4.6 Radiation of Sound from Brass Instruments
183
The velocity of a surface point on the sphere is along the radial direction and can be
written as
v s = ˆ
v s e
j (ωt) .
(4.81)
Since the surface velocity must match the sound field particle velocity at r = a,
ˆ
v s =
−jA
ρcka 2
,
(4.82)
and
A = jρck ˆ
v s a
2
=
jρckQ
4π
,
(4.83)
where Q = 4πa 2 ˆ
v s is the maximum volume velocity of the air displaced by the
motion of the surface. The pressure in the radiated field can now be written as
p + =
A
r
e
j (wt−kr)
= j
ρckQ
4πr
e
j (wt−kr) .
(4.84)
The pulsating sphere is described as a simple source, since it radiates a monopole
field. The quantity Q is described as the source strength. The imaginary unit j in the
expression on the right-hand side of Eq. 4.84 reflects the phase difference between
the pressure and particle velocity at the surface of the source.
Any oscillating source whose dimensions are much smaller than the wavelength
of the radiated sound can be treated as a simple source. The radiation from an object
too large to satisfy this criterion can be calculated by considering it to be made up
of a set of simple sources, although the interactions between the sources have to be
taken into account (Chaigne and Kergomard 2016, p. 647).
4.6.3 Transition from Internal to External Sound Fields
In Sect. 4.1.2 it was explained that for low frequencies, only plane waves can
propagate along a cylindrical tube. However the sound waves radiated from the open
end of the cylinder are expected to have spherical surfaces when the wavelength
is much larger than the tube diameter (see Sect. 4.6.2). There must clearly be
a transition region in which the flat wavefronts approaching the cylinder exit
gradually develop the curvature which is characteristic of the external spherical
waves. Figure 4.5 indicates schematically the nature of the transition.
The assumption that the wall of the instrument is rigid requires that the acoustic
velocity next to the wall is directed parallel to the surface, which in turn implies that
the wavefront must be perpendicular to the wall. For a conical tube, this condition is
183
The velocity of a surface point on the sphere is along the radial direction and can be
written as
v s = ˆ
v s e
j (ωt) .
(4.81)
Since the surface velocity must match the sound field particle velocity at r = a,
ˆ
v s =
−jA
ρcka 2
,
(4.82)
and
A = jρck ˆ
v s a
2
=
jρckQ
4π
,
(4.83)
where Q = 4πa 2 ˆ
v s is the maximum volume velocity of the air displaced by the
motion of the surface. The pressure in the radiated field can now be written as
p + =
A
r
e
j (wt−kr)
= j
ρckQ
4πr
e
j (wt−kr) .
(4.84)
The pulsating sphere is described as a simple source, since it radiates a monopole
field. The quantity Q is described as the source strength. The imaginary unit j in the
expression on the right-hand side of Eq. 4.84 reflects the phase difference between
the pressure and particle velocity at the surface of the source.
Any oscillating source whose dimensions are much smaller than the wavelength
of the radiated sound can be treated as a simple source. The radiation from an object
too large to satisfy this criterion can be calculated by considering it to be made up
of a set of simple sources, although the interactions between the sources have to be
taken into account (Chaigne and Kergomard 2016, p. 647).
4.6.3 Transition from Internal to External Sound Fields
In Sect. 4.1.2 it was explained that for low frequencies, only plane waves can
propagate along a cylindrical tube. However the sound waves radiated from the open
end of the cylinder are expected to have spherical surfaces when the wavelength
is much larger than the tube diameter (see Sect. 4.6.2). There must clearly be
a transition region in which the flat wavefronts approaching the cylinder exit
gradually develop the curvature which is characteristic of the external spherical
waves. Figure 4.5 indicates schematically the nature of the transition.
The assumption that the wall of the instrument is rigid requires that the acoustic
velocity next to the wall is directed parallel to the surface, which in turn implies that
the wavefront must be perpendicular to the wall. For a conical tube, this condition is