Now it has been shown -either by Prof Hunt or Mr. Walker I don´t know at the moment- that a strip-like membrane shape with a height:width relationship of 8:1 works onto a real acoustic impedance not a complex any more over its full working frequency range.
I would be very interested to know the source of the idea that an 8:1 ratio is the optimum.
The real part of the radiation impedance does change more gradually with frequency for line sources than square source. But, I would think the optimum would be an infinitely long line source, not 8:1.
Calvin’s statement about a rectangular ESL with an 8:1 ratio having a real acoustic radiation impedance over its full working frequency range puzzled me greatly.
But, I think I stumbled on the source for Calvin’s statement. And, once the context of the statement is understood, it makes sense.
From "Telstar Shaped Electrostatic Speaker", by R. J. Matthys, Audio, vol. 48, May 1964:
“The diameter of a single electrostatic diaphragm (round or square) is about 1/24 of the wavelength of sound at the diaphragm’s resonant frequency. Since the air load on a diaphragm is reactive and not resistive when the diameter is less than one-third of the wavelength, a single electrostatic diaphragm will see a reactive air load in the frequency range between the fundamental diaphragm resonance and about eight times this resonant frequency.
Walker has shown that an electrostatic speaker reacts differently to a reactive load than does a moving coil loudspeaker…The best solution is to add speakers of the same size in parallel until the diameter of the speaker array is one-third the wavelength of sound at the fundamental resonant frequency of the diaphragm. This solution solves the reactive air load problem by making the air load resistive at all frequencies above the fundamental diaphragm resonance.
Another way of accomplishing the same thing is to make the diaphragm rectangular, with the length of each diaphragm equal to eight times the width.”
Attachment (1) shows that for ka>1 (ie diameter > 1/3 wavelength) radiation impedance is resistive and constant.
What Matthys said was that it would be possible to combine multiple smaller ESL panels into an array such that the resulting size would make the radiation impedance on each of the ESL panels resistive over their full working range even if this isn't the case when one panel is operated alone. If, as he states for his small ESL panels, the diameter is 1/24 wavelength at their fundamental resonance, an array of 64 panels arranged as in Attachment (2) would provide the desired resistive radiation over the full working range of the ESL panels.
Alternatively, Matthys states(highlighted in red above) that using rectangular shapes ESL strips with ratio of length to width of 8:1 could also be used to achieve the same result.
See Attachment (3).
Note that it is only the array of 8:1 ESL panels that achieves the resistive radiation over the full working range.
One 8:1 ESL panel operated by itself does not.
With just one 8:1 ESL panel, the radiation impedance turns reactive as soon as the wavelength gets longer than the width of the ESL panel, and you get a falling response with reducing frequency below this point. This is the dipole cancellation that we are all familiar with.
