The volume of air, (blue area), from the throat (narrowest point) of the lens back through the transducers' membrane and into the cavity of the speaker, form the Helmholtz resonator. The volume is thus calculated to resonate at c.40Hz. This is precisely where the response of the transducer itself starts to dwindle and where it needs most help. With the aid of the Helmholtz, the entire system's response reaches down to 40Hz and even somewhat lower. Note the Helmholtz resonator characteristic: it resonates at its designed frequency and cuts off pretty sharply below that. So does the Beveridge system.
The main function of the lens is to transform the planar waves generated by the transducer into a cylindrical wavefront dispersing the sound - and all the frequencies contained therein - uniformly, across a 180o arc. Those who read the Patents will be familiar with the way they accomplish that. It should be appreciated that a lot of thinking and development work went into the design of those lenses since many mechanical and acoustical problems had to be ironed out. For instance, despite the lenses being braced at short intervals by horizontal holders, their walls also resonate. The problem is mostly solved by using a relatively inert acoustic material. Further, the inner guides experience an equal amount of sound pressure on either side, a fact which also tends to cancel out any vibrations. The outer walls though, are only exposed to sound pressure on one side only (the inner one) through half of the membrane cycle; the other side is exposed to almost the same pressure during the second half of the cycle, which is out of phase. To that end, the outer guides' walls had to be highly reinforced with resin to render them solid.
Since the transducer itself tends to emphasize somewhat the higher frequencies, Beveridge sought to tone down this effect. He solved the problem in an ingenious way, by ignoring the final inch or so of the transducer's extremities and devised the lens to cover only part of the transducer's area. He thus traded a small amount of SPL loss for accurate, more balanced frequency response. The outer inch of the radiating area is really firing into a proximity board, the higher frequencies being directional and absorbed within the enclosure. The low frequencies, which disperse omnidirectionally, find their way out through the wave guides, thus reinforcing the low frequency component of the total sound. A nice photo of Beveridge in front of an unfinished speaker illustrates the arrangement.

The top, as yet unmounted transducer, reveals the back
of the lens structure. Note how much narrower it is from
the actual lower, mounted transducers. The clearance
of the lens from the transducer is less than half an inch,
enough to allow low frequencies to escape through the
wave guides.
The other major piece of acoustic hurdle was to design the lens so that despite its increasingly curving guides toward the outer extremities, with their associated different path lengths (see photo above), the sound should reach the lens mouth with no time delays, or phase shifts. This is where the clever piece of acoustic design comes in: the goal is achieved by employing the throat (narrowest point in the lens' guides) together with calculating an equal arc-like path for the sound waves, an arc whose base is an invisible plane stretching between the outer walls of the most extreme channels and whose middle point forms the radius of that arc. Once this arc is achieved, the waves will propagate uniformly, in phase, into the room forming the legendary 180o wavefront.
Contrary to popular conceptualization of the sound waves emanating from a loudspeaker, the acoustic propagation is not a flow of air but rather a rapid to-and-fro movement of air particles, in sympathy with the diaphragm's positive and negative cycles. It's more of a piston-like movement (of air particles) and this is the reason why the volume of air in the Beveridge system that comprises the Helmholtz resonator begins, and includes, that of the "trapped" air between the throat and the transducer (diaphragm). Note that when the diaphragm's motion is inward, "into the cabinet" phase, so are the particles of air in the wave guides, and for this purpose the membrane is a "transparent" device; the air particles in the lens move in the same direction as those behind the membrane, i.e. toward the inner enclosure.
The pressure in the lens' throat is considerably higher than that in any other part of the lens' guides (a phenomenon that in itself can, and does, give rise to wave guide walls' flexing, had they not been thoroughly braced). On the "room side" of the throat the air particles have almost infinite room to expand, hence no pressure can accumulate. At the base of the "inward side" of the throat, adjacent to the transducer, the volume is larger than that in the throat, hence the pressure will be lower. Given that the throat narrowing is about 1/7th that of the area covered by the lens immediately adjacent to the membrane, it would appear that the momentary pressure of the air in the throat will be 7 times higher. This is not entirely the case for the law of diminishing returns is at play here. (Taking that line of thought to the extreme would mean that if we narrowed the throat to almost total closure, the pressure would raise to enormous values, but that is plainly not the case.) There are those who suggested the pressure ratio to be 10:1 in the throat as there are those who suggested that by narrowing the waves guides the system becomes a Venturi device and thereafter (at the throat's exit) it becomes a horn device. Neither is the case. The acceleration of the air particles achieved in the throat is matched by a deceleration immediately following it, due to the lens' spatial expansion. The horn effect is neutralized since the guides' walls theoretically contribute nothing to the augmentation of the sound, i.e. they don't (or shouldn't) resonate. (It is recalled that any resonant column of air must have a material around it to be set in sympathetic resonance). Also note that the dispersion angle into which the waves propagate into the room is identical - with, or without, the wave guides in place - and is determined by the compass of the outer guides' walls; the lens merely aid the uniform dispersion of the entire audio range into that angle.
The last but by no means least piece of engineering "wizardry" in Harold Beveridge's arsenal is the construction of the transducers themselves. Those who had the chance to inspect them would have noticed that the "ribs" (the horizontal prongs or tines) that flank the membrane are quite thick. Moreover, a sectional cut through one of them will reveal it to be of a trapezoid shape, the wider base facing the mylar while the narrower one faces outward. This is not incidental. That short, but widening path of the sound through the tines was chosen for several reasons. The one drawback is the fact that the minute air column "trapped" between the tines has a resonance of its own and that resonance had to be kept outside the audio spectrum (above 20KHz). The actual open air area faced by the membrane is the distance between the middle points of the trapezoidal tines. However, the electrical force exercised over the mylar is that of the wider, closer base of the trapezoid - and even wider than that since, to a point, the electrical field will follow the widening path dictated by the trapezoid shape. Thus, a wider area of the mylar will be activated than the actual air gap (which, we recall, is the middle point of the trapezoid) - as is the case in virtually all other electrostatic transducers. Neat and clever.