Introduction to Acoustics

tributing significantly to the sound of a plucked, bowed
or struck string.
The additional stiffness energy required to bend the
string will also affect wave propagation on the string
and the frequencies of the excited modes. Assuming
sinusoidal wave solutions varying as e i(ωt±kx) , the modified
wave equation (15.45) gives modes with resonant
frequencies
ω 2 n = c2 k 2 n
�
1 + δ 2 k 2 �
n . (15.46)
Waves on a real string are therefore no longer dispersionless,
but travel with a phase and group velocity
that depends on their frequency and wavelength. Any
Helmholtz kink travelling around a real string will therefore
decrease in amplitude and will broaden with time.
To maintain the Helmholtz slip–stick bowed waveform,
with a well-defined single kink circulating around the
string, the bow has to transfer energy to the string to compensate
for such broadening each time the kink moves
past the bow (Cremer [15.29], Chapt. 7 and Sect. 15.2.2.
If a rigidly supported string is free to flex at its
ends (known as a hinged boundary condition), solutions
of the form sin(nπx/L)sin(ωt). However, the mode
frequencies remain are no longer harmonic;
ω∗ �
n
= 1 + Bn 2� 1/2
∼ 1 + 1
2 Bn2 , (15.47)
ωn
with B = (π/L) 2 δ 2 , where the expansion assumes
Bn 2 ≪ 1.
When a string is clamped (e.g. by a circular collet), it
is forced to remain straight at its ends. Fletcher [15.43]
showed that this raises all the modal frequencies by
an additional factor ∼[1 + 2/π B 1/2 + (2/π) 2 B]. For
a real string supported on a bridge, connected to another
length of tensioned string behind the bridge, the boundary
conditions will be intermediate between hinged and
clamped.
Kent [15.44] has demonstrated that finite-flexibility
corrections raise the frequency of the fourth partial of
the relatively short C5 (an octave above middle-C) string
on an upright piano by 18 cents relative to the fundamental.
The inharmonicity would be even larger for the
very short, almost bar-like, strings at the very top of
the piano. However, the higher partials of the highest
notes on a piano rapidly exceed the limits of hearing,
so that the resulting inharmonicity becomes somewhat
less of a problem. The inharmonicity of the harmonics
of a plucked or struck string results in dissonances and
beats between partials, providing an edge to the sound,
which helps the sound of an instrument to penetrate
Musical Acoustics 15.2 Stringed Instruments 563
more easily. This is particularly true for instruments like
the harpsichord and the guitar when strung with metal
strings.
Finite-rigidity effects are particularly pronounced
for solid metal strings with a high Young’s modulus.
To circumvent this problem, modern strings for musical
instruments are usually composite structures using
a strong but relatively thin and flexible inner core, which
is over-wound with one or more flexible layers of thin
metal tape or wire to achieve the required mass (Pickering
[15.45, 46]). The difference in sound of an acoustic
guitar strung with metal strings and the same instrument
strung with more flexible gut or over-wound strings is
illustrated in .
15.2.2 Nonlinear String Vibrations
Large-amplitude transverse string vibrations can result
in significant stretching of the string giving a timevarying
component in the tension proportional to the
square of the periodically varying string displacement.
This leads to a number of nonlinear effects of considerable
scientific interest, though rarely of musical
importance.
Morse and Ingard [15.42] and(Fletcher and Rossing
[15.5], Chap. 5) provide theoretical introductions to
the physics of nonlinear resonant systems and to nonlinear
string vibrations in particular. Vallette [15.47]has
recently reviewed the nonlinear physics of both driven
and freely vibrating strings.
The Nonlinear Wave Equation
Transverse displacements of a string result in a fractional
�
increase of its length L by an amount
L
1/L0 0 1/2(∂ξ/∂x)2 dx and hence to a similar fractional
increase in tension and related frequency of
excited modes. For a spatially varying sinusoidal wave,
the induced strain and hence tension will vary with
both position and time along the string. Any spatially
localised changes in the tension will propagate along
the string with the speed of longitudinal waves. As
this is typically an order of magnitude larger than for
transverse waves, cL/cT ∼ √ L/∆L, where∆Lis the
amount that the string is stretched to bring it to tension,
such perturbations will propagate backwards and
forwards along the string many times during a single
cycle of the transverse waves. Hence, as pointed
out by Morse and Ingard [15.42], to a rather good
approximation, transverse wave propagation is determined
by the spatially averaged perturbation of the
tension.
Part E 15.2