Introduction to Acoustics

e described by treating the instrument as a superposition
of monopole, dipole, quadrupole and higher-order
multipole acoustic sources, with the directional radiating
properties shown schematically in Fig. 15.2.
A monopole source can be considered as a pulsating
sphere of radius a with surface velocity ve iωt resulting
in a pulsating volume source 4πa 2 ve iωt = Q e iωt .
Equations (15.10) and(15.11) describe the sound field
generated by such a source. Equating the velocities on
the surface of the sphere to that of the induced air motion
gives, at low frequencies such that ka ≪ 1,
p(r, t) = iωρ
4πr Q ei(ωt−kr) . (15.12)
The radiated power P is then given by 1 2 p2 /ρc0 integrated
over the surface of a sphere at radius r, so
that
P(ka ≪ 1) = ω2ρQ 2
. (15.13)
8πc0
In the high-frequency limit (ka ≫ 1), when the acoustic
wavelength is much less than the size of the sphere,
P(ka ≫ 1) = ρc0
8πa2 Q2 2 1
= 4πa
2 z0v 2 . (15.14)
Equation (15.12) is a special case of the general result
that, at sufficiently high frequencies such that the
size of the radiating object ≫ λ, the radiated sound is
simply 1 2 z0v2 per unit area, though the sound at a distance
also has to take into account the phase differences
from different parts of the vibrating surface. Note that
P(ka ≪ 1)
P(ka ≫ 1) = (ka)2 . (15.15)
The radiated sound intensity from a monopole source
therefore initially increases with the square of the frequency
but becomes independent of frequency above the
crossover frequency when ka > 1. Hence members of the
violin family and guitar families are rather poor acoustic
radiators for the fundamental component of notes
played on their lowest strings, as are wind and brass instruments,
which radiate sound from the relatively small
open ends and side holes. However, it is only because
of such low radiation efficiencies, that strong resonances
can be excited in the air columns of brass and woodwind
instruments.
A dipole source can be formed by displacing two
oppositely signed monopoles ±Q a short distance along
the x-, y- orz-directions. For a dipole aligned along
the x-axis of strength qx = Q∆x. The sound pressure
is simply the difference in pressure from monopoles of
Musical Acoustics 15.1 Vibrational Modes of Instruments 539
opposite sign a distance ∆x apart, so that in the far field
(kr ≫ 1)
p(θ)dipole = p(θ)monopole ×(ik∆x)cosθ. (15.16)
A polar plot of the sound pressure from a dipole is illustrated
schematically in Fig. 15.2, with intensity and
radiated power now proportional to ω4 and q2 x .Ingeneral,
any radiating three-dimensional object will involve
three dipole components (px, py and pz), with radiation
lobes along the three directions.
A quadrupole source is generated by two oppositely
signed dipole sources displaced a small distance along
the x-,y-, or z-directions (e.g. of the general form qxy =
Q∆x∆y). The pressure is now given by the differential
of the dipole radiation in the newly displaced direction,
so that, for example, the pressure from a quadrupole
source qxy in the xy-plane is given by
�
pdipole = pmonopole × −k 2 �
∆x∆y cos θ sin θ,
(15.17)
as illustrated in Fig. 15.2. Note that each time the order
of the multipole source increases, the radiated pressure
depends on one higher power of frequency, while the intensity
increases by two powers of the frequency. The
radiated power from multipole sources therefore decreases
dramatically at low frequencies relative to that of
a monopole source. At low frequencies, radiation from
most musical instruments is dominated by monopole
components.
In general, six quadrupole sources (qxx,... ,qyz)
would be required to describe radiation from a threedimensional
source. However, because the acoustic
power radiated by a quadrupole source at low frequencies
is proportional to ω 6 , one need often only consider
the monopole and three dipole components to describe
the low-frequency radiation pattern of instruments like
the violin and guitar family, as described in a recent study
of the low-frequency radiativity of a number of quality
guitars by Hill et al. [15.10]. However, at high frequencies,
when λ is comparable with or less than the size
of an instrument, the above simplifications break down.
The directionality of the radiated sound then has to be
computed from the known velocities over the whole surface,
taking into account phase differences and baffling
effects from the body of the instrument.
Radiation from Surfaces
Many musical instruments produce sound from the vibrations
of two-dimensional surfaces – like the plates
of a violin or the stretched membrane of a drum. Imagine
first a standing wave set up in the two-dimensional
Part E 15.1