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The article I’m talking about: https://www-ee.stanford.edu/~dabm/146.pdf

In this article, the author solves the main problem with Huygens Principle, which is that you have to ignore the backward wavelets, despite the sources being point sources on the wavefront. This solution is also on Wikipedia prominently as the legitimate correction.

The author solves it by substituting the point sources by a dipole, in which the left source is delayed in such a way that there’s no backward wave, only forward.

I have no doubt that it works mathematically, but physically speaking, isn’t it just another arbitrary correction to the original principle? Like, he just put another source there in conditions that would erase the backward wave. Does this for any of you, however, make sense physically speaking? Why would a dipole be more appropriate in this case than a point source?

I’d appreciate little math, more conceptually speaking.

Qmechanic
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    Yeah that does seem arbitrary. But the point of that paper isn't in making it make physical sense. Indeed, its abstract already says that Kirchhoff's solution is a full rigorous one. The point of the article is to make a model which has the simplicity of Huygens principle but doesn't have one of its shortcomings. – Ruslan Oct 27 '17 at 19:28
  • True indeed. That way we still don’t have an intuitive answer to problems like these: https://www.google.com.br/amp/s/amp.reddit.com/r/Physics/comments/5yogr0/do_waves_travel_backwards_or_not/ — this correction though, I think could explain this problem, because, as the sources are separated by a distance, the part that’s diffracting backwards could not be destructively interfered by the wave from the delayed source, only from the first one. Does that make sense? –  Oct 27 '17 at 19:55
  • I'm not sure what you mean, but if you're talking about diffracting "to the side and then backwards" as in the picture referenced in your link, then there I suppose it's just the expansion of the spherical wave resulting from the dipole to the side, where there's no other source to continue shaping the wavefront. – Ruslan Oct 27 '17 at 21:27

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Here in "Huygens Principle" Kevin Brown gives an interesting historical outlook and explains that the Huygens principle is valid only in the odd number of dimensions.

This however, can be interpreted a bit differently. Perhaps a better interpretation would be that the Huygens principle only establishes any point of the wave as a source of a new wave. This way we could say that the Huygens principle is valid in any number of dimensions, but the retarded wave is canceled only in the odd number of dimensions.

A few interesting examples of how this works. In 3 dimensions the retarded wave is canceled. As a result, we don't hear the echo of our voice reflected from the air (but only as reflections from distant objects). Similarly, a flashlight doesn't blind us, because light does not reflect back from the empty space. However, it would do so in 2 or 4 dimensions.

Another odd-dimensional example is a whip. A wave induced in a long whip is 1-dimensional. Once sent, it does not reflect back. (A half a century ago I used to make a gunshot-like sound with a 30-feet whip to control a herd on a pasture.)

A common example in 2 dimensions is a water surface. Drop a rock in the lake. The rock drowns and triggers a circular wave. While it is expanding in a circle, it does not leave the inside area of the circle quiet. When we turn off a light bulb in 3 dimensions, the darkness is instant. However, on the surface of the lake, the wave in the center of the circle persists for a long time.

The above link also explains that the statement in the article you quote is incorrect that the Kirchhoff solution proves the cancellation of the retarded wave. It does not. It only links the Huygens principle to the Maxwell equations, but these equations do not prohibit the retarded wave.

safesphere
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  • What does it mean that Huygens doesn’t work in 2 or 4 dimensions? In the odd dimensions the backward wave cancels? What’s the difference between these odd and even dimensions? –  Oct 29 '17 at 15:39
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    @RobertoValente Yes, "doesn't work" means that the back wave doesnt cancel. In 1D it cancels, because the forward wave of the previous point cancels the back wave of the next point. They have nowhere to escape in 1D. In 2D the waves have more room to propagate and the math works out that they don't cancel completely. And then in 3D even more room, but the math magically works out that they exactly cancel anyway. And so on. The difference between dimensions is in formulas like $2\pi R$ in 2D, but $4\pi R^2$ In 3D and so on. – safesphere Oct 29 '17 at 17:12
  • Thanks for the answer but are you sure that’s the way to interpret the odd/even number of dimensions issue? In this question https://physics.stackexchange.com/questions/129324/why-is-huygens-principle-only-valid-in-an-odd-number-of-spatial-dimensions on the comments someone claims that Huygens Principle doesn’t work for even number of dimensions because the wave equation presents infinite many velocities for the wave, which would invalidate Huygens Principle. There’s no mention of backward waves being canceled –  Nov 02 '17 at 11:51
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    Yes, all answers and comments there talk about the back wave canceling as "the Huygens principle". A wave dependence on the future wave or on the wave inside the "light" cone or a variable velocity is equivalent to not canceling the back wave. However nothing anywhere contradicts the fact that any point of the wave is a source of a new spherical wave regardless of the fact if the new waves cancel in the reverse direction. So my interpretation is correct. – safesphere Nov 02 '17 at 13:59
  • Thanks again, though I can’t say I’m still not a little conflicted about it. It’s just that, by different velocities I understand that, from the same point, there are many waves, all going forward, just different velocities. Backward wavelets would mean a different thing entirely for me, as the direction of propagation would be opposite direction (opposite side really) from where the wavefront is going. But thanks anyway! –  Nov 02 '17 at 14:35
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    If a wave has one constant speed, like light, it is located on the light cone. Otherwise it is located on and inside the light cone (where the cone represents the highest speed). The wave from each point is circular (or spherical, etc.) going forward and back. If all speeds are the same and all the wave is on the cone, the back wave from the next point is canceled by the forward wave from the previous point. If the speeds are different, the cancellation is not complete, as the wave is not exactly on the cone. Thus each point depends on the future wave = the back wave. – safesphere Nov 02 '17 at 14:52
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    If you think of the waves going forward and the waves going back as different waves, you are not applying the Huygens principle. According to it (the way I interpret it), each point of the medium oscillates sending a sirvulsr/spherical wave in all directions. Then the wave becomes the "forward" wave only because the part of the circle/sphere set back from the next point was canceled by the part of the circle/sphere sent forward from the previous point. – safesphere Nov 02 '17 at 14:59
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    Consider this analogy (not exact, but very intuitive). Consider a straight line of billiard balls with gaps between the balls. Hit the first one, it hits the second and stops, the second hits the third and stops and so on. The "wave" travels only forward. Now consider each next ball slightly larger than previous one (a rough equivalent to different wave speeds). The first ball hits the second and bounces back a little and so on. The "wave" travels forward, but also partially back. – safesphere Nov 02 '17 at 15:05
  • Typo one comment over above "sirvulsr" should read "circular". – safesphere Nov 02 '17 at 15:07