Introduction to Acoustics

y bowed, plucked and struck stringed instruments and
all percussion instruments.
Time-Domain Measurements
The vibrational characteristics of an instrument can also
be investigated in the time domain. For example, by
striking a stringed instrument with a light hammer or
exciting the vibrational modes of a woodwind or brass
instrument with a short puff of air, the frequencies of
free vibration of the vibrational modes and their damping
can be determined from their time-dependent decay.
Provided the damping is not too strong (Q ≫1), the
modes will decay with time as
ξn(t) = ξ0 e −t/τn e iωnt , (15.5)
where τn = 2Qn/ωn = Qn/π fn. The frequency fn of
a given mode can be determined from its inverse period
and Qn from π× the number of periods for the amplitude
to fall by the factor exponential e. The Q-value
of strongly excited modes of a musical instrument can
be estimated from τ60dB = 13.6τ, the Sabine decay time
(Chap. 10 Concert Hall Acoustics). This is the perceptually
significant time taken for the sound pressure to fall
by a factor of 10 3 – from a very loud level to just being
detectable. Hence, Qn = π fnτ60/13.6 ∼ 0.23 fnτ60.
For example, the sound of a strongly plucked cello
open A-string (220 Hz) can be heard for at least ∼ 2s,
corresponding to a Q-value of ≈ 100 or more.
Damping results in a loss of stored energy given by
dEn
dt =−ωn En =−2
Qn
En
. (15.6)
τn
Hence, the power P required to maintain a constant
amplitude at resonance is ωn
Qn En, whereEnis the energy
stored. This tends to be the way that Q is defined
and measured by physicists, whereas in acoustic spectroscopy
it is more usual to define and measure Q-values
from either the width of resonances in spectroscopic
measurements or from decay times after transient excitation.
As illustrated above, all such definitions are
equivalent.
15.1.2 Radiation from Instruments
Although a large number of vibrational modes of a musical
instrument may be excited simultaneously, they will
not be equally important in radiating sound, which has
important consequences for the quality of the sound.
This section therefore provides a brief introduction to
the radiation of sound from the vibrational modes of
musical instruments.
Musical Acoustics 15.1 Vibrational Modes of Instruments 537
Sound Waves in Air
In free space, the longitudinal displacement ξ(x, t) =
ξ0 e i(ωt−kx) of plane sound waves satisfies the wave
equation
∂2ξ 1
=
∂x2 c2 ∂
0
2ξ . (15.7)
∂t2 The dispersionless (independent of frequency) velocity
of sound c0 = √ γ P0/ρ, whereγ (≈ 1.4) is the ratio
of specific heats at constant pressure and volume, P0
(≈ 10 5 Pa or N/m 2 ) is the ambient pressure and ρ
(≈ 1kg/m 3 ) is the density (the brackets give the values
for air at ambient pressure and temperature). The ratio
of acoustic pressure p =−γ P0∂ξ/∂x to the particle velocity
v = ∂ξ/∂t is referred to as the specific impedance,
z0 = p/v = ρc0.
The appearance of γ in the expression for the velocity
of sound reflects the adiabatic nature of acoustic
waves. This arises because acoustic wavelengths are
far too long to allow any significant equalisation of
the longitudinal temperature fluctuations arising from
the compressions and rarefactions of a sound wave. In
free space longitudinal heat flow between the fluctuating
regions is only important at very high ultrasonic frequencies
(MHz), where it leads to significant attenuation. The
major source of attenuation of freely propagating acoustic
sound waves arises from the water vapour present.
However, both viscous and transverse thermal losses to
the side walls of woodwind and brass instruments can
result in significant attenuation, as described later.
The above expressions neglect first-order, nonlinear,
corrections to the compressibility, proportional to
∂ξ/∂x, and other inertial correction terms in the nonlinear
Navier–Stokes equation. This approximation breaks
down at the very high intensities in the bores of the
trumpet and trombone when played very loudly [15.9],
which results in shockwave propagation, with a transition
from a relatively smooth to a very brassy sound (son
cuivré in French). For the present, such corrections will
be neglected.
The speed of sound in air depends on the temperature
θ (degrees centigrade) and, to a lesser extent, on the
humidity. For 50% humidity,
c0(θ) = 332 (1 + θ/273) 1/2 ≈ 332(1 + 1.710 −3 θ) ,
(15.8)
giving a value of 343 m/s at20 ◦ C. Note that the air
inside a woodwind or brass instrument, once the instrument
is warmed up, will always be warm and humid,
which will affect the playing pitch.
Part E 15.1