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?
Best Answer
The link you gave set me searching for a more detailed answer, and I learnt an interesting fact:
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.