Music and Design TM
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The Influence of Boundary Reenforcement om
Monopole, Dipole and Cardioid Woofers.
It is fairly well known that when a monopole woofer is placed close to a reflecting surface there is reenforcement of
the radiated sound pressure. "Close" is interpreted as meaning a distance significantly less than a wave length.
Compared to a woofer in free space, when a monopole woofer is placed close to or near an infinite wall the radiated
sound pressure at low frequency will increase by 6dB and the efficiency will increase by 3dB. When placed at the
intersection of two infinite planes which intersect a 90 degrees as a floor and a wall, the pound pressure will increase
by another 6dB and efficiency increases by another 3 dB for net gains of 12dB SPL and 6dB efficiency compared to
the woofer in free space. When placed in a corner, we see another 6dB SPL and 3dB efficiency for a net gain of 18dB
SPL and 9 dB efficiency. Of course, this assumes that there are no other walls present.

The behavior discussed above applies to monopole woofer, however, what is the result for dipole and cardioid
woofers? If we assume that the axis of the dipole and cardioid are aligned so as to be parallel to a side wall and floor,
which would not be unusual, and perpendicular to the wall behind the woofer system we find that the reenforcement
by the side wall and floor is identical to that for a monopole. On the other hand, the wall behind the woofer system
behaves somewhat differently. The figures below show how monopole, dipole and cardioid woofer behave when
placed near a wall directly behind them. The distances are not specific but chosen for demonstrative purposes.
The first figure shows the relative
positioning of the different woofer system.
In each case the distance, D, from the wall
is measured with reference to the front of
the woofer. The dipole woofer has a dipole
moment, d, equal to the separation between
front and rear sources. The cardioid is
similar to the dipole but the signal to the
rear woofer is delayed by a time increment
of d/c, where c is the speed of sound. The
grayed section of the figure shows the
phantom sources which result from the
reflection off the wall behind the woofers.

The next three figures show the impact
of the reflected sound on the sound
pressure far from the sources.  
The figure to the right shows the behavior for a
monopole. The red trace is for the woofer in free
space. The green trace includes the reflection from the
rear wall. At low frequency, below 100 Hz, the sound
pressure is 6dB greater than the free space value.
However, as the frequency increases a series of peaks
and dips appears in the response. These occur because
the phase of the reflected sound varies with frequency
due to the additional distance the sound wave must
travel and, depending on frequency, may be in or out
of phase with the direct sound. The dips are not
complete nulls because the reflected wave must travel
further and is therefore slightly lower in amplitude
when the "listening distance" is not infinite. A similar
behavior would be caused if the wall was not 100%
reflective.
Monopole
When the source is a dipole, equalized for flat free
space response, the result is as shown here, to the left.
The free space response, given by the red trace, is flat
up to the frequency of the first dipole null. The green
trace shows the effect of the rear wall. A peak in the
response is observed. Above the peak a series on
secondary peaks and dips are seen. Below the peak the
response rolls off a 6dB/octave. This should not be
confused with the "dipole" roll off as the dipole was
equalized to flat free field response. The behavior is
because the phantom and direct sources form a linear
quadrupole which is a second order gradient system
with 12dB/octave roll off at low frequency. The
flattening of the response at very low frequency is due
to the differences in distance the reflected waves must
travel.
Dipole
The last figure shows the result for a cardioid woofer.
Here it is observed that the cardioid is unaffected by the
rear wall. Thus, while placement of a cardioid woofer
close to the side wall or floor will provide the expected
boost is SPL and efficiency, the influence of the rear
wall is negated.
Cardioid