Yes, but as in Olbers' paradox, I expect only logarithmically towards an asymptote, as amplitude dissipates at the rate it does even without any collision.
Ok, the idea that, in a perfectly reflective and closed chamber (even one which bounces light around incoherently), amplitude would dissipate as it does in open space is more of a weak and probably wrong assumption than an expectation. Regardless, the important mystery is not whether the energy leaks or not, but whether it would leak more in one direction than another as efficiently as it appears to.
I think the resolution to the mystery is to be found in Shawyer’s discussion. Shawyer talks about how it’s long been well demonstrated that light (group speed) slows down as the aperture it is travelling through decreases in size, and indicates that the end result motion of the Emdrive is in the direction of the small end. So it seems to me that the reason momentum decreases overall as aperture decreases (e.g., as light travels towards the small end), is that the momentum of the light is converting into momentum of the aperture in the same direction increasingly efficiently as aperture decreases. Reports also show the large end hotter, so maybe that’s evidence that the transfer of linear momentum onto the large end is more converted into omnidirectional thermal radiation than transfer of the same amount of energy onto the small end. So maybe a good mechanical analogy would be throwing two ice balls towards the center of a wall at a right angle to the wall. One ice ball smacks into the wall still frozen and without shattering, and so imparts almost all of its momentum into linear momentum of the wall in the same direction; the other ice ball passes through a furnace on its way to the wall, and so hits the wall at the same speed but as a liquid, spreading out laterally on impact, converting some of its momentum towards the wall into lateral motion of droplets dispersing radially and parallel to the wall, and thus imparting less momentum to the wall.
Here’s an even better way of looking at it. For simplicity, imagine the cone is a right circular cone, with the lateral surface reaching from apex to base at a 45 degree angle. A wave within the cone is travelling perpendicular to the base, toward the base. When that wave hits the base, 100% of the radiation pressure of that wave impacting the base pushes the base directly away from the apex. Then the wave bounces back in precisely the direction it came from. When it hits the lateral surface, it does so at a 45 degree angle, so half the impact of the radiation pressure pushes the lateral surface in the direction of the apex, and the other half pushes the lateral surface directly away from the center of the cone in a direction parallel to the base. Then the wave travels to the opposite inner side of the lateral surface, and reverses the process of the previous impact, which means half the impact pushes the cone in a direction parallel to the base and opposite the direction the previous impact pushed. Then the wave again heads toward the base, perpendicular to the base, and again dumps all its impact into a direction directly away from the apex. Instead of just throwing the wave out the back end as in a normal photon rocket, the process then repeats indefinitely, with the light dissipating in amplitude as it transfers it to the cone through radiation pressure. I may have the exact fractions more or less wrong, but the point is the radiation pressure is more widely distributed when striking the lateral surface than when striking the base.
Now you might think that this results in motion of the chamber from apex to base because you have two half momentum hits towards the apex on the lateral surface versus two full momentum hits towards the base on the base, but it doesn’t, because, in a complete circuit of the impacts, the half-momentum impacts occur twice as often. HOWEVER, the fact that momentum hits more or less hard DOES make a difference. So the base gets hit half as often but twice as hard, that’s fine, but the impact from the hits on the lateral surface are twice as distributed (or perhaps more, since one can see the distribution as occurring over a 90 degree arc when hitting a lateral surface, versus some smaller arc, or maybe no arc, when hitting the base), which means that the hits on the base express themselves more as radial structural damage to the base, i.e. heat. For example, a tow truck driver drives up to assist your stalled vehicle, and says, “I can either push your car with my hand for a few minutes, or compact all that energy into one rifle shot at your car. It’s the same amount of energy, so it shouldn’t matter which one I use on your car, right?” Wrong, of course. The rifle shot will convert more of its energy into lateral momentum expressed as tearing a hole through your car, and won’t do much to move your car. The long hand push will move your car though, because it converts less into lateral momentum of your car. Now overall, the shape of the cone prefers light paths in which light hits the base more directly than the lateral surface, so the base receives more heat, and the lateral surface receives more momentum. No fictitious or obscure forces, all forces accounted for.
