In this update, I'll talk about what I found by using solid plugs and a thicker-walled enclosure. Plus some surprising findings and maybe a directive for a new approach.
I ended up building a thicker speaker box made from 1/2 inch pine. I used one of the ramps above to build a cap that is removable. The idea being to make comparative studies between a 90 wall and a ramped wall at the start of the line. My theory was that the ramp could serve as a waveguide and save energy by keeping the sound waves from reverberating and the top of the line. I had also observed that the previous box (left side in the picture below) was radiating a significant amount of sound. It turns out the latter as very important, but probably not because of what you expect.
I made a cap for the box using a piece of wood and the wood ramp. For the results I present today I did not smooth over the ramp with stucco as shown in the picture, and I just left it in raw wood, but I could also flip the cap upside down and that would be my "no ramp" test subject. The reason for covering the ramp with stucco is to smooth the curve (the wood ramp is not curved it's three segments with different slopes), and to rule out any possible effects due to sound reflection off a semi-flexible surface during testing. I will test with the ramp with the stucco on it next time.
The comparison would be more perfect if I build a block of wood with the same volume as the ramp to take out the volume of air that the ramp needs during the "no ramp" trials - maybe I'll do that for the next round of tests, but I didn't for this informal test, knowing that a change of +/- 5 cm in the length of the waveguide doesn't seem to give me that much of a difference (unmeasurable, < 1dB according to previous tests). The length (hence volume) of the new thicker box is adjusted to include the volume of the ramp and the volume of the extra wood in the walls (the channel is 4mm narrower in one direction).
I did some preliminary studies, but I don't have plots for that yet, because it was just an informal exploration. Nothing was calibrated. I was just anxious to see what I would get. The comparisons were made against the Sony Bass Reflex speaker which is my baseline. The trials also compared between ramp and no-ramp caps and I measured sound levels for sine wave inputs (57, 67, 77. 87 Hz) and also measured sound levels for white noise. To compare between 90 degree wall ("no ramp") versus ramp wall performance all you have to do is flip the cap. I can do that even while listening to some material in real time to make a comparison with my ears. This turned out to be important as well.
Between the frequencies of 57 and 87 Hz white noise profiles were indistinguishable between the old speaker box and the new one, and between ramp and no ramp caps. So basically no new information could be gathered from the white noise plots; all profiles are flat.
However I did find significant differences between the no-ramp and the ramp caps. I found that using a ramp yields output levels 3 dB higher than a 90 degree ("no ramp") cap, quite consistently (+- 1dB time -dependent drift/error) throughout the 57-87 Hz bandwidth. So this settles the question; I had theorized that perhaps having a 90 degree wall against the speaker would be beneficial to standing wave formation, in direct contravention of the energy analysis that states that you need a smooth curve to direct the energy into the horn - it's basically an argument for energy savings versus accumulation of energy by resonance. I just wasn't sure which was the right answer and I had to test it. Normally wave-guide speakers don't use ramps like these, they just use 90 degree walls and deal with the consequences by adding batting, so I had to test it out. But it turns out that 3 dB is a significant difference in favor of using a ramp, especially since this is a logarithmic scale. Why curved ramps are not used by others is a question mark. It could still be that a 90 degree wall is needed for standing wave formation, but at a ramp length of 10 cm and a wavelength in the order of 4 meters, it's hard to see how you would impede standing wave formation.
Unfortunately for sine wave results in the 57-87 Hz range, I was much surprised to see that I *lost* 1 dB of performance across the band quite consistently, when compared to the Sony Bass Reflex baseline. This baffled me. I did everything right. The ramp is rigid. I made the box with plenty of mass. The box is more rigid and dense. By all accounts energy loss is mitigated by the new design, and yet I lost 1 dB, so what gives?
Let's analyze what I have learned so far:
1. The resonance frequency is 57 Hz. At this frequency a 5 cm difference in the length of the waveguide did not make much difference (wooden extension ring, I talked about above). At 2,5 cm of added length, this new box should not change that equation.
2. The shape of the ramp turned out to be significant. A no ramp configuration lost 3 dB versus a ramp design. Shape and smoothness in general is very important near the top of the line (speaker).
