4.6 Radiation of Sound from Brass Instruments
191
internal conical surface. The pressure amplitude in the shock wave is greatest in
the axial direction and falls off with increasing angle; the black lines recording the
measured wavefronts end when the pressure has fallen so far that the trace is no
longer detectable. The radiated sound intensity is largely concentrated in the solid
angle defined by the internal surface of the cone.
The behaviour of the shock waves emanating from the piccolo trumpet bell is
rather different. The apparent centre of curvature of the wavefronts is not fixed, but
moves significantly as the wavefront expands. Although the conical funnel and the
piccolo trumpet have similar exit diameters, the solid angle containing most of the
radiated sound energy is much smaller in the case of the trumpet, confirming that
for very loud playing, the flaring bell directs the radiation strongly in the forward
direction.
4.6.6 Far Field Directivity in Brass Instruments
In the far field region, all acoustic sources are characterised by spherical wavefronts
and a pressure amplitude which is inversely proportional to the radius (Sect. 4.6.1).
The pressure amplitude is not necessarily the same at every point on a spherical
wavefront, however, but may vary with the angle θ between the bell axis and the
direction of radiation. In Fig. 4.83 the acoustic centre is shown in the plane of the
bell, but the experimental measurements of López-Carromero et al. (2016) suggest
that it is more likely to be approximately one bell diameter inside the bell.
The far field pressure from an axisymmetric acoustical source can be expressed
as
p + =
A
r
H (θ, f )e
j (wt−kr)
(4.85)
where the directional factor H (θ, f ) describes the angular dependence of the
radiation for points on the spherical wavefront with radius r (Kinsler et al. 1999,
p. 188). H (0, f ) is the pressure on the bell axis. For a monopole source, H (θ, f ) =
Fig. 4.83 Illustrating the
definition of the directivity
angle θ
191
internal conical surface. The pressure amplitude in the shock wave is greatest in
the axial direction and falls off with increasing angle; the black lines recording the
measured wavefronts end when the pressure has fallen so far that the trace is no
longer detectable. The radiated sound intensity is largely concentrated in the solid
angle defined by the internal surface of the cone.
The behaviour of the shock waves emanating from the piccolo trumpet bell is
rather different. The apparent centre of curvature of the wavefronts is not fixed, but
moves significantly as the wavefront expands. Although the conical funnel and the
piccolo trumpet have similar exit diameters, the solid angle containing most of the
radiated sound energy is much smaller in the case of the trumpet, confirming that
for very loud playing, the flaring bell directs the radiation strongly in the forward
direction.
4.6.6 Far Field Directivity in Brass Instruments
In the far field region, all acoustic sources are characterised by spherical wavefronts
and a pressure amplitude which is inversely proportional to the radius (Sect. 4.6.1).
The pressure amplitude is not necessarily the same at every point on a spherical
wavefront, however, but may vary with the angle θ between the bell axis and the
direction of radiation. In Fig. 4.83 the acoustic centre is shown in the plane of the
bell, but the experimental measurements of López-Carromero et al. (2016) suggest
that it is more likely to be approximately one bell diameter inside the bell.
The far field pressure from an axisymmetric acoustical source can be expressed
as
p + =
A
r
H (θ, f )e
j (wt−kr)
(4.85)
where the directional factor H (θ, f ) describes the angular dependence of the
radiation for points on the spherical wavefront with radius r (Kinsler et al. 1999,
p. 188). H (0, f ) is the pressure on the bell axis. For a monopole source, H (θ, f ) =
Fig. 4.83 Illustrating the
definition of the directivity
angle θ