Tech Design.....

The U-Frame Woofer System
The U-frame is an unconventional open backed woofer system. A correctly designed and damped U- frame woofer
potentially achieves 6dB greater on axis sensitivity than an H frame dipole woofer with the same bulk dimensions
or foot print. The U-frame woofer does not have a dipole radiation pattern, but rather a cardioid* like pattern.

The concept of cardioid speaker systems and U-frame type woofers is not new. Olson described the
characteristics of 1st order bidirectional (dipole) and unidirectional (cardioid) speakers in a presentation at the
43rd Convention of the AES in 1972 entitled
Gradient Loudspeakers. The analysis of U-frame or open backed
speakers was described by Holmes,
The Acoustic Resistance Box - A New Look at an Old Principle, J. AES, Vol.
34, No. 12, 1986. As the title implies, even in 1986 this was not a new idea. Holmes gives credit to a 1950 paper
by Kalusche,
Lautsprecheranordnung mit einseitiger Richtwirkung (Loudspeaker arrangement with one-sided
directive effect),
Z. angew. Physik,  vol. 2, 1950 as being the earliest description of this type of speaker. Below are
show figures form the Olsen and Holmes papers. Below, the figures present a pictorial illustrating the operation of
a U-frame speaker system with comparison to an H-frame dipole.

*A true cardiod pattern is obtained only under ideal conditions when both front an back radiation are
identical in amplitude. Other factors, such as the 1/4 wave resonance in the U frame, possible
differences in the radiation impedance seen by the front and rear of the driver and impedance
mismatches at the terminus of the U frame can  lead to deviation for a true cardioid polar response.
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The figure to the left show the
configurations for a dipole, an
H-frame dipole and a U-frame.
Below the differences in
propogation delay to a point
in front of the system due to
geometric considerations is
shown for the different
configurations. In all three
cases the differential delay
due to geometric
considerations is identical.
Based on the geometrical
delay alone the U-frame
would have the same on axis
response as the dipole and
H-frame dipole.
Viewed from behind (pictured below) the system delay due to geometrical considerations is the same as from the front for the dipole
andH-frame dipole, but the geometrical delay goes to zero for the U-frame. If only the geometrical delays were present the U-frame
would produce a
cardioid radiation pattern with non sound radiated directly to the rear of the woofer.
In this next figure below we look at  the driver's raw response and the effect of the transmission line in front and/or behind the driver for the H and
U frame. As indicated, when undamped the transmission line has a transfer function that lools like a modeate Q resonance at the 1/4 wave
frequency of the duct. The driver's response, signified by T(D), is multiplied by the "frame's" transfer function, T(F), to obtain the response at the
exit plane of the transmission line, fornt and or rear. It is also noted that T(F) introduces a group delay (GD) which quickly becomes negative
below the resonant frequency. This negative GD tends to cancel the delay associated with the internal propagation at low frequency. Since the
H-frame is symmetrical, this is of no consequence and the net delay to a listener in front or to the rear of the system remains D/C. However, for
the U-frame this negative GD is only introduced to the rear wave and since the internal delay is canceled  the net propogation delay becomes
D/(2C) to the front and rear. Thus the undamped U-frame starts to look like a dipole of length D/2. However, this is not quite the case since the
front and rear radiation are not symmetrical. The undamped U-frame reduces to a dipole of length D/2 only as the frequency approaches zero.
When damping is added to the H- and U-frames two factors come into play. First the damping decreases the amplitude and Q of the
resonances. If properly controlled these two effects drive the negative GD towards zero. If it were possible to completely damp the resonance
without introducing any other effects the GD associated with the resonance of the frame would be eliminated and the U-frame would behave as
a true cardioid. However, this is generally not possible using acoustic damping (stuffing). Thus there is a second effect which is the attenuation
of higher frequencies to a greater extent than the lower ones. This gives rise to a low pass filter effect as shown by the transfer function of the
damped transmission line, T(Fd). The low pass filer effect introduces an addition positive GD at low frequency further canceling the negative
GD of the undamped duct. Again, since the H-frame is symmetrical, this is of little consequence. But by proper control of the damping it is
possible to drive the negative GD originating with the duct resonance to zero at low frequencies restoring the roll of the internal delay in the
case of a U-frame. Optimal damping is obtained when the GD is as close to zero over as wide a region of the low frequency response area as
possible, with additional consideration given to the symmetry of the front and rear response of the U-frame.
The figure below shows the on axis and polar response of undamped H (RED) and U (BLUE) frame woofers (upper) and optimally damped H
and U-frames (lower). The H- frame is of length
D and the U-frame is of length D/2. In the undamped case, shown in the upper portion of the
figure, the on axis response of the U-frame looks very much as would be expected for a dipole of length D/2; 6dB below the H-frame dipole of
length D. Above resonance, however, the undamped U-frame response is identical to the H-frame. This is because above resonance the GD of
the undamped frame goes to zero and delays are those determined by geometric considerations (see the GD plots for the undamped case
above). The polar plots reveal significant differences, however. At 20 Hz the undamped U-frames does exhibit a dipole like response but as the
frequency rises the response is neither dipole or cardioid like. This is due to the asymmetry between the front and rear radiation as the
resonant frequency is approached.

