Introduction to Acoustics

a) Force on mass
b)
1
1
0
0 1 2
Time
0.5
0
0 5 10 15 20
Fig. 15.26 (a) Time dependence of the upward force acting
on a light hammer impacting a string in arbitrary units of
time, and (b) amplitudes of Fourier coefficients of force
acting on end supports for a hammer after hitting the string
1/7 of the length from an end-support. The continuous line
shows the continuous spectrum for a string of semi-infinite
length
off the string. However, because the mass cannot change
its velocity instantaneously, any returning wave will be
partially reflected, so that the mass acts as a source of
secondary reflected waves travelling outwards in both
directions. The total force acting on the hammer is then
given by any residual force from the first impact plus
the subsequent forces created by the succession of reflections
from the end-supports. This problem was first
correctly solved by Hall, in the first of four seminal papers
on the string–piano hammer interaction [15.38–41].
Hall showed that the first reflected wave exerts an
additional decelerating force g(t ′ ) ∼ (1 − t ′ /τ)e −t′ /τ on
the mass, where t ′ is the time after arrival of the first reflection.
This is illustrated in Fig. 15.26 for a relatively
light mass impacting the string at a position 1/7-th of
the string length from an end. Provided the mass is sufficiently
small, the force from the initial impact will
have decayed significantly by the time the first reflection
returns, so that the force acting on the mass will
become negative (the dotted section in Fig. 15.26a), and
the mass will detach itself from the string. The string
will then move away from the mass and will vibrate
freely, provided the hammer is prevented from falling
back onto the string. An elaborate mechanism is used on
the piano to prevent this from happening (see Rossing,
Fletcher [15.5], Sect. 12.2), while the zither or dulcimer
player quickly lifts the hammer well away from the string
after the initial impact using much the same striking action
as a percussionist playing a drum, where the same
considerations apply.
The heavier the mass, the longer it will remain in
contact with the string. Hall showed that it may then
take several reflections from both ends of the string and
sometimes several periods of attachment and detach-
Musical Acoustics 15.2 Stringed Instruments 561
ment before the mass is finally thrown away from the
string. A sufficiently heavy mass will never bounce back
off the string.
In general, the waveforms excited on the string
will therefore be rather complicated functions of the
properties of the string, hammer and striking position.
However, for a very light mass (≪ mass of the string),
which is thrown off the string by the first reflected wave,
the Fourier coefficients of the induced velocity waveform,
and hence the force on the end-supports, are given
by vn ≈ � 1 + e −1+inπα� sin (nπα), whereα = a/L, illustrated
in Fig. 15.26b for an impact 1/7-th of the way
along the string. Note that the seventh harmonic is missing,
as again expected from general arguments, since no
work can be transferred to a particular mode of string
vibration for a force applied at a nodal position.
In practice, the spectrum is affected by the finite
size of the hammer, multiple reflections occurring before
the hammer is thrown from the string, and the
elastic and often hysteretic properties of the hammer
material [15.38].
Striking Tangent
On the clavichord (Fletcher, Rossing [15.5], Sect. 11.6),
a string is struck by a rising end-support, or tangent,
which remains in contact with the string, exciting transverse
vibrations of the string on both sides of the tangent.
If we assume a simplified model in which the rising
tangent moves with constant velocity until its final displacement
a is reached, there is again a simple Helmholtz
wave solution. In practice, the length of string on one
side of the tangent is damped, so that free vibrations are
only excited on one side of the striking point. We therefore
need only consider the length of string between the
tangent and the end connected to the soundboard. The
discontinuities ±v in the tangent velocity, occurring on
initial impact and on reaching its final displacement after
a time ∆t, generate propagating kinks and discontinuities
of velocity of opposite sign separated in time by
∆t. The striking therefore excites waves with kinks, velocities
and displacements along the string shown in
Fig. 15.27a. The solutions are again Helmholtz waves,
but now with two kinks of opposite signs travelling
around the string in the same direction.
The Fourier coefficients of the velocity waveform
showninFig.15.27b can be written as
cn ∼ 1
n
a
L ′
�
1 − e in2πβ�
, (15.44)
where β = ∆t/T1 is the fraction of the period T1 = L ′ /2c
of the freely vibrating length of string during which the
Part E 15.2