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CORONA: A Big Metamaterial Enclosure

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  • CORONA: A Big Metamaterial Enclosure

    “BIG META”

    A Big “Metamaterial”
    Broadband Acoustic Resonator Array Enclosure
    For a Midrange Driver

    Designed and Built by
    Meredith A. Cargill

    October 14, 2020


    To implement in midrange the same kind of device as
    KEF has introduced for the tweeter in its LS50 Meta

    To finish before the end of the year a novel design that
    commemorates 2020 as the year of the Novel Coronavirus


    Use many discrete plastic pipes to construct a
    broadband acoustic resonator behind the driver,
    with the pipes radiating magnificently out from the center.

    The build is 70.5” high and 88.5” wide. The stand lifts the driver to 42” from the floor, making the whole structure 78.5” tall.

    Before I give the details of how I built it, and the acoustic measurements, let me give the theory.

    Broadband Acoustic Resonator Array

    Researchers in acoustic engineering have been exploring, over the past fifteen or twenty years, devices they have come to call “metamaterials” and “metasurfaces.” These are structures that can be used to control the transmission or reflection of sound in place of traditional solutions that involve thick layers of material or large, heavy structures, such as fiberglass bats and quadratic diffusers. The engineering goal is to reduce the size of the devices and constructions needed for passive reduction of unwanted sound in rooms and other spaces.

    They typically, if not universally, implement these solutions using multiple resonating tubes or volumes. Their obsession is proving that their structures are thinner or smaller than the traditional solutions while being as effective at reducing acoustic energy. For instance, in a 2017 paper in Physical Review X, “Ultrathin Acoustic Metasurface-Based Schroeder Diffuser,” Yifan Zhu, et al., describe connecting multiple small Helmholz radiators to form a sheet or panel that scatters sound in the same way as a Schroeder quadratic diffuser, but is only one-tenth as thick. In a 2017 paper in Annual Review of Materials Research, “Sound Absorption Structures: From Porous Media to Acoustic Metamaterials,” Min Yang and Ping Sheng describe the “realization of broadband absorbers with discrete arrays of resonators” (107). Instead of Zhu’s pockets with openings that produce Helmholtz resonances, Yang and Sheng use a package of tubes that resonate according to their length. The out-of-phase reactive behavior of the tubes cancels the sound impinging on the device, so that the sound is “absorbed” by it, with the same effect as a much larger volume of porous material that absorbs sound by impeding it. Even though each tube individually produces a very narrow-band effect, they get a broadband effect (at least two octaves) by using an array of 16 tubes of different length (110).

    The engineers at KEF have used the principles described by Yang and Sheng to build a device to absorb the energy coming from the back side of the tweeter diaphragm in their concentric Uni-Q driver, which has the dome tweeter mounted in the center of a mid-woofer. Sebastien Degraeve and Jack Oclee-Brown describe the device in their June, 2020 paper for the Audio Engineering Society, “Metamaterial Absorber for Loudspeaker Enclosures.” It is a disk 11mm thick, the diameter of the magnet structure of the driver (about 5"). It thus resembles in physical shape a sheet of sound-absorbing mat or felt that they might have used to damp that energy by mounting it to the inside of a conventional sealed box enclosure. But it is instead an injection-molded array of tiny tubes cleverly folded to fit into the neat disk shape. The disk mounts to the back of the magnet structure of the driver. The rearward output from the tweeter is directed by a channel through the midrange motor to the center of the disk, where it meets the mouths of all the tubes. The device absorbs the energy by cancellations due to reactive resonances, and is more effective at doing so than any known resistive absorbing material of comparable thickness would be. Thus, it fits the description of what the researchers call a “metamaterial.”

    Because the word “metamaterial” is ultra-technical and mystifying and unfamiliar to any audience outside acoustics research, I don’t like it. You can find my rant in the TechTalk thread about KEF’s device, here: Once it crosses over from acoustics research to KEF’s marketing, it sounds less like science than like hype. We could look to a quotation like the following for help understanding the idea: “Merely by relating himself to what is, man places and faces beings in their Being. Seen in the light of what is, the facing, the idea of beings always goes beyond beings. … But the presence of what is present is not finally and also something we face, rather it comes before. Prior to all else it stands before us, only we do not see it because we stand within it. It is what really comes before us. The facing, the idea of what is, judged from what is, is always beyond what is--µetá.” That is Martin Heidegger trying to turn phenomenology into metaphysics (What Is Called Thinking? Harper, 1968, 97-98). Does it help?

