H-frame and U-frame open-baffle woofers
The electrical circuit diagram below models the essential acoustic parameters of the H-frame. Reducing the length of the left transmission line to zero (T1 => 0) and increasing the length of the right transmission line to d2 = L, such that T2 = L/c, leads to the U-frame.
The front and rear voltages are shown below for the case that L = 0.6 m, (24"), the transmission line or waveguide impedance Zw = 1000, and the impedance seen out the open end is Rr = 400.
Addition of the front and rear outputs for distant observation points at different angles a yields magnitude and phase of the radiated signal. Added to the graphs below are magnitude and phase of the H-frame at zero degree for comparison purpose. At low frequencies the U-frame analyzed has 3.6 dB higher output on-axis than the H-frame. At 1800 off-axis the output is about 9 dB down. The phase shift of the U-frame radiation is -900 on-axis and +900 degrees towards the rear of the frame, relative to the source. Magnitude and phase are quite frequency dependent as the resonance frequency is approached.
The polar diagram (3) has been sketched for different frequencies using the magnitude graph above. It indicates somewhat unidirectional radiation, whereas the H-frame is bidirectional (1).
The introduction of losses in the U-frame opening in the form of an acoustic flow resistor RF tends to give the polar diagram a more uniform appearance (4). With RF + Rr = Zw = 1000 the waveguide is terminated and exhibits no resonance. The rear output voltage, VR, has almost the same magnitude as VF. Magnitude and phase of the radiated signal for this case are shown below. Graph (4) is drawn from these data.
I have seen before an oversimplified analysis of the U-frame and H-frame. While it works for the H-frame due to the complete symmetry of the structure and the resulting cancellation of effects from both waveguide openings, it leads to incorrect answers for the U-frame. The implied assumption in this analysis is that there is no change in impedances. It is as if the waveguide continued in front and rear all the way to each observation point and only the length of the front and the rear waveguide changes as a function of the angle a. In the electrical circuit model this corresponds to setting Rr = Zw = 1000. There is no resonance anymore, VF = VR, the two voltages only differing by 1800 in phase. The resulting "radiated" magnitude and phase for different angle a are shown below and the corresponding H-frame on-axis response for comparison.
The polar diagram in this flawed analysis is a cardioid (5) with 6 dB higher output than the H-frame on axis.
If on the other hand the radiation resistance is assumed to be even less than the value 400 of the previous examples and set to Rr = 100, then the polar diagram (6) of the U-frame tends more towards that of the dipole at low frequencies. The lower value for Rr would be indicated by a 14 dB resonance peak in the rear opening of the U-frame.
None of this leads to a cardioid polar response, though it would be nice if a simple U-frame could provide it. If the objective were to build a unidirectional woofer, then a different approach is necessary. One way is to use two closed box woofers back-to-back, driven 1800 out of phase (A). The electrical signal to the rear woofer is delayed by a time T corresponding to the time it takes the acoustic output from the front woofer to travel the distance l to the rear. A commercial example of such cardioid woofer can be seen at Meyer Sound.
Rather than using an electrical delay T it is possible to derive the necessary delay from lumped acoustic elements as sketched in (B). The compliance of the air volume inside the box corresponds to an acoustic capacitor. In combination with the flow resistor ZF it forms an RC lowpass filter. Below its cut-off frequency the lowpass has nearly constant delay. The difficulty in this approach lies in finding a practical flow resistor of the necessary value, linearity and flat frequency response. It is difficult to obtain a resistor that can handle the large volume flows required for a woofer and (A) becomes the best solution. For higher frequencies variations of the concept in (B) can lead to cardioids with very wide bandwidth. By proper dimensioning of various parameters the roll-off of the rear radiation, due to the lowpass filter, can be made to coincide with the decrease in rearward radiation from the front side of the box, due to front baffle size and driver diameter becoming acoustically large. When combined with a coaxial tweeter, that takes advantage of the horn like nature of the midrange cone, it becomes possible to extend the unidirectional frequency range even further. A commercial example is the Revolution loudspeaker by Gradient Ltd. .
To go deeper into unidirectional speaker design you may
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