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Consider a spherical secondary source (``point source'')
located at
as shown in Fig.15. Then the
pressure amplitude along the listening line
for all
is given by

(9) 
where
denotes the pressure amplitude observed from the point
source at
, and
denotes the angle of the line from the source center to the linearray
point at
, i.e.,
. This polarpattern slice is
plotted in Fig.16 for the triangular
array+listener geometry such as described in §4.10 on page , with
. The top plot shows
over a 12 m width
centered about a 6 m wide array, and the bottom plot shows
for
, thereby covering the
entire axis containing array.
Figure 16:
Pointsource
polarpattern slice for a listeningline
away from the
source. Top: Gain
versus position
. Bottom: Gain
versus angle
.

In addition to the gain variation
in Eq.(9), we also
have phase effects that we don't have when keeping radius
constant and only varying
. The envelope
in Fig.16 is due only to spherical
spreading loss according to
. The difference in propagation
distance between
and
is

(10) 
The resulting frequencyresponse
re
is
plotted in Fig.17 for 1 kHz (wavelength about a
foot (1.126 ft)) and
, where
is the wavenumber (spatial radian frequency), which has
made its appearance for the first time in these formulas.^{34}
Figure 17:
Real part of pointsource
frequencyresponse for a listeningline one wavelength away
(
). Top: Gain
versus position
. Bottom: Gain
versus angle
.

For small
, we can use the approximation
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