Tech Design.....
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A novel configuration for subwoofers in a rectangular room
yielding characteristics of dipole bass.
Before starting this discussion let me state that the intent is not to optimize in room
bass response by placement of multiple subwoofers as suggested by Welti [1] or to
eliminate the contribution from standing waves with complex woofer arrays on the
front and rear walls with additional time delay (
see for example this AVS post   and
the  paper from
Klein+Hummel). The intent here is only to show that  through
correct placement of two monopole woofers, with one woofer is connected with
inverted polarity, the woofer system (i.e. the summed response form both sources)
fails to excite at least the same room modes as dipole woofer system aligned with a
room axis. Thus, such a configuration should therefore yield bass response which
sounds subjectively similar to dipole a dipole woofer system while avoiding the
complexity of dipole equalization and the need for high excursion woofers.

It is often reported that dipole woofers sound subjectively better than monopole
woofers. This remains a matter of contention, and there is little objective evidence
to support this position. However, the commonly stated reason that lends support
to this subjective observation is that dipoles excite fewer room modes than
monopole woofers. This statement itself is somewhat flawed. In
another section it
was shown that only when a dipole woofer system is oriented in a listening room
with the dipole axis aligned with a modal direction are fewer modes excited. For
example, consider Figure1 where the dipole axis is aligned with the X axis. If, as in
Figure 1, the X axis is taken along the length of the room, the Y axis as the width,
and the Z axis as the height, then axial modes in the Y and Z directions as well as
the tangential modes in the YZ plane are not excited (we shall refer to these modes
as the Y, Z and YZ modes). This is because no matter where we are positioned in
this YZ plane shown in Figure 1, or where the YZ plane intersects the X axis of the
dipole, the contributions to the in room SPL of both sources of the dipole from
these modes is equal in strength but opposite in polarity. Thus the summed
contribution from the two sources cancel and the Y,Z and YZ modes do not
contribute to the in room SPL.  Additionally, a dipole, regardless of orientation, can
not excite the DC mode. Thus, room pressurization effects are also excluded from
the response of a dipole woofer in a room.

We can take this a step further by noting that in a rectangular room a monopole
source placed near the front wall and a similar source placed near the rear wall
excite the same axial Y and Z modes and the same YZ tangential modes (see
Figure 2). Thus we see that regardless of the X location of the source the same Y,
Z and  ZY modes will be excited by the sources if the Y and Z positions for the
sources are the same.

This leads us to consider a rather unconventional woofer arrangement for a
sub/satilite speaker system. Typically, such a system with dual  monopole woofers
would be set up with the woofers placed near the main speakers, or perhaps in
room corners as shown in Figure 3. However, there may be advantages to
Figure 1
Figure 2
Figure 3.
Figure 4.
setting up the woofer system as shown in Figure 4 and 5. Here one woofer with
normal polarity is positioned in the corner near the wall behind the main speakers.
The second woofer is placed in the corner behind the listen, along the same side
wall and connected with inverted polarity, also shown in Figures 4 and 5. If in free
space, this would represent a dipole woofer system with separation equal to the
room length. In free space the on axis response of such a dipole would have the
dipole peak at the same frequency as the axial mode in the X direction (front to
back direction) and a comb filter appearance above that frequency. However, we
are not in free space. We are in a room and must consider the interaction of the
woofer with regards to room modes. Despite the wide separation, in this
configuration the Y, Z and YZ modes still will not be excited, at least in theory.
This is because while individually each woofer will excite the same Y, Z and YZ
modes, the inverted polarity of one woofer results in cancellation, just as it does
for a closely spaced dipole woofer. We also recognize that the woofer system will
not excite the DC room mode. Furthermore, since the dipole peak is at the first
axial mode in X direction, no dipole equalization is required since the usual, free
space dipole roll off occurs below the room fundamental. Thus, if we accept that
dipole woofers subjectively sound better than other woofer formats, and if it is true
that the reason is because fewer room modes excited by a dipole when aligned
with a room axis, then the novel woofer positioning suggested here should render
the same effect, without the need for the complexity and inefficiency of dipole
equalization at low frequency.

To get an idea of how this woofer configuration will perform consider a room with
X = 7M, Y = 5M and Z = 2.5M. Place the listen at X = 5.5M, Y = 2.5M and Z =
1.5M. When the woofers are positioned more conventionally, as shown in Figure
3, the low frequency response at the listening position might resemble that shown
in fuchsia in Figure 6. When positioned and connected as in Figure 4 the
response might be as shown in black. The response should also less dependent
on listing distance. Figure 7 shows a similar result as that of Figure 6 when the
listening distance is 4M. In fact, the response is relatively insensitive to listing
distance accept for level, and it must be recognized that, do to symmetry, the
response is theoretically null when sitting exactly half way between the front and
back walls.
Figure 5
Figure 6.
Figure 7
In summary, for suggested configuration it can be shown, under the rigid wall
assumption, that the following modes will
not contribute to the in room SPL at
any listing position:

Axial modes of the form M(Nx,0,0) where Nx is even.

Axial modes of the form M(0,0,Nz) or M(0,Ny,0) for any Ny or Nz.

Tangential modes of the form M(0,Ny,Nz).

Tangential modes of the form M(Nx,0,Nz) or M(Nx,Ny,0) where Nx is even.

Oblique modes of the form M(Nx,Ny,Nz) where Nx is even.

The DC mode, M(0,0,0).

Looking at a room which is 19' long, 12' wide and 8' high the modes which can
contribute to the in room SPL below 150 Hz are

Lengthwise, axial

29 Hz, 89 Hz, 148Hz

Tangential, L x W

55 Hz,  98Hz, 100 Hz, 113 Hz, 129 Hz, 144Hz

Tangential, L x H

76 Hz, 144 Hz

Oblique

89 Hz, 121 Hz, 122 Hz

If the subwoofer has a high frequency cut off of 80 Hz there are effectively only 5
modes (highlighted in red) which contribute to the in room SPL form the woofer
system. Tangential modes in the L x H  directions have little effect since listener
ear height is relatively constant for a seated listener. Additionally, some of the
remaining modes have zeros when the listener is centered, left to right, canceling
their contribution to the in room response. These include the 55 Hz L x W mode,
and the 89 Hz oblique mode. Thus the primary contributions to the in room
response for a listener centered, left to right, are the two lengthwise axial modes
which leads to a response shape that is relatively independent of listening
distance. However, sound level does change with distance. If the listener sits at a
position 83% of the room length form the front wall the second lengthwise axial
mode will also have a zero and the in room response is dominated by the 1st
axial mode. While such an arrangement may not be suitable for a large home
theater with wide spread separation of listeners, for music or a small home
theater with one or two listeners such an arrangement can provide excellent bass
response with out complex equalization. The boost by the 1st room axial mode
can be compensated for by choosing woofers which naturally roll off slightly
above this frequency.

I have experimented with this woofer configuration with the NaO Mini and the
results bear out the theoretical considerations. The only other considerations
which I would suggest are that the woofer system be limited to below 100 Hz and
that the low pass filter of the woofer be sufficiently steep that higher frequencies
do not give away the location of the woofer behind the listening position. If you
have multiple subs for your system this is an easy experiment to conduct for
yourself to see if you can realize the subjective benefits of a dipole woofer system
without the need to build a closely spaced dipole woofer.

[1]
Welti, T. Subwoofers: Optimum number and locations, Harman International
Industries, Inc.