What is the basis of Huygens principle?

When we were studying mechanical waves like sound waves, and waves on strings in class, we never studied Huygens' principle with these - and nor did we really derive the laws of reflection or refraction with them. We always in any case took them to travel linearly, and did not talk of any wavefronts then. But in studying geometric optics recently, we've begun by proving the laws of reflection and refraction by using Huygens' principle. In doing so we looked at the behaviour of wavefronts - not linear waves - which we had not with sound waves previously. In real life, I understand there must exist mostly wavefronts - there can hardly be any isolated linear wave disturbances, so reflection and refraction occur the way they do.

But if we were to talk of wavefronts of sound, will they still follow Huygens' principle? Alternately, will linear light waves behave like sound waves - for example, will they set up standing waves as well? And is Huygens' principle any more than a simple method of analysis of how wavefronts propagate? It must just a consequence of the wave behaviour - and must be obtainable from the wave theory. Surely it isn't an independent principle?

• Yes, all wavefronts follow this principle. TO be blunt, Huygens proposed it and as a formalism it works perfectly. BTW, his name includes the trailing "s" so watch those apostrophes :-) Commented Sep 14, 2015 at 13:43
• see researchgate.net/publication/316994209 for a geometrical derivation Commented May 19, 2017 at 22:15

The grounding of Huygens's principle is essentially the observation that the Green's Function for the Helmholtz Wave Equation is the spherical wave source

$$\psi_G(\vec{r})=\frac{e^{i\,k\,r}}{r}$$

Since approximately monotonal sound waves also fulfill the Helmholtz equation, the reasoning below, and thus Huygens's principle, applies exactly to them as well.

A simple calculation to show that one can sum up the effect of such sources on a wavefront and get approximately the right answer runs as follows. We look at a semi-infinite region $V$ with boundary $\partial V$. Only part of that boundary - an aperture $A$ - has a significantly nonzero disturbance $\psi(\vec{r})$. We wish to find $\psi$ at some position $\vec{r}_0$ away from the aperture. We consider two functions $\psi(\vec{r})$ and the Green's function $\psi_G(\vec{r}-\vec{r}_0)$ and form the vector field $\psi(\vec{r})\,\nabla \psi_G(\vec{r}-\vec{r}_0)-\psi_G(\vec{r}-\vec{r}_0)\,\nabla \psi(\vec{r})$ and then plug this little beast into Gauss's divergence theorem for a surface comprising (1) the boundary $\partial V$, which, by assumption, for this calculation is the same as simply the aperture $A$ (because the field is by assumption small elsewhere) and (2) a small sphere of radius $\epsilon$ that excises the singularity in $\psi_G(\vec{r}-\vec{r}_0)$ at $\vec{r}_0$. So we are applying the divergence theorem to the part of the volume within $\partial V$ that is outside the little "quarantining" sphere. The divergence theorem yields:

$$\int_{\partial V} (\psi(\vec{r})\,\nabla \psi_G(\vec{r}-\vec{r}_0)-\psi_G(\vec{r}-\vec{r}_0)\,\nabla \psi(\vec{r}))\cdot \hat{n} \,\mathrm{d} S = -4\,\pi\,\psi(\vec{r}_0) + \int_V (\psi(\vec{r})\,\nabla^2 \psi_G(\vec{r}-\vec{r}_0)-\psi_G(\vec{r}-\vec{r}_0)\,\nabla^2 \psi(\vec{r}))\,\mathrm{d} V + O(\epsilon) = -4\,\pi\,\psi(\vec{r}_0) + O(\epsilon)$$

where the $-4\,\pi\,\psi(\vec{r}_0)$ term (recall that $\psi(\vec{r}_0)$ is what we want to find) comes from the surface integral of the little quarantining sphere and the volume integral vanishes because both $\psi$ and $\psi_G(\vec{r}-\vec{r}_0)$ fulfill the Helmholtz equation outside the quarantining sphere. So you can see that what we're left with is that (after we take the limit as $\epsilon\to 0$):

$$\psi(\vec{r}_0) \propto \int_A (\psi(\vec{r})\,\nabla \psi_G(\vec{r}-\vec{r}_0)-\psi_G(\vec{r}-\vec{r}_0)\,\nabla \psi(\vec{r}))\cdot \hat{n} \,\mathrm{d} S \approx \int_A \frac{\exp(i\,k|\vec{r}-\vec{r}_0|)}{|\vec{r}-\vec{r}_0|}(i\,k\,\psi(\vec{r})\,\cos(\theta(\vec{r}))-\nabla \psi(\vec{r})\cdot \hat{n}) \,\mathrm{d} A$$

which, under various approximations, leads to Huygen's principle (witness the first term, which sums up the spherical wave $\exp(i\,k|\vec{r}-\vec{r}_0|)/|\vec{r}-\vec{r}_0|$ weighted by the value of the field $\psi(\vec{r})$ on the aperture. $\cos(\theta(\vec{r}))$ is responsible for the so-called obliquity factor.

