Introduction to Acoustics

558 Part E Music, Speech, Electroacoustics
Part E 15.2
a)
c
b)
c
c
v bow
Fig. 15.19a,b Helmholtz waveforms for (a) a centrally
plucked and (b) a bowed string. The horizontal arrows indicate
the directions that the kinks are travelling in and the
vertical arrows the directions of the moving string sections.
The different colours represent string displacements at different
times. P indicates a typical bowing position along
the string
slope between adjacent straight-line sections (equivalent
to a δ function in the spatial double derivative) can there
be any acceleration. From our earlier discussion, any
such kink must travel backwards and forwards along
the string at the transverse string velocity c, reversing
its sign on reflection at the ends. As the kink moves
past a specific position along the string, the difference
in the transverse components of the tension on either
side of the kink results in a localised impulse, which
changes the local velocity of the string from one moving
or stationary straight-line section to the next. In general,
there can be any number of Helmholtz kinks travelling
along the string in either direction, each kink marking
the boundary between straight-line sections either at rest
or moving with constant velocity. Similar solutions also
exist for torsional and longitudinal waves.
We now consider the Helmholtz wave solutions for
the plucked, bowed and hammered string in a little more
detail.
Plucked String
Consider an ideal string initially at rest with an initial
transverse displacement a at its mid-point, as illustrated
in Fig. 15.19a. On release, kinks will propagate away
from the central point in both directions with velocity c,
but points on the string beyond the kinks will remain at
rest. When the kink arrives at a particular point along the
string, the associated impulse will accelerate the string
from rest to the uniform velocity of the central section of
the string. After a time t, the solution therefore comprises
P
a) Force
b) 1
t
0.5
0
0 5 10 15
Fig. 15.20 (a) Square-wave time dependence of transverse
force acting on the bridge from a string plucked at its centre
and (b) the corresponding amplitudes of the odd Fourier
components n varying as 1/n (dotted curve)
a straight central section of the string of width 2ct moving
downward with constant velocity c (2a/L), with the
outer regions remaining at rest until a kink arrives. After
a time L/2c, the kinks separating the straight-line sections
reach the ends and are reflected with change of sign.
After half a single period L/c, the initial displacement
will therefore be reversed and will return to the original
displacement after one full period 2L/c. In the absence
of damping, the process would repeat indefinitely.
Now consider the transverse force acting on the
end-support responsible for exciting sound through the
induced motion of the supporting bridge and vibrational
modes of the instrument. The initial transverse force
on the bridge is 2Ta/L, where we assume a ≪ L. This
force is unchanged until the first kink arrives. On reflection,
the direction of the force is reversed and is reversed
again when the second kink returns after reflection from
the other end of the string. The two circulating kinks
therefore cause a reversal in sign of the force on the endsupports
every half-cycle, resulting in a square-wave
waveform, as illustrated in Fig. 15.20. The spectrum of
a square wave has Fourier components at odd multiples
n of the fundamental frequency with amplitudes
proportional to 1/n,Fig.15.20b.
1
0.5
0
0 5 10
m=4
15 20
1
0.5
0
0 5 10
m=7
15 20
Fig. 15.21 Normalised Fourier amplitudes for the force on
the bridge for a string plucked 1/4 and 1/7 of the string
length from the bridge. The dashed curves show the 1/n
envelope of the partials of a sawtooth waveform