    A more appropriate, less confusing, more informative term for KEF’s device is a broadband acoustic resonator array. To quote Degraeve and Oclee-Brown, it is “a structure containing many high-Q resonators optimized to provide a wide bandwidth of overall absorption” (2). The authors go overboard in the direction of mystification when they say “acoustic metamaterials can deliver unconventional effective properties without the constraints normally imposed by nature” (2). Without the constraints imposed by nature? They make it sound as if a metamaterial is miraculous or supernatural. (Let the marketing metaphysics begin!) The principle it works on is entirely natural. The principle also is not anything new, because resonators are familiar in speakerbuilding. We use Helmholtz resonators when we build boxes with ports in them, and we use quarter-wave resonators when we build “transmission line” cabinets. Neither is there any new material involved. Not only is injection-molded plastic not new, the same device could also be made out of cast metal or carved wood. It is misleading to suggest, by calling it some kind of “material,” that the idea is about the material. What is new about it is implementing an array of a large number of resonators to achieve a broadband effect.

    In a thread at, Sin Phi claims to have produced an equivalent to KEF’s disk, suitable for a tweeter, using 3D printing.

    What I have done in this design, Corona, is simply scale up KEF’s device so that it functions at longer wavelengths, appropriate for a midrange driver. The same concept could be used for a woofer, if one is willing to accommodate the necessary pipe lengths.

    One thing I hope to demonstrate is that the concept is not about a material that can be used to line or stuff a conventional box, or even build a box out of; it is instead about an enclosure design that damps internal energy by its structure. It is, thus, not an alternative to fiberglass or Dacron fiber or denim felt or long-fiber wool, which we use to absorb and impede sound inside an enclosure; it is instead a whole different kind of enclosure, which makes it an alternative to a sealed box, ported box, open baffle, band pass, etc. As such, it strikes me as something novel.


    I have kept KEF’s idea of laying the tubes out in a plane, radiating outward, with their mouths forming a circle in the center. Rather than try to fold the tubes to be compact, I have chosen to let them fly outward, something like the points of a crown. Since the Latin word for a crown, ‘corona,’ has been used to describe other things with this shape, I have used it as the name for this design. One can see, for instance, the Sun’s corona during a total eclipse.

    And one can see coronas painted behind the heads of the divine beings in Christian icons and religious paintings.

    More familiar to everyone this year is the corona virus, especially the new one that appeared in 2019. A corona virus gets its name from how it looks under a microscope—like a circle with spikes sticking out all around it. In 2-D silhouette, it looks like a crown.

    Once I had the idea of a corona-shaped loudspeaker, I could not resist building it while in the year of the Novel Coronavirus (Covid-19) pandemic.

    Since the pandemic has prevented us from holding shows for our speaker designs, such as the contest at Parts Express Midwest Audiofest, I presume this will be an online-only presentation. I am thankful that this means I can get by with building only one, as proof of concept, rather than having to build a stereo pair, and be judged by its stereo imaging. I also did not have to worry about whether it would fit in my van, or be damaged in transport, or how difficult it would be to set up in the show space.

    Build details, and acoustic measurements, will come in additional posts.
    Last edited by mcargill; 10-17-2020, 02:03 AM.

  • #2
    CORONA, by Meredith Cargill

    I chose the Dayton Audio RS100-4. It is a 4-inch metal-cone driver known for being a low-distortion midrange. Using DATS, I measured its resonance frequency at 95.55Hz and Qts at 0.59. Its high-frequency response is adequate to use it without a tweeter, as long as one is willing to listen on axis and tolerate a sharp response peak above 13KHz, or listen 10-20 degrees off axis and tolerate the absence of the top octave. I take advantage of that feature by dispensing with a tweeter and crossover in this design, so that I can concentrate on the midrange as my innovation.


    I have adopted several design parameters straight from KEF. For instance, KEF uses thirty resonators in their array. KEF was not satisfied, for their purposes, with the ragged results produced by Yang and Sheng when they used only 16, so KEF experimented with using more resonators. They achieved a consistent 99% absorption rate throughout the design band, with 30 tubes and the help of a little conventional absorptive material, meaning there is no need for more than 30. I see no reason to second-guess their decision.
    The Materials

    I chose ½” PVC pipe as a readily-available tube appropriate to the dimensions of my driver. My array therefore consists of thirty pieces of ½” PVC, each a different length and closed on the outside end with a PVC cap. Installed in two layers, there are fifteen pipes per layer.

    If I crowd a circle of fifteen of these pipes around the rear of the driver, radiating outward, the diameter of the ring is 4”. I prefer to have the ring of mouths encroach on the driver basket as closely as possible, to reduce the effects of any other spaces or volumes of air in the enclosure besides the device. I want the device to be as much as possible the entire enclosure, rather than something added to a “sealed box.” I therefore chose to grind down the width of the pipes, to bring the mouths closer to the driver diaphragm. The smallest ring permitted by the basket of the driver would be 3” in diameter, but grinding the ends of the tubes to fit that closely would change the shape of the mouth, possibly affecting the resonating behavior of the pipe. Grinding the ends down only to the inner diameter of the pipe resulted in a ring 3.25” diameter.

