148
4 After the Lips: Acoustic Resonances and Radiation
Fig. 4.41 EFP values for the first 12 resonant modes of the cylindrical tubes with (squares) and
without (circles) a trombone mouthpiece
Fig. 4.42 Relationship
between plane wavefront
cutting the horn axis at x and
quasi-spherical wavefront,
perpendicular to the wall at x
and cutting the horn axis at z
1
S
∂
∂x
S
∂p
∂x
=
1
c 2
0
∂ 2 p
∂t 2 ,
(4.56)
where S(x) = πr 2 is the cross-sectional area of the pipe at position x along the
bore and c 0 is the speed of sound propagation in open air. This is usually called
the Webster horn equation. This formulation assumes plane wave propagation in
the tube. As was already noted when discussing sound propagation in conical tubes
(see Fig. 4.27), the wavefronts in a tube of varying radius are in general curved
surfaces. It is possible to modify the plane wave treatment to take the curvature of
the wavefronts into account by redefining S as the area of the wavefront surface
which cuts the bore axis at z (see Fig. 4.42) (Benade and Jansson 1974; Fletcher and
Rossing 1998). An effective bore radius can then be defined as
a =
S
π
1
2
.
(4.57)