So far we have provided the equations describing the behaviour of the modes of uniform ducts with circular and rectangular cross-section. As mentioned before, the aim of this chapter is to enable the calculation of acoustic variables in a duct of varying cross-section. The method employed here is to discretise the smoothly varying duct into a large number of concentric cylinders (or rectangles). While we have already seen equations which describe propagation within each cylinder, we still need to analyse how the modes of the duct are effected by changes of cross-section. This section deals with this problem.
Consider again the typical discontinuous join between two sections
of tube of differing cross-section shown in figure 2.2.
The pressure field at either side of the discontinuity must be equal on the
section of air they share.
However, the th mode on
will not match the
th mode on
because the cross-sections are different. This means that
when the
th mode is incident on the discontinuity, the pressure on the
other
side must consist of the sum of the contributions of many modes. We say that
the wave experiences mode conversion at the discontinuity.
Now this will be put into our mathematical framework. We recall
is the vector of modal pressure amplitudes on
the surface
and define
as the vector of modal
pressure amplitudes on the surface
.
In circular
cross-section, when
,
can be found from
by projection.
This procedure can also be performed in rectangular geometry
when
and
.
The following expression relating the pressure
vectors on either side of the discontinuity is derived in appendix B
using the orthogonality of Bessel functions:
The vector can be projected by equating the axial velocity on
the air shared by
and
. Also the axial velocity is
required to be zero into the wall surface perpendicular to the
axis which
results from
not equalling
.
For
the axial velocity on either side is therefore matched on
and set to zero on the part of
which is not in contact with
.
The calculation is performed in detail in appendix B to give