    All this trouble minimizing the distance of the mouths from the diaphragm probably makes no difference, since the distances are so small, but it emphasizes my theory behind this design—that the “metamaterial” is not something added to a conventional design, but is itself the entire enclosure design. The array of pipes is the enclosure; it is not added to, or installed in, or attached to, or fitted into, or stuck on what would otherwise be called a sealed box.

    Two layers of pipe are less than the mounting depth of the driver, so the back of the device is enclosed with a piece of plywood with a cup cut out to fit over the driver’s magnet. The result is an enclosure that completely seals around the rear of the basket, but has almost no open space; it is entirely a set of mouths for the thirty tubes.


    • #3
      To fit the pipes to the rear of the driver flange, the rear side of the driver’s flange must be flush with the rear side of the baffle. This called for a multi-layered baffle—a thin layer with the cutout for the driver’s flange, plus a front layer to which the driver is effectively rear mounted.


      • #4
        CORONA, by Meredith Cargill
        The Calculation

        The pattern of pipe lengths KEF developed for their design can be inferred from Figure 7 in the paper by Degraeve and Oclee-Brown.
        • For the lowest 1.5 octaves, there are resonators every 1/12 octave. So the first (longest)18 pipes cover that band with their first (fundamental) harmonic. While 1/12-octave spacing comes close to providing uniform response at any random frequency, KEF added some conventional sound absorbing material in front of the tube mouths to make the response more uniform. I left this out, to measure instead the effect of the pure array on its own.
        • For the next octave, resonator fundamentals are every 1/6 octave (pipes #19-#24). In this frequency band, the longer pipes (#1-#18) are producing their second harmonics.
        • KEF’s graph in Figure 7 does not show how far apart the upper six resonator fundamentals are (pipes #25-#30), but they appear to be farther apart than the previous set, so maybe 1/4 or 1/3 octave. This band is 1.5-2 octaves. Fewer fundamentals are needed, because at these higher frequencies pipes #1-#18 are resonating at their third and fourth harmonics, while pipes #19-#24 are at their second. I chose a pattern that makes this band 1.8 octaves and the range of fundamentals in the whole array 4.3 octaves.
        • At even higher frequencies, the entire array of thirty pipes is producing higher harmonics so densely spaced that the combined absorption is practically uniform at all frequencies. The KEF graph or text does not indicate any upper limit for the effectiveness of yet higher orders of resonance. It might extend until the wavelengths get so small that other features of the structure, such as edges and corners, become reflective.
        KEF claims theirs works from 620Hz to at least 5000Hz, so it is reasonable to expect similar devices to have a bandwidth of a decade or more. Given that the resonance band of a single pipe is on the order of 1/24 octave or narrower, it is remarkable to find a device made of such pipes having a uniform effect over such a broad band. The broadband effectiveness depends entirely on the large number of pipes in the array and their progressively different lengths.

        I created an Excel spreadsheet to calculate how long the pipes must be to resonate at these frequencies. Plugging in the lowest frequency for the device’s effect generates the 30 different lengths of pipe to cut. I used the formula for determining the sequence of frequencies in the well-tempered chromatic scale, which is also 12 divisions per octave. The length of each successively shorter pipe in the 1/12-octave band (pipes 1-18) is calculated by dividing the length of the previous pipe by the twelfth root of 2. In other words, starting with the length (L0) of the longest pipe, and counting (n) how many twelfths of an octave a given pipe frequency is above that one, the length (Ln) of that pipe will be the length of the longest pipe divided by 2 to the n/12 power.

        Ln = L0/2n/12

        The lengths of the higher bands, where there are fewer than twelve pipes per octave, can then be calculated by incrementing n by 2 or more for each successive pipe.

        The design length of a pipe is ¼ of the wavelength at which it is expected to resonate. Other factors can affect the accuracy of this calculation, such as the width of the pipe, any taper in the pipe, and any constrictions or nearby boundaries at the pipe’s mouth. Also, a plug in the end of a pipe would call for cutting the pipe longer, to start with, to end up with the right interior length with the plug installed. The physical length of a pipe will therefore probably be different from its design length for a given target frequency. Or, the resultant frequency will be different from the target frequency for a given length. But this might not matter, because in this design the exact length of any given pipe is irrelevant. What is relevant is that the whole set of pipes is of incrementally different sizes, with the longest pipe determining the lowest effective frequency. Because that lowest frequency is expected to be well out of the pass band of the driver, it need not be determined precisely.

        In practice, it is not necessary to be so precise about exact pipe lengths that you need to calculate each one, as long as they progress in three subsets, more or less by the general outline given above. One could cut them by first identifying which ones are one octave apart, and measuring them to be progressively shorter by half. Pipe #13 should be ½ the length of pipe #1. Pipe #24 should be ½ of pipe #18, which should be ½ of pipe #6. The pipes between these can probably be cut by guessing, as long as the result looks kinda regular.
        The Size

        The longest tube in KEF’s array resonates at 620Hz. They do not say how they chose this frequency, leaving me to figure out how I will choose the lowest effective frequency for my design.

