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I was reading French's Vibrations & Waves where he discusses Huygens-Frensel Principle.

The principle talks about how secondary sources give rise to secondary wavelets to form the displaced wavefront. However, any secondary source can form two wavelets one moving forward & other towards the original source as pointed by French:

[...] The Huygen's construction would define two subsequent wavefronts , not one. In addition to a new wavefront farther away from the source, there would be another one corresponding to a wavefront back toward the surface, But we know this does not happen.

Then he writes:

If the Huygens way of visualizing wave propagation is to be acceptable, it must introduce the unidirectional property of a traveling wave. This can be achieved by requiring that the disturbance starting out from a given point in the medium at a given instant is not equal in all directions. Specifically, if $O$ is the true original source ,& $S$ is the origin of Huygens wavelet, & $P$ is the point at which the disturbance is being recorded, then the effect at $P$ due to the region near $S$is a function of $f(\theta)$ of the angle $\theta$ between $OS$ & $SP$.

I am not understanding how his reasoning actually averts the possiblity of formation of back-waves. Can anyone help me visualise what he is talking? How does his reasoning maintain the unidirectional propagation of waves? Can anyone explain his argument?

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    $\begingroup$ Without having the original in front of me it's hard to give an authoritative answer. Does he appear to be claiming "any" $f(\theta)$ will satisfy this, or just "some" function? Because in the latter case you could simply posit that $f=0$ for $|\theta|>\pi/2$. $\endgroup$ – Floris Aug 28 '15 at 19:19
  • $\begingroup$ I thought the Huygens principle was out of fashion because it is somewhat vague and non rigorous, leading to problems as the above. $\endgroup$ – Sebastian Riese Aug 28 '15 at 23:15
  • $\begingroup$ @Floris: You can check books.google.co.in/… $\endgroup$ – user36790 Aug 29 '15 at 3:34
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The link you gave set me searching for a more detailed answer, and I learnt an interesting fact:

Huygens' construction works in 1 and 3 dimensions, BUT NOT IN TWO!

The theory behind this is was first derived by Fresnel and later by Kirchoff - the math is explained in detail in this article. It all comes down to the fact that the wave equation for waves propagating from a point can be written as

$$\frac{\partial^2 \phi}{\partial r^2}-\frac{(n-1)(n-3)}{4r^2}\phi = \frac{\partial^2\phi}{\partial t^2}$$

where $n$ is the dimensionality. For $n=1$ or $n=3$ the second term on the left vanishes, and the expression becomes like that of a one-dimensional wave propagating outwards. For $n=2$, like for the ripples on a pond, there is in fact another term that results in waves "traveling backwards". In that case, the usual Huygens construction does not (quite) work.

Quite subtle stuff - and the fact that the mathematical treatment happened more than 100 years after Huygens' initial publication of his ideas (1679: publication of "Traité de la Lumière"; Fresnel was born in 1788) is something that I had not previously appreciated. The casual treatment it is given in French's book makes sense in the context of a big book covering a lot of ground quickly - but I appreciate you brought this question to my attention; it's more interesting than it looked at first sight.

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    $\begingroup$ I'm not sure if this answers the question. The equation you wrote says that if $n=2$ there are infinite ripples (Bessel functions). But aren't backward solutions allowed in $n=3$? If you replace $\phi(r-t)$ with $\phi(r+t)$ it still obeys the equation. And in the article you linked "Nevertheless (and despite statements to the contrary in the literature), it does not actually resolve the question about "backward" propagation of waves, because Maxwell's equations themselves theoretically allow for advanced as well as retarded potentials." $\endgroup$ – jinawee Nov 21 '15 at 11:22
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Huygens' principle of taking the convex hull of the spherical waves emitted by all points on a wavefront indeed gives you two new wavefronts: One "forward" and one "backward". Those are both meaningful: The "forward" wavefront corresponds to the retarded solution of the wave equation, the "backward" wavefront corresponds to the advanced solution of the wave equation.

Both those wavefronts have physical meaning: The "backward" one is where the wave was (if it wasn't emitted this instant - this is crucial), the "forward" one is where the wave will be, since the retarded Green's function propagates solutions forward in time and the advanced Green's functions propagates them backward in time. This you must simply deduce from the wave equation itself, it does not follow from Huygens principle.

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I'd like to add to ACuriousMind's Answer. His/Her answer emphasizes that Maxwell's equations and other equations that give rise to the Huygens spherical wave kernel in their Green's functions are inherently acausal and we must force causality by hand through the appropriate boundary conditions.

In antenna problems, we simply discard the advanced wave part of the solution to force a causal relationship between source current / voltages and the field.

In Kirchoff diffraction theory, one adds an obliquity factor $(1+\cos\theta)/2$ to kill the advanced wave. Kirchoff theory "derives"the obliquity factor from equivalent boundary conditions on a diffracting slit, but the boundary conditions are equivalent to a forcing of causality.

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Addressing your final question:

"I am not understanding how his reasoning actually averts the possibility of formation of back-waves. Can anyone help me visualise what he is talking? How does his reasoning maintain the unidirectional propagation of waves? Can anyone explain his argument?" where 'his' refers to Huygens:

Refer to "Treatise On Light", Christiaan Huygens, 1678 (Forgotten Books 2012). In it Huygens deals with the backward wave on page 21: ".., but it will thereby merely generate backwards towards the luminous point some partial waves incapable of causing light, and not a wave compounded of many as CE was". Here CE is the expanding wave front (the envelope of the wavelets) shown in the figure on page 19. Huygens considered the propagating medium to be the ether which consisted of particles of etherial matter, and apparently the propagation took place via the particles colliding with each other. I think his reasoning on the backward waves was that the particle motion was primarily outward, away from the luminous point (the motion's point of origin), and so any particle motion in the inward direction would be 'feeble'. (Could conduct an experiment on a pool table to verify this.) Finally as to your question, this is how he 'averts the possibility of formation of back-waves'.

A major resource on Huygens Principle as it has developed since his time is "The Mathematical Theory of Huygens' Principle", Baker and Copson, Chelsea 1987. Here Poissons Formula, page 14 thro 20, is used to show that no backward wave will be formed. Later using results from Fresnel and Helmholtz he shows the same for monochromatic waves, pages 20 - 28. However, It is perhaps questionable whether as yet the final work on this, or other Huygens Principle issues, has been done.

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According to Voigt and Kirchoff, the contribution of wavelet depends on an angle ($Y$) that it forms with the normal of wavelet is $\frac12(1+\cos Y)$. We know the wavelet forms $180$ with the normal, so its contribution is zero, as per mathematical formula given by Voigt and Kirchoff.

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