4.6 Radiation of Sound from Brass Instruments
187
the wavefront travels through the bell towards the exit plane is evident in Fig. 4.78a.
The contour of equal real pressure at the bell exit is approximately a circular arc
with its centre around 5 cm inside the bell. Once into free space, the contours are
also close to circular, but the common centre is now near the (x = 0, y = 0) point
where the bell exit plane cuts the axis. There is no evidence in either theoretical or
experimental map of an angular dependence of the radiation pressure amplitude at
500 Hz. The (0,0) point could therefore be described as the acoustic centre at this
frequency.
The radiation field looks a little different at a frequency of 1000 Hz, as illustrated
in Fig. 4.78c and d. There is a slight suggestion of flattening of the contour lines
and a reduction of the strength of the radiation at angles approaching 90 ◦ . The
flattening of the wavefronts and the concentration of the radiated power in the
forward direction is increasingly evident as the frequency is raised to 2000 Hz
(Fig. 4.78e and f) and then to 4000 Hz (Fig. 4.78g and h). In Fig. 4.78g a wave crest
is just passing the (x = 0, y = 0) point; a further three crests cut the bell axis at
values of x which are multiples of the free space wavelength 0.086 m. Since the
wavefronts are not spherical and the radiation is far from isotropic, the idea of an
acoustic centre is of limited use at these higher frequencies.
4.6.5 Visualising Wavefronts with Schlieren Optics
When a brass instrument with a high proportion of narrow-bored tubing is played
very loudly, shock waves can be generated within the air column of the instrument.
This phenomenon, which is responsible for the ‘brassy’ timbre of fortissimo
trumpets and trombones, is discussed in detail in Sect. 6.1. The crest of the wave
which leaves the mouthpiece travels down the tube slightly faster than the trough;
eventually the crest catches up with the trough in front of it, resulting in an almost
instantaneous pressure rise. If the mouthpiece signal is a sine curve, the dependence
of pressure on time inside the tube just after shock formation resembles the idealised
‘N-wave’ shown in the upper part of Fig. 4.79. When such a shock wave reaches the
bell of the instrument, the rapid pressure rise is followed by a precipitous drop as
the wave expands into free space; the pressure signal registered by a microphone in
the radiation field is characterised by a sequence of sharp spikes as illustrated in the
lower part of Fig. 4.79. The external pressure signal resembles the derivative of the
internal N-wave.
The technique of schlieren photography registers gradients in the refractive index
of a fluid as patterns of light and shade in an image (Settles 2001). Since the change
in pressure due to the passage of a sound wave is accompanied by a change in the
density and therefore refractive index of the air, it is in principle possible to make
sound wavefronts visible by this technique. The density gradients in the radiation
fields shown in Fig. 4.78 are too small to be visualised using this method: the
distance between maximum and minimum pressure contours is half a wavelength,
which even at 4000 Hz is 43 mm, and the pressure amplitude was restricted to a
187
the wavefront travels through the bell towards the exit plane is evident in Fig. 4.78a.
The contour of equal real pressure at the bell exit is approximately a circular arc
with its centre around 5 cm inside the bell. Once into free space, the contours are
also close to circular, but the common centre is now near the (x = 0, y = 0) point
where the bell exit plane cuts the axis. There is no evidence in either theoretical or
experimental map of an angular dependence of the radiation pressure amplitude at
500 Hz. The (0,0) point could therefore be described as the acoustic centre at this
frequency.
The radiation field looks a little different at a frequency of 1000 Hz, as illustrated
in Fig. 4.78c and d. There is a slight suggestion of flattening of the contour lines
and a reduction of the strength of the radiation at angles approaching 90 ◦ . The
flattening of the wavefronts and the concentration of the radiated power in the
forward direction is increasingly evident as the frequency is raised to 2000 Hz
(Fig. 4.78e and f) and then to 4000 Hz (Fig. 4.78g and h). In Fig. 4.78g a wave crest
is just passing the (x = 0, y = 0) point; a further three crests cut the bell axis at
values of x which are multiples of the free space wavelength 0.086 m. Since the
wavefronts are not spherical and the radiation is far from isotropic, the idea of an
acoustic centre is of limited use at these higher frequencies.
4.6.5 Visualising Wavefronts with Schlieren Optics
When a brass instrument with a high proportion of narrow-bored tubing is played
very loudly, shock waves can be generated within the air column of the instrument.
This phenomenon, which is responsible for the ‘brassy’ timbre of fortissimo
trumpets and trombones, is discussed in detail in Sect. 6.1. The crest of the wave
which leaves the mouthpiece travels down the tube slightly faster than the trough;
eventually the crest catches up with the trough in front of it, resulting in an almost
instantaneous pressure rise. If the mouthpiece signal is a sine curve, the dependence
of pressure on time inside the tube just after shock formation resembles the idealised
‘N-wave’ shown in the upper part of Fig. 4.79. When such a shock wave reaches the
bell of the instrument, the rapid pressure rise is followed by a precipitous drop as
the wave expands into free space; the pressure signal registered by a microphone in
the radiation field is characterised by a sequence of sharp spikes as illustrated in the
lower part of Fig. 4.79. The external pressure signal resembles the derivative of the
internal N-wave.
The technique of schlieren photography registers gradients in the refractive index
of a fluid as patterns of light and shade in an image (Settles 2001). Since the change
in pressure due to the passage of a sound wave is accompanied by a change in the
density and therefore refractive index of the air, it is in principle possible to make
sound wavefronts visible by this technique. The density gradients in the radiation
fields shown in Fig. 4.78 are too small to be visualised using this method: the
distance between maximum and minimum pressure contours is half a wavelength,
which even at 4000 Hz is 43 mm, and the pressure amplitude was restricted to a