        The published specs for the LS50 say the crossover point is at 2100Hz. Thus, the resonator array begins to function (its low-frequency “cutoff” point) at a frequency that is a ratio of 1/3.387 compared to the crossover point. If I were to choose a similar design principle, I would need to first determine the crossover point for my RS100. If I chose, for instance, the common estimate of one octave above its resonant frequency, then the crossover point would be 191Hz and the longest pipe would be cut to resonate at 191Hz/3.387 = 56.4Hz, which would be a design length of 60”. A 5’ pipe is pushing my limit for how big a thing I am willing to build. I could accommodate it if I were willing to bend, coil, or fold it, but since I am committed to a straight-outward radial array (the corona), I have to keep the pipe straight and still end up with a result that will fit in a room and through a doorway.

        In their white paper, “Q Series Technologies Explained,” KEF publishes the frequency-response curve for their unfiltered tweeter. It is flat above 4000Hz. At the LS50 crossover point (2100Hz), response is about 2dB down. (KEF explains that they prefer to achieve a 12dB/octave acoustic slope using a first-order electrical filter combined with the natural low-frequency rolloff of the driver.) By this response curve, their resonator array starts to function when the natural output from the tweeter is down about 25dB. Looking at the published response curve for the RS100, a design based on the same points in its frequency response would put the crossover point at around 120Hz and the lowest resonator around 29Hz. To resonate at 29Hz, a pipe would be 9.7’, definitely too long for me (this time).

        KEF’s description of the tweeter does not include an impedance curve or Fs. Dealing with a widerange driver, however, resonance becomes more of an issue. Just as we use ported boxes and quarter-wave transmission lines to control driver resonances, I would hope that one benefit of this design would be a flattened impedance curve around the driver’s resonance (95.55Hz). I therefore prefer to have one of the pipes in the array targeted at that frequency precisely. I prefer also to have the array extend low enough to cover the whole band where driver resonance matters, which, according to the published curve, is down to 20Hz-40Hz. But once again, the pipe will be too long for my practical tolerance (40Hz is 7’).

        I ended up choosing a maximum tolerable length for the longest pipe—approximately 4’. Then I used the spreadsheet to find, by trial and error, a precise length that generates one pipe with a target resonance at 95.55Hz (Fs of the RS100). It turns out to be 49.95” (67.6Hz) for pipe #1, with pipe #7 targeted at 95.55Hz. This means the array will start to function ½-octave below Fs. Using KEF’s ratio of 1:3.387, this makes the device appropriate for a crossover point at 229Hz with a 12dB/octave slope.

        Having chosen the length of the longest pipe, and the spacing of fundamental resonances in the upper range, pipe #30 turned out to be 2.5”.

        Being a sealed system, the performance of the driver will be affected by the amount of “acoustic suspension” provided by the air confined in it. The pneumatic pressure of the interior volume is the sum of the volumes of all the tubes, plus a few cubic inches of space around the diaphragm and basket. I used the spreadsheet to calculate this total volume. It comes to 0.147 cu ft, which is large enough that I expected the driver to operate pneumatically in an infinite baffle, with resonance and Q the same as in free air. I was mistaken about this, as I will describe when I give the measurements, below.

        I cannot upload the actual spreadsheet, but here is a PDF of what it looks like:
        BARA Dayton RS100 v5.pdf


        • #5
          CORONA, by Meredith Cargill

          This post starts the details of construction.


          The rear layer of the baffle is the same thickness as the driver’s flange. I used 1/8” Masonite hardboard, to fit this driver. I cut a hole to fit the driver’s outside diameter.

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          The first layer of pipes, the shorter set (#16-#30), is glued to the back side of this material, with their mouths extended past the edge of the hole, overlapping the rear side of the driver’s flange. Fifteen pipes radiating in a full circle results in the pipes being 24 degrees apart. I used a protractor to draw the radiating lines, but I could also have used a piece of wood, cardboard, heavy paper, or other material cut to a 24° angle. A right triangle with a ratio of 7-unit rise to 16-unit run has a 24° angle. Such a triangle can be made by cutting an 8” length of 1X4 (3.5” wide) on the diagonal.

          I cut the outer dimension of this piece of hardboard so that the cap of each pipe will be beyond the edge.

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          This allows the pipe to lie down flat against the board. I ended up with a very irregular shape for this piece, because I used a scrap piece of hardboard to start with. For a less scrappy build, I would need to start with a piece of hardboard as large as the front layer of the baffle.

