Tech Design.....
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Design of woofer with varying directionality.
© 2006 John Kreskovsky
In the  discussion of power matching of midranges to woofers it was shown that for a speaker with dipole or cardioid
midrange operating in a 4 Pi environment and a woofer system operating in a 2 Pi environment, the directivity factor
of the woofer, DF, must be one half that of the midrange on the speakers axis if flat on axis response is to be
achieved while at the same time radiating constant total power vs frequency. A novel solution was presented for
matching dipole mids and woofers in which the woofer was rotated by 45 degrees relative to the system axis.
However, it was also shown that if the radiation pattern for the woofer is to be aligned with that of the system, then
the woofer system can not have a dipole radiation pattern, but must have a radiation pattern with DF equal to 1.5
along the axis of maximum intensity. It was suggested that such a woofer system could be developed by combining
a dipole woofer with a monopole. The chart below shows how the radiation pattern of such a combined woofer
behaves as a function of the relative strength of the on axis radiation of the dipole and the monopole. From left to  
right the chart shown the variation in radiated power (left vertical axis) and on axis DF (right vertical axis). The
relative on axis strength of the monopole source varies form 60 dB greater than the dipole source at the extreme
left to 120dB below the dipole at the extreme right. The total radiated power of the combined sources is shown in
pink  and the on axis DF in blue. As we move from left to right, at the point where the monopole and dipole are of
equal strength the radiation pattern, shown at the far right, is a cardioid with DF = 3 and total radiated power -4.77
dB relative to a pure monopole source with same on axis intensity. Moving further to the right we find a maximum DF
of 4 and a minimum in radiated power of -6dB at the point where the monopole component of the combined source
has an on axis SPL is -9.54 dB (0.33333x) that of the dipole component. Note the shape of the radiation pattern. As
the dipole strength continues to increase DF comes down and asymptotes to 3, the same values as for a cardioid,
and the radiated power is -4.77 dB. Examining the curve for DF we see that a DF of 1.5 occurs at the point where
the monopole on axis SPL is approximately 12.56dB  (4.25x) greater than the dipole. The radiation pattern is shown
second from the bottom. While difficult to see in the plot, the sound pressure level at 180 degrees is just over 4 dB
below the on axis SPL for this case. A woofer with this radiation pattern, operating in 2 Pi space (close to a floor),
would produce the same total radiated power, when the on axis SPL was matched, as a dipole operating in 4 Pi
space. Such a woofer could be constructed using a single monopole woofer and a an H frame dipole with the same
on axis response. The dipole woofer could likely be smaller then the monopole woofer due to the significantly lower
SPL levels required. Such a woofer would likely require two channels of amplification since the monopole and dipole
woofer must have the same on axis response and would probably require application of different response shaping
networks in addition the required dipole equalization of the H-frame. Potentially, a low power amplifier could be used
to drive the H-frame due to the low level of the required output. However, it is questionable that the benefits of
match the radiated power out weigh the complexity of using a single monopole woofer would  result in only a 1.76
dB mismatch between woofer and midrange.