Introduction to Acoustics

couraged a world-wide school of violin makers, who
use scientific measurements and plate tuning in particular
as an aide to making high-quality instruments. The
comprehensive monograph on the Physics of the Violin
by Cremer [15.29] provides an invaluable theoretical
and experimental survey of research on instruments of
the violin family, with particular emphasis on the bowed
string, the action of the bridge, the vibrations of the body
and the radiation of sound.
The production of sound by any stringed instrument
is based on the same acoustic principles. The player
excites the vibrations of a stretched string by bowing,
plucking or striking. Energy from the vibrating
string is then transferred via the supporting bridge to
the acoustically radiating structural vibrations of the instrument.
The radiated sound is then conditioned by the
performing environment.
There are many different types of stringed instruments
formally classified as chordophones. Harp-like
lyres appear in Sumerian art from around 2800 BC.
However, more primitive instruments, like a plucked
string stretched over a bent stick and resonated across the
mouth, probably date back to soon after the emergence
of man the hunter [15.30, 31].
Stokes [15.32] was the first to recognise that the
vibrating string was essentially a linear dipole, which
radiated a negligible amount of sound at low frequencies
(see also Rayleigh [15.3] Vol. 2, Sect. 341). To produce
sound, the vibrating string has to excite the vibrations
of a much larger area radiating surface. For bowed and
plucked instruments, such as members of violin, lute and
guitar families, almost all the sound is radiated by the
shell of the instrument, with the acoustic output at low
frequencies usually boosted by the Helmholtz resonance
of the air inside the instrument vibrating in and out of
Player
Brain
Fingers
Hands
Arms
Body
Ears
Bowing
Plucking
Striking
Tactile
Aural
Instrument
Acoustic
Listener
Ears
Brain
Fig. 15.17 A schematic representation of the complex feedback
and sound radiation systems involved in the generation
of sound by a bowed string instrument
Musical Acoustics 15.2 Stringed Instruments 555
the f- or rose-holes cut into the front plate. On larger
instruments, such as the piano and harp, the sound is
radiated by a large soundboard.
For any continuously bowed (or blown) instrument,
the sound is conditioned by a complex feedback loop involving
the instrument, player and surrounding acoustic,
illustrated schematically for the violin in Fig. 15.17.The
expert string player controls the intonation and quality of
the sound produced using slight adjustments of the position
of the left-hand fingers stopping the string, and the
pressure, velocity and position of the bow on the string,
in response to the sound heard from both the instrument
and the surrounding acoustic. In addition, there is direct
tactile feedback through the fingers of both the left hand
controlling the pitch of the note and the right hand controlling
the bow. A similar overall feedback system is
also involved in playing woodwind or brass instrument.
The perception of the sound by both player and listener is
also strongly influenced by the performing acoustic and
the way the brain processes the sound received by the
sensory organs in the ears, as illustrated schematically in
Fig. 15.17. All such factors are involved in determining
the perceived quality of the sound produced by a musical
instrument. However, for simplicity and physical
insight into the various mechanisms involved, it is convenient
to consider the acoustics of musical instruments
in terms of their component parts, like the vibrating
string, the supporting bridge and shell of the instrument.
Nevertheless, it is important not to lose sight of the
fact that the sound produced by any instrument will involve
the interactions of all such subsystems and, even
more importantly, the skill of the player in exciting and
controlling the vibrations ultimately responsible for the
sound produced.
15.2.1 String Vibrations
The transverse vibrations ξ(x, t) of a perfectly flexible
stretched string, of mass µ per unit length and tension T,
satisfy the one-dimensional wave equation (d’Alembert,
1747)
∂2ξ 1
=
∂x2 c2 ∂
T
2ξ , (15.32)
∂t2 where the velocity of transverse waves cT = √ T/µ.The
tension T = ES∆L/L,whereEis Young’s modulus, S
is the cross-sectional area of the string and ∆L/L is the
fractional stretching of the string over its length L. For
the relatively small transverse displacements of bowed
and plucked strings on musical instruments, changes
in tension can be ignored. However, at larger ampli-
Part E 15.2