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Analysis
Expressing (3.13) in cylindrical coordinates for a cylindrical duct
of radius :

(3.14) 
where

(3.15) 
As discussed in section 2.4.1, we will be treating
cylindrically symmetric modes only.
From equation (2.48), the mode profile on the surface is

(3.16) 
where is the th zero of the Bessel function and is
tabulated in appendix A.
Now we will give in terms of , a dummy variable of
integration. We will show that all the other variables of integration will
then have an analytic solution. Sonine's integral from Watson [52]
p.416 gives:

(3.17) 
The integrand is imaginary when and real when .
Care must be taken when choosing the sign of
with the negative and imaginary interpretation taken here when .
Notice that we have been using the
opposite sign convention from Zorumski [37] for the imaginary part
throughout because we are assuming a time factor
of
rather than
.
Neumann's addition formula [52] p.358 is

(3.18) 
which can be substituted into (3.17) to give

(3.19) 
Now substituting (3.19) into (3.14) we get







(3.20) 
Note that
unless , in which
case it is equal to .
Integrating by and then gives a factor of .
When rearranged, the integral can be reduced to



(3.21) 
where

(3.22) 
The integration in equation (3.22) can be found analytically
(see equation (A.1) in appendix A):

(3.23) 
The four dimensional integral has now been reduced to a one dimensional
integration, with the variable (and therefore the singularity mentioned at
the end of section 3.3.1) eliminated.
Noticing that in equation (3.21) the integral is real for
and imaginary for , we split the integral into real
and imaginary parts with variables
and
respectively.







(3.24) 
Now
and
.
Remembering that the negative imaginary interpretation should be
taken for the resulting
we get







(3.25) 
The first integral can be performed by numerical integration using
Simpson's rule or an equivalent.
In the second integral, however, the range extends to infinity.
The integrand is an oscillatory function of whose amplitude of
oscillation decays exponentially to typically at around .
Numerical integration can then be performed from 0 and 10 without
incurring any significant numerical errors.
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Jonathan Kemp
20030324