from body and cavity will now be in
phase, resulting in a net volume flow and strong acoustic
radiation. The Helmholtz resonance serves the same
purpose as mounting a loudspeaker in a bass-reflex cabinet,
with the air cavity resonance boosting the intensity
close to and above its resonant frequency.
A number of authors have considered the influence
of the enclosed air on the lowest acoustically
important modes of the violin (Beldie [15.97]) and guitar
(Meyer [15.98], Christensen [15.99] andRossing
et al. [15.100]) using simple mechanical modes of interacting
springs and masses with damping and their
equivalent electric circuits. Figure 15.60 shows the mechanical
and equivalent electrical circuits and resulting
admittance curve for the top plate for the illustrative
three-mass model used by Rossing et al., which accounts
for the qualitative features of the first three most important
resonances of a guitar body. To emphasise a number
of important points, we have calculated the admittance
for a cavity with identical front and back plates with
uncoupled resonances at 300 Hz, coupled via a cavity
Helmholtz resonance at 250 Hz. The closeness in frequencies
of the coupled resonators has been chosen to
emphasise the influence of the coupling on the modal
frequencies.
Without concerning oneself with mathematical detail,
one can immediately recognise an unshifted normal
mode associated with the uncoupled body resonances