Can your explanation please by simplified to high school level? I apologise, but I cannot follow the maths.

Apologies on my part then. It's a good idea to state one's level in the question: your own question is well and thoughtfully put, so you actually come across as someone more advanced than high school. Describe your level in terms of what you know rather than age as we have several teenage physicists on this site who approach graduate level.

Anyhow, you will need to be content with the explanation that Huygens's principle was postulated by Huygens as a "guess" to explain the nature of waves. He understood the linear superposition principle: that the disturbance caused by the sum of waves is the sum of the individual disturbances and thus understood, for example, that a line of point radiators could sum together to yield a plane wavefront. In the 19th century, mathematicians came up with a rigorous description of these thoughts - the Green's function method is essentially the building up of general solutions to linear equations by linear superposition of "fundamental solutions". Lo and behold, if you work out what the "fundamental solution" for the monochromatic wave equation (Helmholtz's equation) is, it turns out to be a spherical wave point emitter, exactly as Huygens had guessed.

So Huygens's principle works for any phenomenon that is described by the Helmholtz equation. This includes light, sound and water waves (in certain approximations). The Helmholtz equation is another form of the D'Alembert wave equation that is valid when the waves are roughly monochromatic or monotonal. Even in modern quantum field theory, where the relevant equations aren't wave equations, certain path integrals are still calculated by Huygens-like methods.

• Can your explanation please by simplified to high school level? I apologise, but I cannot follow the maths. Commented Sep 14, 2015 at 14:48
• the above description is an excellent summation of the origin of the Huygens' principle. It is not on the "high school" level because Maxwell's equations or the Helmholtz' equation are not high school level subjects but you asked for the origins. Huygens' principle is, at some level certainly, is high school level, but to go deeper you need calculus. Good luck! Commented Sep 14, 2015 at 21:16
• @Charles Sorry for that. Please see the updates at the end of my answer. Commented Sep 14, 2015 at 22:40
• Huygens original explanation of his principle does not use calculus and is interesting and useful: "Treatise on Light", Christiaan Huygens, Forgotten Books, 2013 {there are two 'a's in his first name} Commented Jun 3, 2016 at 21:16
• See "Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math". The derivation involves rigging secondary sources so as to yield the same wave function as the primary source(s) in the region of interest. That gives precise mathematical form to Huygens' principle. Commented Oct 15, 2022 at 9:26

I have two links to external resources.

(I am aware of the stackexchange policy that answers should not be link only. Preferably the answer should be standalone. At the very least enough information should be provided such that the reader can decide beforehand whether it is worthwhile to follow the link.)

Article by David A. B. Miller, 1991
Huygens's wave propagation principle corrected

The abstract of the article:

Huygens's principle that each point on a wave front represents a source of spherical waves is conceptually useful but is incomplet; the backward parts of the wavelets have to be neglected ad hoc, otherwise backward waves are generated. The problem is solved mathematically by Kirchhoff's rigorous integration of the wave equation, but the intuitive appeal of Huygens's simple principle is lost. I show that, by using spatiotemporal dipoles instead of spherical point sources, on can recover a simple principle of scalar wave propagation that is correct whenever the conept of a wave front is meaningful.

Forest L. Anderson, 2021
Huygens’ Principle geometric derivation and elimination of the wake and backward wave

The abstract of the article:

Huygens’ Principle (1678) implies that every point on a wave front serves as a source of secondary wavelets, and the new wave front is the tangential surface to all the secondary wavelets. But two problems arise: portions of wavelets that exist outside of the new wave front combine to form a wake. Also there are two tangential surfaces so wave fronts are propagated in both the forward and backward directions. These problems have not previously been resolved by using a geometrical theory with impulsive wavelets that are in harmony with Huygens’ geometrical description. Doing so would provide deeper understanding of and greater intuition into wave propagation, in addition to providing a new model for wave propagation analysis. The interpretation, developed here, of Huygens’ geometrical construction shows Huygens’ Principle to be correct: as for the wake, the Huygens’ wavelets disappear when combined except where they contact their common tangent surfaces, the new propagating wave fronts. As for the backward wave, a source propagates both a forward wave and a backward wave when it is stationary, but it propagates only the forward wave front when it is advancing with a speed equal to the propagation speed of the wave fronts.