Here are a couple more simple mechanical/particle variants of my explanation which improve on my understanding of my explanation in previous posts.
To stabilize motion, there are two identical propulsion chambers in a spaceship arranged bilaterally symmetrically to each other. Each chamber has a propellant launch site at the front and near the center of the ship, where a person throws a racquet ball directly aft simultaneously with the person in the other chamber, causing the ship to move forward. The racquet balls travel for a while, then each ball simultaneously hits a separate 45 degree angled deflector at the rear of the ship. This causes ¼ of each ball’s momentum to continue aft, reducing but not eliminating the ship's forward momentum, ¼ of each ball’s momentum to push laterally on the opposite chamber, resulting in pressure on the ship’s structure itself from both balls directed at each other, and the remaining ½ of each ball’s momentum to change direction 90 degrees to directly away from the opposite chamber, towards the outer side edge of the ship. The balls continue travelling away from each other for a while, and then each hits another 45 degree angled deflector, this time deflecting each ball at half the NEW ball speed again after venting ¼ of the NEW momentum aft again, and ¼ of the NEW ball momentum into the outer edge of the ship’s structure. The balls travel forward until each ball reaches the front of the ship and hits yet another 45 degree angled deflector, this time directing the ball back inwards towards its launch site after reducing its speed by half yet again and directing ¼ of the ball’s momentum forward, and ¼ of the ball’s momentum into the outer side edge of the ship again. At the launch sites, the throwers recapture the balls, and prepare to repeat the process, but the ship is still moving slightly forward relative to to when they prepared to throw the balls the last time.
The reason this works where something simpler like throwing a ball at a wall perpendicular to you and letting all its momentum go into first impact with that wall does not work is that you are instead changing the direction of some of that momentum.
If you can see how the previous variant works, then you can see how two hollow rings fixed next to each other in the same geometric plane, with each having a fan in its side closest to the other ring, and each fan blowing air directly aft inside its ring, would also work. When a fan pushes the air immediately around it, and some of that air leaves the fan blades and heads directly aft, the pressure on the fan from that air moving directly aft is an equal and opposite momentum directly forward. However, the fans keep all of their equal share in forward momentum, while the air in each ring keeps bumping into the outer walls of its ring at all angles, and thus dissipates some of its momentum omnidirectionally, resulting in more momentum forward than backwards overall. The result would be very inefficient on Earth, where propellant and friction are both in abundance, but in the micro/zero-g vacuum of space, where propellant and friction are both scarce, it works out much better.
The problem with Newton’s third law of “equal and opposite reaction” is that it deals only with relatively simple interactions and consequences, and does not take conversion of momentum into change in direction fully into account. I have long been suspicious of Newton’s use of fictitious forces for rotating systems, since I can replicate a circle with a stone on a string by, instead of swinging the stone from a central point, simply walking in a large circle with the stone dragging behind me, where it is clear that the stone follows a circular path only because I am constantly applying energy into changing direction while I walk. The problem is that it is easy to see the inconspicuous source of the force at the center of a stone whirling in a circle as doing nothing but pulling the stone inward, easy to obsess on the tension in the string as strictly an inward force, but the source of the pull on the stone is actually not just pulling the stone, but also constantly changing the direction in which it is pulling the stone. So the force countering the inward pull of the string is not any of Newton’s fictitious forces, rather it is simply the very real force of inertia from his first law. In Newton’s bucket experiment too, as I have described elsewhere, no absolute frame of reference is necessary, but only because all motions can be discerned by simply looking at one history or another of the inertias of the objects associated with that motion. Newton defines inertia, and then fails to fully understand the implications of that definition, leaving the door open for him and others to bring in all manner of fictitious forces.
“Hai Yai Forces!”
- Susumu Hirasawa