3. According to the latter study, the added mass of the box walls and ramp were not beneficial, but detrimental. That doesn't mean that energy saving is not important, but a loss of 1 dB in measurements can be attributed only to addition of the mass/thickness, because everything else is basically the same versus the mesh-stucco ramped box
4. For the mesh stucco ramp box , I had detected a significant amount of sound coming from the walls of the speaker box. That, in fact was my impetus for building a new wall and ramp, more solid that before
It's as plain as the nose in your face that that extra decibel I'm missing is the radiation coming from the thin-walled box, a combination of radiation from the mesh-stucco ramp (which is very thin) and the thin pine walls.
Traditional speaker theory states that the walls of a speaker should not radiate. The box should not resonate, like a violin or a guitar, and to that effect you make the box of the densest most rigid material possible, to both absorb any sound attempting to escape and reflect it back into the system.
The reason why you do that is that the sound waves coming from the back of the driver's cone are 180 degrees out of phase with the sound waves coming from the front of the cone. The box's function is to separate the rear waves from the front waves, and either make those waves disappear (infinite baffle) or harness them in a suspension speaker (sealed cavity), or in a bass reflex system (Helmholtz resonator).
But there is no reason you can't have a spherical radiator. In a spherical radiator, a speaker is entirely surrounded by a thin wall, very much shaped like a balloon. all the waves coming from the back of the speaker are trapped in that balloon, but when the driver of the cone pushes inward, the whole balloon expands, creating waves. If the driver is producing sound waves that are very limited in spread (the cone "throws" sound into one narrow region of the room) the sound cancellation in that region is limited to a narrow angle about the speaker, and most of the sound coming from the passive radiator is useful sound. Also you can design the mass density of the balloon to have enough inertia to force the balloon surface to move out of phase with the sound inside the balloon, avoiding sound cancellation. In other words, the whole box becomes the driver and the driver becomes part of the motor.
It is possible to design such an enclosure, but more often than not you will sometimes see membranes used as "dummy speakers" or "Passive radiators." Basically they are made the same way as a regular driver but without a magnet, and they behave exactly the same way as a driver with a magnet, but are using the pressure energy from the enclosure and their mass dictates how far away in phase they are from the sound waves in the enclosure.
https://en.wikipedia.org/wiki/Passive_radiator_(speaker)
You can always design a piston located strategically on the box that is tuned mechanically (spring-mass-damper) to resonate at the desired frequency, thus helping you radiate more sound. But it's a difficult thing to get right, because that piston or "passive radiator" needs to be tuned properly, oriented properly, and be designed mechanically and acoustically as part of the whole system. In spite of the probability of sound cancellation, I think that my lost decibel was being produced by the the stucco/wood box itself acting as a spherical/ passive radiator of sorts. Maybe that stucco ramp? Where is Stradivarius when you need him?
So at this point I'm not sure what to do. I have a working theory of what is happening, and that gives me some ideas. But barring any further theory, the horn setup with the mesh-stucco ramp seems to be the best that I can achieve so far. And I need to treat it as a "sacred geometry" because I can't readily calculate the resonance and phase behavior of that particular stucco/pine box without using a good structural dynamics program - even a very old one such as the venerable NASTRAN
It makes no sense to add more weight to the system by adding solid ramps and thick walls trying to save more energy, unless I see an increase in performance in some other way, be it 1) placing the drivers at the endpoints which produced very strong - out of control- resonance, 2) making a new horn design (I suspect my horn's exit slot is too small it should be at least 1/9 of the input area) or 3) by designing a passive piston/spherical radiator setup, which significantly increases the amount of physics and math I have to calculate, but it's definitely not going to look the same.
If I had another pair of large cones, a passive radiator design seems like a very interesting idea to me, and a way to put my stamp on the design, but I'm not sure I want to go that way. I don't feel like I have mastered the problem of transmission line speaker yet, because my horn design is no longer ideal, having been modified so many times.
As an aside, I must state that the particle board box of the Sony Bass Reflex was unusually thin for a speaker enclosure (the board is only 8mm thick).
Secret sauce?
