Introduction to Acoustics

560 Part E Music, Speech, Electroacoustics
Part E 15.2
Force
t
1
0.5
0
0 5 10 15
Fig. 15.23 Sawtooth time-dependence of force on the end
supports from Helmholtz bowed waveform and corresponding
amplitudes of the normalised Fourier spectrum with
partials varying as 1/n
20 dB markers
0 1 2 3
4
(kHz)
Fig. 15.24 The spectrum of the intensity of the lowest
bowed note on a cello, illustrating the very large number of
partials contributing to the sound of the instrument
waveform time dependence, as shown in Fig. 15.23.
Each time the kink is reflected at the bridge, the transverse
force acting on the bridge reverses in sign. It then
increases monotonically with time until the process repeats
again. The sense of the sawtooth motion reverses
with bow direction. The spectrum of the force acting
on the bridge includes both even and odd partials, with
amplitudes varying as 1/n.
The spectrum of the sound produced by the lowest
plucked and bowed notes on stringed instruments can
typically involve 40 or more significant harmonic partials,
as illustrated in Fig. 15.24 by the spectrum of the
sound produced by a bowed cello open C-string (C2 at
∼64 Hz, audio ). The FFT spectrum is plotted
on a dB scale to illustrate the large range of amplitudes
of the partials (Fourier components) excited. The amplitudes
of the individual partials depend not only on
the force at the bridge exerted by the plucked or bowed
strings, but also on the frequency dependent response
and radiative properties of the supporting structure, as
discussed later.
Struck String
Many musical instruments are played by striking the
string with a hammer. The hammer can be quite light
and hard, as used for playing the dulcimer, Japanese koto
and many other related Asian instruments, or relatively
heavy and soft, like the felted hammers on a piano.
Some time after the initial impact, the striking hammer
bounces away from the string, leaving the string in a free
state of vibration. There are a few instruments, such as
the clavichord (Thwaites and Fletcher [15.37]), where
the string is struck with a metal bar (the tangent), which
remains in contact with the string, defining its vibrating
length and hence the note produced.
Consider first a point mass m moving with velocity
v striking an ideal stretched string of infinite extent.
In any small increment of time, the moving mass will
generate a wave moving outwards from the point of
impact. This will result in a decelerating force on the
mass equal to 2Tv/c = mcv, as illustrated in Fig. 15.18.
The displacement of the mass will then be described by
the following equation of motion
m d2ξm =−2T
dt2 cT
dξm
dt
. (15.42)
The transverse velocity of the impacting mass therefore
decays exponentially with time as
dξm
dt = vm exp (−t/τ) , (15.43)
with τ = mc/2T. The is identical to the dynamics of
a trapeze artist dropping onto a stretched wire, with
waves of displacement and velocity travelling outwards
in both directions away from the point of impact, as
illustrated in Fig. 15.25.
In general, the string will be struck at a distance a
from one of its end-supports. Hence, in a time (a/2L)T0,
a reflected wave will return to the mass and exert an additional
force, which will tend to throw the mass back
a) Displacement b) Velocity cT cT
cT
vm(t)
cT
Fig. 15.25a,b Time sequences of (a) string displacement
and (b) string velocity for a mass striking a string