          The center part of the front layer of the baffle is a disk of ¾” MDF . It has a hole sized to fit tightly around the driver’s surround (3”), so the disk covers the front of the driver’s flange. It has a horn-like contour around the driver to provide a smooth transition between the driver and the air. Although it is shaped like a waveguide, its intended function is only to make a relatively smooth, non-diffracting transition for a rear-mounted driver, so I’ll call it the transition disk. A more thorough design effort would pay attention to this part, to make it properly functioning waveguide or horn. Adding a rim wide enough for a ring of screws to attach the disk to the rear layer of the baffle brought the outside diameter of the transition disk to 6.5”. I cut the horn/waveguide shape by starting with a big 45° chamfering router bit, and finishing by hand with a wood rasp.

          The outer part of the front layer of the baffle is a disk of ¾” plywood. The outer dimension is enough to allow the second layer of pipes to bend down to be attached to it with strap clamps, without resorting to heat to bend the pipe. I judged by experimenting that this required a disk at least 29” in diameter. I cut a hole in the center to fit exactly around the transition disk. When I decided to cut the outer edge of the disk with a sabre saw instead of a router with a circle jig, I faced the question of just how perfect I wanted the circle to be. I decided to go with a scalloped outer edge, for aesthetic reasons, and for the practical reason that it relieved me from having to try to make the circle perfect. The resultant disk is 29-31 inches diameter.

          Putting the driver temporarily in the hole of the rear layer of the baffle, I positioned the transition disk in front of the driver and attached it with screws. With that central disk in place, I put the outer part of the baffle around it and secured it to the rear layer with wood glue and tacks driven from the rear into the plywood, being careful not to get glue on the driver or the transition disk. I removed the central transition disk immediately, before any errant glue could make removing it trouble.

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          I fashioned a stand by screwing a scrap of 2X4 to the back of the baffle, and two pieces of scrap 1X4 to the other end of it. I am not investing in a fancy stand for this build—just something strong enough to hold it for testing and photos. If anyone were to build one of these for real, it would make more sense to mount it to the wall than to put it on a stand. Mine turned out to be only 6.75" deep, and I could have made it only 3.625" deep if I had (like the acoustic engineers I mentioned above) made utmost thinness a goal and reduced, for instance, the length of the bolts holding it together.

          I made room for the 2X4 among the array of pipes by arranging the three shortest ones (#28, #29, and #30) to point downward. The other pipes would splay to either side of it.

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          I made a dummy plug to insert in the driver’s cutout, to help align the pipe mouths in a neat circle.

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          It is a disk (two layers of ¾” MDF) the same diameter as the circle of pipe mouths (3.25”). But it also has a flange to match the thickness of the rear layer of the baffle. I used a router and circle jig to cut a disk-with-a-flange. It would have been easier to cut a disk of hardboard and glue it to the two disks of MDF while I was gluing them together. I treated the plug with Danish oil, in hopes of preventing any glue or caulk sticking to it. Not feeling confident that this would be enough protection, I also coated it with petroleum jelly.

          I inserted the dummy plug into the driver’s cutout and held it there by screwing the transition disk in place. I then glued the first layer of pipes to the back side of the baffle, using latex caulk underneath the length of each pipe and also between adjacent pipe mouths.

          Any pipe long enough to extend beyond the front layer of the baffle I attached to the baffle with a pipe clamp. I ought to have used a clamp on every pipe, just to hold them all in place tighter than the caulk can be expected to, and to hold them still while the caulk dried. While the caulk was still wet, I removed the plug. I put a weight on the ends of the pipes to clamp them down and let the caulk in this layer dry a day before attaching the second layer of pipes.

          I ran the hookup wire between a couple of pipes, amidst the mass of caulk between two pipes.

          I re-inserted the dummy plug to position the second layer of pipe mouths.

          Before installing the second layer, I drilled the holes between pipes in the first layer, for the four machine screws that hold the rear disk on.

          Having those screws sticking up, in their place, allowed me to position the pipes in the second layer around them, even if it meant deviating from the regular 24° spacing. Since the pipes in the second layer all (except #29) extend beyond the edge of the baffle, I secured each one with a metal clamp near the edge of the baffle.

          I covered the top/back of the top-layer circle of mouths with a layer of caulk and completed the enclosure by clamping it together with the rear disk, using the long machine screws through the baffle, between the pipes.

          I then could remove the dummy plug, and allow the caulk that oozed out between the pipe mouths to dry, before peeling it away to clear the ends of the pipes.


          • #6
            CORONA, by Meredith Cargill

            I painted the front to suggest a sunburst. I tried to make it look like shining gold, using orange, yellow, and white paint, plus some spray-paint gold glitter.

            The final step of assembly was to insert the driver (from the front), with a little caulk under the flange to seal the rear enclosure, then insert the transition disk over it. I was satisfied that the transition disk would stay put by friction, without screws, so I didn’t worry about whether any of the pipes were covering the screws holes I had used earlier.

