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We denote the pressure field on surface 1 as
and the pressure
field on surface 2 as
. In plane wave
propagation we saw that the pressure is taken to be the same on both sides of
the discontinuity. In the multimodal case the pressure field is matched
at either side. For the case shown where
the
pressure is matched on the air they share,
.
 |
(B.1) |
Now the concept of orthogonality of the modes is used.
Recalling our orthogonality relation from equation
(2.47):
 |
(B.2) |
it follows that the integration of the mode profile
with a general
pressure field,
will extract the component of the
th mode in the pressure
field. Using the fact that the pressure may be expressed as a sum of
the modes from equation 2.26
(ignoring the
time factor):
 |
(B.3) |
We will use
and
to denote the complex mode
amplitudes on surfaces 1 and 2 respectively.
and
are the corresponding mode profiles on surfaces 1 and 2.
Combining equations (B.1) and
B.3) gives
 |
(B.4) |
which may be written as
 |
(B.5) |
where
 |
(B.6) |
Using matrix notation,
is a column vector whose entries are
given by
and
 |
(B.7) |
where F is a matrix with elements
. We have now proved
equations (2.79) and (2.80) from chapter 2.
We now have a formula giving the pressure on a smaller cross-section
at a discontinuity from the pressure on a larger cross-section on the other
side. Since
and
are assumed to not be separated by any distance
along the
axis the formula holds whatever side the largest cross-section
is on. Consider if
. Now section 1 is the larger cross-section
meaning that we just have to interchange the labels 1 and 2 in
equation (B.7):
 |
(B.8) |
where
is a matrix with the elements defined by
 |
(B.9) |
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Jonathan Kemp
2003-03-24