Bruno Ogorelec wrote:The Short answer (pun intended) to your questions is, â€˜I donâ€™t knowâ€™. Now, letâ€™s see exactly how little I know.
The location of the hottest part of the engine is truly curious. I have two possible explanations. One is that this is where the focus of your concave mirror on the front end of the engine is located. Ideally, that surface should be parabolic, so that it reflects all the energy that hits it in a straight line towards the exhaust. Further shaping of the energy flow should be done by the nozzle shape of the entire chamber. Your mirror is obviously not parabolic but what I call â€˜potatoidâ€™, meaning that it probably concentrates the reflected energy in some relatively narrow area. I am familiar with some experiments in this direction and I know that focusing does work to a certain extent.
Bruno, Steve et al -
I agree with the logic of everything else you've said, Bruno, but when you talk about parabolic reflectors [actually, paraboloidal, technically speaking], you're talking about old friends of mine, and I really need to speak up. Yes, I'm sure that focusing reflectors can do something in pulse combustors. However, I am 99% sure that they don't do what you seem to be expecting. Bear with me, here -- this will take a while.
A paraboloid, used the way you describe it, is performing the work of a big searchlight mirror. The mirror has a precise 'focal point'; that is, the point at which the image of some 'infinitely' distant object, like a star, would be formed if it were used as a telescope mirror. Here, the situation is exactly reversed; waves are propagated at a nearly infinitesimal point very close in, radiating out in a spherical pattern. Some of these waves are intercepted by the mirror's surface, and reflected back. The "magic" of the paraboloid is that, IF the wave source is at the exact focal point, the reflected waves are perfectly re-shaped to absolutely flat wave fronts that proceed forward in a perfectly straight path whose direction is at a right angle to the fronts, just like in those classic textbook photos of straight waves moving in a wave tank. Think of all those old war movies where searchlights are used: those things were so perfectly adjusted that the 'beam' [actually a slug of moving wave fronts] maintained a five-foot width from the light housing practically to infinity. Only a paraboloid is capable of doing this.
However, a paraboloid is NOT the correct shape for other geometries of wave flow. Suppose, for example, that we want to propagate waves from a point in space back to a point right beside it. We could take our paraboloid and move it gradually farther and farther from the point of light propagation, and what we would see [if the air has a little smoke or dust in it, as it almost always does] is that the projected 'beam' of light starts to narrow in as a long cone. As we move our mirror farther and farther back, the cone of reflected light becomes shorter and shorter. Finally, when we reach a point where the light source is precisely 2x the focal length from the central spot of the mirror's surface, the point of this reflected cone will be at [or right beside, if we just tweak the aim of the mirror a bit] the point of propagation. However, now we have a problem.
The paraboloid turns out to be the wrong curve, in this case. If our point of propagation is as small as we can make it [say, a tiny pinhole in an opaque housing with a bright light inside it], we may be surprised to find that the reflected image is not very good. If we would analyze it further, say, by masking off parts of the mirror, we'd find that the reflected cones of light from the outer edges of the mirror 'focus' to a point a little beyond the pinhole, while reflected cones from the central zone focus a bit short. On a big short-focus reflector like a 60-inch searchlight mirror, the deviation from edge to central reflection would not be small -- probably an inch or two!
While we're pondering this, we might notice something by further experimentation: if we mask off more and more of the surface by working in from the outside edge of our mirror, the remaining area in the center produces better and better images of our pinhole source; in fact, we can narrow it down to where we don't see any distortion of the image at all; that is, the whole central zone that's left seems
to bring all the light back to a perfect point image. At this point, we could try to make some physical measurements of the surface of the central zone that's left. It might astonish us to find that, if we take a small enough section of the center of the paraboloid, we can't tell that it isn't simply a segment of a sphere! Of course, we know mathematically that it can't be spherical, but if the piece we're studying is a small enough central region, you can't measure the difference with ordinary tools.
Suddenly, it hits you: if you want to focus a perfect image at the same location as the wave source
, the perfect shape would be a sphere! This stands up under rigorous scrutiny: A wave front radiates out from a point source as a spherical front. If a portion of that wave front is reflected by a spherical surface whose center of curvature coincides with the point source, ALL of that front will be reflected by the spherical surface at precisely the same moment in time
, and further propagation of the wave will be along paths of convergence toward that point! A paraboloid cannot to that -- its outer zones are too far away, and will reflect the incoming spherical wave front too late for them to converge at the same distance!
It is this phenomenon that was quantified theoretically and validated empirically by the great French physicist Lean Foucault, and has come down to amateur telescope makers everywhere as the 'Foucault Test'. In skilled amateur hands, it can measure differences between a sphere and the nearest matching paraboloid of four millionths of an inch or better!
Now, suppose we have a different problem yet: we'd like to use our 5-foot 150lb paraboloid as a photo enlarger. Let's say we want to enlarge an image 2x its original size. So, we start moving our pinhole around to see what we can do. Intuitively, we realize that to do this, we need to move our point source back in closer to the mirror -- not as close as when it was a searchlight, but closer than when the light was reflected back to a point in space right beside the source. Let's say we now drill out our pinhole to a 'porthole' 1 mm in diameter. We can move it in until the reflected image is right at 2mm diameter -- a 2:1 enlargement. What we will find is that the distance from the source to the center point of the paraboloid is precisely half the distance from that central point out to the reflected image. Except for one little thing: Once again, the image isn't perfect, with the outer zones of the mirror throwing their light cones further than the central zone. If we again start to 'mask down' our mirror from the edge inward, we again finally get to a place where the image seems perfect; we notice, however, that the uncovered central zone is now a lot larger than it was for a perfect image at a 1:1 ratio.