Looking at the lower figures, for the
damped case, the on axis response of the D/2 length U-frame and the D length H- frame are almost
identical below the resonance. Above resonance, however, the U-frame shows an oscillatory response about a flat level while the H-frame rolls
off. This is a result of the low pass filter effect of the damped transmission line and the lack of such low pass filtering of the front radiation of the
U-frame. The polar response plots now indicate that the U-frame is behaving as a cardioid at low frequencies. As the frequency rises the
cardioid pattern is lost but there is still substantial reduction in the rear radiation.
The last figure below shows a real world comparison between a 36" damped H-frame and 18" damped U-frame along with an 18" damped H-
frame for comparison. Since it is difficult to make far field measurement of such systems the means of producing the response curves was to
measure the SPL data using near field techniques just off the driver's cone or at the exit planes of the U or H frame, as appropriate. Since the
SPL was measured at the exit plane of the U or H frames, all information regarding the roll off and the internal delays from geometric and
acoustic effects (damping) are embedded in the response data. This data was imported to a simulation code without modification and assigned
to sources positioned geometrically at a distance of 18" apart for the U-frame data and 36" or 18" for the respective H frames. It is clear that the
application of optimal damping has resulted in the U-frame having an approximate cardioid response at low frequency and the 18" U-frame has
similar response to the 36" H-frame on axis. The 18" H-frame has significantly lower on axis response. It should also be noted that the woofer
systems were driven through an active equalization/low pass crossover. Finally, it should be stress that the resulting polar plots are very
sensitive to the damping and it is also possible to obtain "hyper-cardioid" response patterns where a secondary lobe emerges out the back of
the U-frame. Even so, such secondary lobes are typically much lower in amplitude than the main front lobe. The on axis response is much less
sensitive to optimal damping. Also see the discussion of the
NaO II woofer system.
A Simplified look at U- and H- frame woofers........
All to often the analysis of different ideas leads to overly complex mathematical explanations which serve more to
confuse and/or mystify the simple physical reasoning behind the idea rather than clearly demonstrate the principles
upon which it is based. These mathematical models often make used of simplified linear models which neglect important
frequency dependent effects such as the frequency dependence of the acoustic resistance of damping materials used
in real systems. Neglect of such nonlinearities can lead to overly simplified and incorrect results.  Here I make an effort
to remove the mysticism of the design of U-frame woofers and comparison to H-frame dipoles through a step by step
building block approach to the various factors which effect the design. The ultimate objective of the U-frame woofer is to
achieve similar max on axis SPL capability as a dipole is a smaller, less expensive package.