            Here is a photo with two shop assistants, whom some of you may remember helping me with my Easter Eggs at MWAF 2019.


            • #7
              CORONA, by Meredith Cargill

              The Baffle, First Issue

              Before addressing the effect of the resonator array, I need to address the transition disk and front baffle. I would have preferred for the driver to be flush mounted. But it had to be flush in back, too, to meet the pipes. The only way to do this would have been to have a baffle only 1/8” thick, which would not have provided enough structure, and would have resonated loudly. I therefore resorted to a ¾” front baffle as a necessary evil, and the transition horn/waveguide as an effort to make a bad situation tolerable.

              To see the effect of this arrangement, let’s first compare the driver in this baffle with the same driver flush mounted in an infinite baffle. For this test, I mounted the RS100 in my infinite baffle with the back of the driver left open.

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              My infinite baffle is ¾” plywood fit snugly into a doorway. (Some of you saw this same panel used at MWAF in 2016 when I installed an infinite-baffle woofer in a doorway.) So the rear chamber is a whole room, 10.5’X13’X11’ (1501 cuft) (42,534 liters). The front of the baffle is even with the wall of the room where the test microphone is, though there is a little opportunity for diffraction around the door frame on the left, and reflection from the stub of adjacent wall on the right. On the back side of the baffle the door frame and an adjacent wall interfere by reflecting sound back to the driver, so my baffle cannot be taken as adequate to judge the performance of a driver. It can only be used to compare two measurements taken in this same apparatus.

              After measuring the flush-mounted driver at 50cm, I screwed the Corona transition disk and baffle to the front of the infinite baffle (before any pipes were attached to the back, obviously) and measured again.

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              The graph shows increase from 350-3kHz, especially 900-3kHz, and a trough at 10kHz. Some of this effect may be due to diffraction off of the outer edge of the baffle; other due to the transition-disk horn/waveguide.


              • #8
                CORONA, by Meredith Cargill
                The Baffle—Second Problem

                Some of you immediately recognize a big problem with my design—I have put a driver dead center in a virtually circular baffle. This is a worst-case scenario for interference from edge diffraction. Edge diffraction is a distraction in this write-up, because the point of the design is to address an entirely different issue, so if you already are familiar with edge diffraction and how to address it, you may want to skip to the next post. But it is also an opportunity to show how edge diffraction can affect a speaker's response.

                As the sound radiating from the driver travels along the surface of the baffle, when it hits the edge, the sudden change in impedance generates a reflection—a sound wave that travels toward the listener. Compared to the sound arriving directly from the driver, this additional sound is delayed by the time it takes to travel to the edge of the baffle, and by the additional time it takes to travel the additional distance from the edge to the ear. It produces a wavy alteration of the frequency response, as this secondary sound goes in and out of phase with the direct sound, according to the wavelength. It is easiest to show in a measurement when any other reflected sound is eliminated, by using a gated impulse response (3 or 4 milliseconds, in this case).

                In this graph, the different curves are taken on axis, but at different distances from the driver. The interference of the edge reflection shows as a ripple that moves up in frequency as the listening distance increases. The waves in the FR get smaller with increasing frequency because the driver is becoming more directional, so less energy is radiating across the baffle to power the reflected sound wave, until the diffraction artifact disappears above 5kHz because the driver is beaming.

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                Notice that the bottom point of the dips appears at even multiples of frequency. The purple curve, for instance, dips at 800, 1600, 2400, and 3200. The yellow curve dips at multiples of 700Hz.

                One correction for this problem would be to make the baffle less like a circle, such as making it spiky, with deep, deep notches (like a crown!) instead of the gentle undulations I have around the edge. That way, the sound would arrive at the edges at various times rather than at once in every direction. Another would be to make the baffle spherical, so that there is no edge. Without rebuilding the baffle, the only thing to do is to listen off axis, where the secondary sound does not all arrive in phase, because the edges of the baffle are no longer at an equal distance from the ear.

                Here are two graphs showing the ripples measurable from diffraction, comparing the on-axis condition to off-axis. The first is in 5-degree increments. The second compares on axis (blue) to 10, 15, and 20 degrees off axis.

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                Already by 10 degrees off axis, the response is considerably flatter. For this reason, this design calls for being listened to off axis, though listening off axis sacrifices some high-frequency response. But 10 degrees is not far enough off axis to take care of that deep dip below 1kHz. As the previous graph shows, one would have to listen at 25 or more degrees off axis for the response to flatten out that low, and by then the response is rolling off above 2kHz.
                Last edited by mcargill; 10-19-2020, 06:43 AM.