If we now make careful physical measurements of the exposed central mirror surface, we are surprised to find that we cannot fit it closely to any sphere -- the deviation is just too large. We are seeing the effect of the parabolic cross-section; yet, the paraboloid can't be quite right, because we did have to trim off the outer zone ... hmmm ...
Again, intuition finally kicks in, and we measure the distance between the source, image, and central zone of the mirror. The maths of analytic geometry will verify that what we have is a classic ellipsoid: its section is an ellipse with one focus at the point source and the other at the image point. And, there is some portion of the center of a parabola that we cannot sensibly distinguish from this elliptical curve -- it happens to be a much larger section than the part that seems to be a sphere.
It is a truism that the laws of wave mechanics hold for acoustics exactly as they do in optics, as long as appropriate media are available. So, it seems natural to use reflection and other 'optical' properties in dealing with acoustic phenomena. The only difference is in the effect of the HUGE differences in frequency and wavelength. However, in practice for our present case, this turns out to be of monumental importance.
There is a single basic reason that we are able to easily measure the differences in the ideal curves in our searchlight mirror example using light: the diameter of the reflector, no matter how tightly we mask it down, is MILLIONS of wavelengths. When we deal with our pulsejets, no matter what the specific geometry, we are dealing with FRACTIONS of a wavelength. This is not a mere technical detail; it changes everything, radically.
You may remember Mike E, Bill, Bruno and others mentioning the 'Rayleigh Criterion', which has to do with exactly what condition has to exist for explosive combustion to occur. What you probably don't know is that the same Lord Rayleigh provided another Rayleigh Criterion, for us amateur telescope makers to use. It states that, for a perfect image to be formed, a reflective surface must have a curve that falls within one quarter wavelength of the theoretically perfect curve at all points. This is why it's so important to be able to measure millionths of an inch on a telescope mirror's surface. What it means in our case is that defects in a reflector of less than one quarter wavelength magnitude have no significant bearing on the accurate focusing of wave energy!
If this is true, how can Bruno's statement, "I am familiar with some experiments in this direction and I know that focusing does work to a certain extent," be correct? The answer is, you have to understand the "to a certain extent" part. The difficulty is that optical problems can be attacked by 'ray tracing', where you think of the direction of propagation as straight-line 'rays' whose change in direction can be deduced crudely by graphic methods or with unbelievable precision by mathematical methods. When people talk about 'reflectors' in engines, they imagine these kinds of 'ray tracing' solutions. Where the explosion happens is a 'source' -- find where you want the energy to converge, and this is the 'image'; draw a curved reflector that makes the lines radiating from one point all converge on the other, and presto! But, it just can't work exactly that way when the clear apertures [our pipe diameters] are so small relative to wavelength!
When you are meters away from a small explosion, it is perfectly acceptable to think of it happening 'at a point' -- but not when you are only millimeters from it. Does conflagration in a confined space start at a neatly contained 'point' and radiate smoothly outward? Not a chance. At a sub-wavelength level, it is more appropriate to imagine a major region where the pressure and temperature suddenly rise dramatically throughout. It is not any more a wave front expanding spherically than it is a quasi-rigid piston moving in a single direction -- you simply can't describe it properly in any 'macroscopic' terms! The same thing applies to wave motion in the pipe. A mental picture of ocean waves moving toward a beach is totally inadequate; it is much more like the action of peristalsis in the esophagus or colon! But even that is incorrect, since mass motion, rather than wave propagation, is suggested. I'm 57 years old and have been interested in Physics most of my life; and the only physical phenomenon I've ever seen [meaning seen
, as with my eyes] that is anything like it was compression waves sent from one point to another through a long Slinky(TM).
I am sure, with Bruno, that reflective focusing works and that, of course, is partly what I'm relying on at the front end of the Focused Wave Engine. But I am not at all sure that we can EVER accurately describe how it happens by simple acoustic theory. I think the only accurate theoretical description of reflection inside pulse combustors would be by an expert with highly detailed knowledge of fluid wave mechanics on a microscopic
level, where you're dealing with the subtleties of pressure distribution, molecular interaction, transfer of molecular momentum between molecules and the pipe wall, etc. I really believe it is inherently impossible to accurately describe it in macroscopic terms.
For the rest of us, all we can do is experiment, observe and document what we see.
Of course, I could just be ... wrong.
My theory of how the Focused Wave Engine chamber should work is that the pressure wave should be concentrated on the region just a little behind the front end of the straight pipe, pressurizing that region uniformly to start the traverse of the wave down the pipe. I feel that most of that will be accomplished by the unbroken conical shape and very gradual slope of the long conical chamber. The front reflector is mostly a pressure cell for the buildup of the initial front formed at the time of 'constant volume expansion'. So what we should visualize, I think, is basically a rapid pressure rise in the front of the chamber followed by a simple 'funneling' of the pressure wave into the pipe, over a finite time interval. That was the idea.
I personally doubt that the shape of the front dome has much to do with the action, and I doubt that subtle changes in it will profoundly affect what happens in there. This is why I didn't press Steve on exactly how to make that part. It's entirely possible that Bruno's description of what's happening in there is basically valid, but at this point I can't buy that the exact shape of the reflector is really responsible for it.
But, somebody out there might convincingly show that I could be wrong about that, too. It has been known to happen ...