Introduction to Acoustics

540 Part E Music, Speech, Electroacoustics
Part E 15.1
xy-plane with displacements in the z-direction varying
as sin(kx x)eiωt . We look for propagating sound
waves solutions radiating from the surface of the form
sin(kx x)ei(ωt−kzz) , which must satisfy the wave equation
and hence the relationship,
k 2 ω2
z =
c2 − k
0
2 �
1
x = ω2
c2 −
0
1
c2 �
, (15.18)
m
where cm is the phase velocity of transverse waves on the
membrane or plate in the xy-plane. Sound will therefore
only propagate away from the surface (k2 z > 0) when
cm > c0. If the sound velocity is greater than the phase
velocity in the plate or membrane, energy will flow from
regions of positive to negative vertical displacements and
vice versa, with an exponentially decaying sound field,
varying as e−z/δ where δ = |kz| −1 .
Typical dispersionless wave velocities for the
stretched drum heads of timpani are around 100 m/s
(Fletcher and Rossing [15.5], Sect. 18.1.2), so that they
are not very efficient radiators of sound. This is particular
relevant for asymmetrical modes, when sound
energy can flow from the regions of positive to negative
displacement and vice versa. However, for even modes,
the cancellation between adjacent regions moving out
of phase with each other can never be complete, so that
such modes will radiate more effectively.
A particularly interesting case occurs for stringed
instruments, where the phase velocity of the transverse
vibrations of the thin front and back plates increases with
frequency as ω1/2 (Sect. 15.2.6). Hence, below a critical
crossover or coincidence frequency, when the phase
velocity in the plates is less than the speed of sound
in air, standing waves on the vibrating plates are relatively
inefficient radiators of sound, while above the
crossover frequency the plates radiate sound rather efficiently.
Cremer [15.11] estimates the critical frequency
for a 4 mm-thick cello plate as 2.8kHz; fora 2.5mm
violin plate the equivalent frequency would be ≈ 2kHz.
Radiation from Wind Instruments
The holes at the ends or in the side walls of wind instruments
can be considered as piston-like radiation sources.
At high frequencies, such that ka ≫ 1, where a is their
radius, the holes will be very efficient radiators radiating
acoustic energy ∼ 1/2z0v 2 per unit hole area. However,
over most of the playing range ka ≪ 1, so that the radiation
efficiency drops off as (ka) 2 , just like the spherical
monopole source. Most of the sound impinging on the
end of the instrument is therefore reflected, so that strong
acoustic resonances can be excited, as discussed in the
later section on woodwind and brass instruments.
15.1.3 The Anatomy of Musical Sounds
The singing voice, bowed string, and blown wind instruments
produce continuous sounds with repetitive
waveforms giving musical notes with a well-defined
sense of musical pitch. In contrast, many percussion
instruments produce sounds with non-repetitive
waveforms composed of a large number of unrelated
frequencies with no definite sense of pitch, such
as the side drum, cymbal or rattle. There are also
other stringed instruments and percussion instruments,
such as the guitar, piano, harp, xylophone, bells and
gongs, which produce relatively long sounds, where the
slowly decaying vibrations produce a definite sense of
pitch.
In all such cases, the complexity of the waveforms
of real musical instruments distinguishes their sound
from the highly predictable sounds of simple electronic
synthesisers. This is why the sounds of computergenerated
synthesised instruments lack realism and are
musically unsatisfying. In this section, we introduce
the way that sound waveforms are analysed and described.
Sinusoidal Waves
The most important, but musically least interesting,
waveform is the pure sinusoid. This can be expressed
in several alternative forms,
a cos(2π ft + φ) = a cos(ωt + φ) = Re(a e iωt ) ,
(15.19)
where a is in general complex to account for phase,
f is the frequency measured in Hz and equal to the
inverse of the period T, ω = 2π f is the angular frequency
measured in radians per second, t is time, and
φ is the phase, which depends on the origin taken for
time.
Any sound, however complex, can be described
in terms of a superposition of sinusoidal waveforms
with a spectrum of frequencies. Figure 15.3 contrasts
the envelopes, waveforms and spectra of a synthesised
sawtooth waveform and the much more complex and
musically interesting waveform of a note played on the
oboe ( provides an audio comparison). Note
the much more complex fluctuating envelope and less
predictable amplitudes of the frequency components in
the spectrum of the oboe.
As we will show later, in defining the sound and
quality of any musical instrument, the shape and fluctuations
in amplitude of the overall envelope are just as
important as the waveform and spectrum.