                • #9
                  CORONA, by Meredith Cargill
                  The Foil—The Cube Box

                  The idea behind this design is to eliminate one of the problems caused by sound that radiates from the back side of the driver diaphragm and is trapped in a rear enclosure. If that sound is not absorbed or cancelled or disrupted, it reflects off the interior walls of the box and can radiate out into the room through the thin diaphragm of the driver, or even through the walls of the enclosure. When the sound emerging from the box is in phase with the sound being emitted by the front of the driver diaphragm, it will cause a bump in the frequency response. When it is out of phase, it will cause a dip. Whether a bump or a dip, it is a delayed repetition of the sound, and will therefore degrade transients.

                  There are two phenomena to watch for. There are the first-reflected soundwaves, when the energy from the back side of the diaphragm echoes off of the interior walls of the box and returns, hitting the diaphragm, and there are also standing waves that form inside the box from multiple reflections. The first is determined by the distance between the driver diaphragm and the reflective surface, the second is determined by the distance between two reflective surfaces.

                  I used the spreadsheet to calculate the total volume of the pipes, and made a box of that same volume to serve as an example of a problematic rear chamber. I made it a shape that would be a worst-case for forming standing waves—a cube. The cube turned out to be 6.33” in each dimension (internal).

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                  A first reflection off the back wall of the box, 180° out of phase with the front energy, would travel 12.7” to arrive back at the cone. Such a reflected wave would cancel whatever is being emitted by the driver at that time when 12.7” is equal to one wavelength, or any even multiple of a half wavelength. We can therefore predict a dip in the FR curve at 1066Hz, 2132Hz, 3198Hz, etc. The reflected wave would boost output when 12.7” is an odd multiple of a half wavelength. We can therefore predict a bump in the curve at 533Hz, 1599Hz, 2665Hz, etc.

                  We can predict standing waves to form when the distance between the walls of the box (6.33”) is a multiple of half-wavelengths of the sound bouncing around in there. This predicts FR anomalies, whether bumps or dips, at 1066Hz, 2132Hz, 3198Hz, etc

                  To see what this box does, let’s measure it compared to the driver with no enclosure. According to DATS, the resonance shifts from 95.55Hz up to 103.6Hz and the Q rises to 0.6893. A slight disturbance also shows just above 1k. (Sorry, no picture of that measurement.) Here is a graph measuring the RS100 mounted in the Corona baffle (on the stand, not in the infinite baffle, but measured only 3cm from the cone) with its back open (no pipes), compared to having the Cube Box mounted as a rear chamber.

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                  This graph shows the change in low-frequency response, as expected from the raised Q, and also some changes in the 900-2kHz band. Note that this graph scale is 1dB per division. The perturbations are not dramatic. The response from 100-2500Hz remains within +/-1dB.

                  Because standing waves, like harmonic distortion, take time to develop, it is easier to measure them with a long (slow) sine-wave sweep. Sadly, in OmniMic, I cannot save a long sweep as an FRD file and overlay it on others, so I give you two separate graphs taken from the Harmonic Distortion function, with the microphone only 3cm from the cone:

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                  Here, it is much easier to see the effect of reflections inside a box, compared to a driver with its back open. A response that was flat +/-1dB from 150Hz to 2.5kHz becomes +/-3.5dB, with sharp troughs and peaks. These sharp deviations are coming in the region where we expected reflections and standing waves to cause problems, given the dimensions of the box.

                  This is the problem that a resonator array tries to solve.

                  I might as well cut to the chase at this point, and show the comparable graph of Corona.

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                  Corona's response, with its +1dB hump 1.5k-2kHz is not quite as flat as the open-back condition, but it is as smooth. It provides a rear enclosure (not dipole, as with an open back) that obviously does not present the aberrations introduced by the foil box. The array solves the problem.

                  This is not to say such a resonator array is the only way to solve this problem to one's satisfaction. It is certainly, in my implementation, a rather impractical solution.

                  I did measure, also, the Cube Box stuffed with fiberglass. I'm sorry I do not have that graph to show you, but I hope you are willing to take at my word that the stuffed box does not produce a curve as flat and smooth as Corona's.
                  Last edited by mcargill; 10-21-2020, 07:48 AM.


                  • #10
                    CORONA, by Meredith Cargill
                    The Single-Pipe Experiments

                    To help understand the potential effects of an array of ½” pipes, I ran the experiment of putting a few different lengths of pipe as appendages to the foil box.

                    To start with, I cut one pipe at ¼ wavelength for the driver resonance in the box (103.6Hz). I measured the driver, with the box behind, in the infinite baffle.

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                    Its effect is most telling in the resultant impedance graph:

                    RS100 IB CubeBox 1 pipe at 103.pdf

                    DATS tells me the effect of that pipe is greatest at 105, which is only 2Hz away from my target. The dip in impedance is not great, because a small tube cannot be expected to have as large an effect as, say, a whole box-volume resonating.

                    That impedance graph shows the aberration at around 1066Hz that I mentioned above, corresponding to the radical dip and peak in the FR near there.

                    I replaced that tube with others of different lengths and found that the pipes at 103Hz and 161Hz had measurable effects on the FR curve, but not the shorter ones, at 286Hz and 810Hz.

