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## Analysis

Consider a rectangular duct of half widths and terminated in an infinite baffle, as shown in figure 3.4.

Expressing (3.13) in rectangular coordinates for a rectangular duct of half-widths and gives:

 (3.26)

where
 (3.27)

and
 (3.28)

with
 (3.29)

 (3.30)

 (3.31)

Changing variables as in Swenson et al. [49] and Levine [51] to , , , and the integral becomes

 (3.32)

where
 (3.33)

The quadruple integral can now be reduced to a double integral by performing integration by and analytically. The first step is to expand the cosines in equations (3.29) and (3.30):
 (3.34)

The second and third terms go to zero since we are integrating over a symmetric interval in . Performing integration gives:
 (3.35)

The integral for the impedance is then:
 (3.36)

Changing variables to and means that the radiation impedance is expressed in terms of the dimensionless variables and :

 (3.37)

Note that if we put into equation (3.35) we obtain
 (3.38)

The radiation impedance from equation (3.37) is then identical to the radiation impedance of a rectangular piston in an infinite baffle [47,48,49,50,51] (note that most authors have used and as widths rather than half widths). Equation (3.37) has a singularity at the origin if and which must be removed if the radiation impedance is to be calculated by numerical integration. To do this the integral is first split into two parts:
 (3.39)

where
 (3.40)

The first part is non-singular and the singularity in the second half may be removed by integration, giving:
 (3.41)

Equation (3.41) may be evaluated by numerical integration to provide the radiation impedance.

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Jonathan Kemp 2003-03-24