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Those were not waves: A bit historical re. Huygens' principle

A few points that might be of general interest:

1. The dates: The date of Huygens' first written down material, which was orally presented to the French Academy of Sciences, is 1678---in contrast to the oft-quoted date of 1690. 1690 was the year of the first, French, publication of these notes (plus other material) in the form of a book.

Surprisingly, Huygens' book was not translated into English until 1912. Reason: The English mechanicians' dogma favored the Newtonian corpuscular theory of light, warts and all, and, being a dogma---not an exercise in Reason---they either attacked, or at least didn't allow for, the wave theory to be taken any serious notice of. (Such practices are not entirely unnoticeable in our times, either!)

Thus, the translation in English of the very beginnings of the first modern wave theory of light occurred some 50 years after the last grand theoretical synthesis involving the wave nature of light by Maxwell (1861 through 1865). 

Also note, the translation/publication occurred in the USA, not in UK. USA was, back then, an "also ran" country when it came to top few frontline research countries. Germany and UK were the top two superpowers in research back then. Clearly, men like Stokes (a Lucasian Professor at Cambridge) were happy to make do with the original French (or possibly a German translation thereof!)  

2. Pulses, not Wave(lets): When Huygens said waves, he meant pulses. His theoretical arguments never did address a *periodic* or *regularly* repeatative phenomenon. He sure had successive rings in his diagrams, but these were not crests/troughs of waves in his imagination---even if he did allude to waves, actual waves, those on the surface of a pond, right in the same writeups. Instead, per his theory, those successive rings simply were the loci, at successive time instances, of what essentially was a pulse (something like the Dirac delta.)

Huygens was concerned with the transmission of momentum from a star to the all-pervading aether surrounding it, in the process of emission of light. Terminology was still evolving back then, and so, the word he used was "movement," not "momentum." He first considered a 1D row of balls, and the propagation of an initially imparted "movement" through that array of balls (rather like in Newton's cradle), in a *finite* amount of time. The Rationalistic philosophic tradition held instantaneous propagation for light. As far as I know, Huygens' argument is the first mechanically based (and hence respectable) departure in asserting finitude to the speed of light.

(The premise of instantaneous action at a distance has not completely died yet; indeed it is thriving well: read up any mathematical proof affirming weirdities of quantum entanglement, esp., the proofs involving variational principles/energetics arguments, e.g., von Neumann's proof, or almost any thing after that.)

Huygens then extended this logic of movement transfers, via intervening balls, from the 1D situation to 2D and 3D. He continued imagining particles of aethers arranged as systematically arranged spheres (as in an FCC lattice). He thought of an expanding circular locus as if in the process of taking a limit for the decreasing radius for the balls (which, he took to be uniform in size). Throughout this development, however, it was the transmission of a single sharp impulse that he was concerned about, not the propagation of waves. (Indeed, this was the reason why eliminating those backwards propagating "waves" came naturally to him.)

3. Phases: It was Fresnel who independently rediscovered the Huygens principle on his own, and introduced phases in the theory.

Fresnel, in fact, also independently formulated the idea of Young's interference experiment. At first, Fresnel was too isolated to even know about Huygens' or Young's work. During the day, he worked as an on-site engineer building roads (or railroads? I am not sure). He worked on optics and mathematics only in the evenings, as a part-time hobby. (Go through the idea of Fresnel integrals, and try to imagine how a young on-site engineer might have built this theory after coming back tired from the field work in the day, living in tents, with only a handful of mathematics books gathered at radom serving as references, writing with the quilt pen, and of course without electric light.) 

It was Fresnel's 1815 paper that, for the first time, treated the Huygens waveslets as waves in the modern sense of the term---as something that can interfere/diffract, as something that can have a *phase*.

BTW, as far as dates go, note that de Alembert had already formulated and solved the wave PDE by then (around 1757); it was not available in Huygens' time.

4. A quirk in the development of mathematics: Huygens was brilliant enough to solve the problem of finding the curvature for the guide-walls of a short pendulum of uniform oscillations. It therefore is especially intriguing that he still didn't think of including anything like a phase in his account of this principle discovered by he himself.

This curious omission is sort of like Einstein publishing original arguments on both Brownian movement (leading in part to Perin's Nobel) and the photoelectric effect (his own Nobel), both in the same year (1905), but without connecting the two, in putting forth an explanation regarding the nature of light.    

5. The Correspondence with the Action/Variational principles: Yet, it is interesting to note that Huygens had, right back then, explicitly noted the correspondence between his principle and Fermat's principle. A local, transient, propagating process had thus already been shown to produce the same results as those by a "static," global---and, IMO, very dull sort of---principles, viz., variational principles, in particular, the action principle.

Do note that this argument was put forth way before even Maupertius' times, let alone those of Euler, Lagrange, or Hamilton.

If you read modern (20th century) accounts, you are likely to come out with the impression (i) that there has been this fantastically fundamental development of the action and variational principles, beginning with Hero's or Fermat's and culminating with Hamilton's (or someone else's still later on), (ii) that the fundamental way to justify Huygens' principle to derive it within the PDE formalism, which itself is to be derived within a topological and variational formalism (with fond allusions to be made to either non-viability or outright triviality of Huygens' principle in comparing 1D vs 2D vs 3D vs nD situations, and (iii) that a rigorous proof of Huygens' principle has been established only in the second half of the 20th century---together with its enormous "limitations" or triviality, of course.  

Nothing is farther from the truth. Variational principles (and their hodge-podge equivalent of the weighted residuals etc.) are among the high favorites of modern mathematicians (esp. the mathematically inclined Indians, Russians, Frenchmen, and Americans) perhaps only because it's so hard (at least for them) to associate a neat physical mechanism with these principles. (It's the same story as with the Fourier/spectral analysis. They love anything for which a physical system (or "picture") is hard or impossible to conceive of.) However, the outright failures of such mathematicians to supply physical systematic explanation does not make variational/action/topological formalism any more fundamental than Huygens'.

It might be best to end this post with a sentence shamelessly lifted from Encyclopaedia Britannica's biographic entry on Fresnel:

"Although his work in optics received scant public recognition during his lifetime, Fresnel maintained that not even acclaim from distinguished colleagues could compare with the pleasure of discovering a theoretical truth or confirming a calculation experimentally." [emphasis mine]

BTW, have a happy Randsday!

--Ajit

[E&OE]

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