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                    This experiment indicates that the pipes in the Corona array may have unwelcome effects below 200Hz. If the crossover point is set above 200Hz, these effects will not be in the pass band. Or it could be that the dips we see in the FR in the Cube Box are due to the pipes interacting with internal volume of the box, in which case the effect will not occur in the Corona array.
                    Attached Files
                    Last edited by mcargill; 10-21-2020, 04:04 AM.


                    • #11
                      CORONA, by Meredith Cargill
                      Final Measurements

                      The most telling measurement has already been posted (post #9)—the long (slow) sweep measurement showing the lack of internal reflections, compared to the foil box. That is the curve that shows that the resonator array does work, and does not make matters worse.

                      Here is a set taken with the microphone at 0cm (inside the driver’s rim) or 3cm, comparing Corona (red) to the foil box in the Corona Baffle (blue) and to the driver in the Corona baffle with back open (black).

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                      Corona's response has a little bit of scallop to it that is probably not audible (No smoothing has been applied to these curves). The array at least does not introduce any deleterious effects measurable by this graph.

                      Once I had the driver mounted in Corona, I could no longer test it in the foil box, or open baffle. But I had another driver like it, that measured almost exactly the same in DATS. I put it in the Cube Box and set it up with a temporary baffle, to make a “mock tower” speaker.

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                      Here is a set of measurements taken in a relatively standard way—2.83 volts, 50cm away, full range. I have included both the on-axis measurement for Corona (black) and the measurement at 10 degrees off axis (blue). The red curve is the mock tower, with the Cube Box as rear chamber.

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                      Given how smooth the response is when measured at 0cm, the deep notch at 700Hz must be produced by the baffle, as part of the family of bumps and dips due to a circular baffle (the dips come in multiples of 700Hz, at 700, 1400, 2100, 2800, etc.). That circular baffle is just not a good idea at all. I had heard that was the case; now I see it dramatically. Then again, the "mock tower" speaker has sharp deviations, too, which are more likely to be from the baffle than from anything internal.

                      Such artifacts distract attention and interfere with my ability to make the case that the resonator array actually does work, because the final response curve looks so not flat.

                      Most interesting, concerning the metamaterial array is the Corona impedance graph, compared to the driver measured loose on the bench:

                      Corona v Bench DATS.pdf

                      Although the same internal volume of air as the Cube Box, the resonator array does not raise the resonance frequency or Q. In fact, resonance is slightly lowered, to 94.21Hz. Q is 0.584. My decision to target 95.55Hz with one pipe was a waste of effort given how little effect any one pipe has, and how the frequency of resonance changed unpredictably.

                      The sum of the resonating pipes did not do much to change the intensity of the resonance. The Zmax is reduced from 13.5 Ohms only down to 11.3 Ohms. If one were counting on the array to substantially control the resonance, as a ported box or QWTL should, one would need larger pipes around that frequency, which would also affect the function of the array when those larger pipes are producing higher harmonics. I can imagine an array designed to kill the driver's resonance entirely (not just knock it down in the middle, as a ported box does), but it might not perform well as a broadband absorber.

                      The scalloped texture of the impedance curve indicates that each pipe’s fundamental resonance affects the driver. From 80Hz to 500Hz you can see the step caused by each successive pipe (pipes #1-#26).

                      Attached Files
                      Last edited by mcargill; 10-17-2020, 08:19 AM.


                      • #12

                        CORONA, by Meredith Cargill
                        ALTERNATIVE ARRANGEMENTS

                        The same concept of a resonator array can be implemented without the showy arrangement of pipes radiating so dramatically. Not every metamaterial has to be a corona. Even if the radial arrangement were kept, the structure could be made more compact by putting all of the longer pipes on one side. Or it could be made shorter, by putting all the longer pipes in the more-or-less horizontal positions. The angles of the pipes could be adjusted so that some arcs are more densely populated than others. Plus, as I mentioned above, the pipes could be folded, bent, and coiled.

                        The pipes could be arranged in parallel to install in or form the side of a rectangular enclosure. Here are photos suggesting how the options would look, if the pipes are grouped in parallel. In two ranks of fifteen:

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                        In three ranks of ten:

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                        That is an 8-inch driver, for size comparison (Parts Express buyout).

                        In four ranks of 7 or 8:

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                        In five ranks of six (the most compact rectangle possible):

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                        The pipes could also be formed in a circle, which is even more compact than a rectangle.

                        A ring of pipes could be made to fit close to the diaphragm from the rear. Here is an example, using a piece of larger PVC pipe as a former that fits over the magnet of an 8" driver:

                        Of course, for a driver used at lower frequencies than the RS100, this particular set of pipes would not be long enough. And for a larger driver, proper effect might also call for the pipes in the array to be larger in diameter.