Music and Design TM

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.
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.
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.