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It was rather strange for me that I used Huygens–Fresnel principle in optics a lot, still not knowing anything about its origin. So I tried to look it up in some books, including Born-Wolf, and did not succeed. In Wikipedia there is a reference which says that it has to do with nothing but initial value problem for the wave equation. Still, there is another one (9), which says that it is a consequence of space homogeneity.

I would be very grateful if somebody tells me if there is any physical reason at all, as far as the mathematical one seems to be right.

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The Huygens-Fresnel principle comes from looking at the Green's function for the wave equation.

If we look at the response of free space to a point disturbance, it turns out to be a spherical propagating wave, hence Huygens principle.

Mathematically, we solve for the Green's function $G$ in the following equation.

$$\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)G(\mathbf{x},t;\mathbf{x}_0,t_0) = \delta(\mathbf{x}-\mathbf{x}_0) \delta(t-t_0)$$

With some math, this can be shown to be a spherically propagating wave.

The idea behind Green's functions is that once you know $G$, you can write any wave in roughly the following way:

$$u(\mathbf{x},t) \sim \int d\mathbf{x}^{\prime} G(\mathbf{x},t;\mathbf{x}^{\prime},0)u(\mathbf{x}^{\prime},0)$$

This is why, given the initial distribution of the wave $u(\mathbf{x}^{\prime},0)$, you can find the wave at a later time by propagating each point of the wave out with the spherical wave Green's function. For the exact form for the wave, see here https://math.dartmouth.edu/~ahb/notes/waveequation.pdf

Edit: As pointed out in the comments, there is a bit of subtlety going on with getting the homogenous solution from the inhomogenous Green's function, but looking at OP's question, that is more of a technicality. Using the inhomogenous Green's function for the inhomogenous case and vice-versa is justified by Duhamel's principle for Partial Differential Equations

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    $\begingroup$ The Green's function is a solution of the inhomogeneous wave equation with the delta function as inhomogeneity and can be used to find the solutions of the inhomogeneous wave equation with general inhomogeneities. How do you arrive at your equation where the Green's function relates the time dependent solution of the homogeneous wave equation to an initial solution of the homogeneous wave equation? Did you derive this yourself? Or can you give a reference where this is derived? $\endgroup$
    – freecharly
    Apr 7, 2017 at 1:51
  • $\begingroup$ Good catch, technically you need the initial wave values and first derivatives as well. I'll fix the equation in my answer. You can read the math math.dartmouth.edu/~ahb/notes/waveequation.pd $\endgroup$
    – KF Gauss
    Apr 7, 2017 at 5:47
  • $\begingroup$ Your link resulted in "The requested URL /~ahb/notes/waveequation.pd was not found on this server." $\endgroup$
    – freecharly
    Apr 7, 2017 at 18:48
  • $\begingroup$ I think the "f" in ".pdf" got cut off. The link in the answer works. $\endgroup$
    – KF Gauss
    Apr 8, 2017 at 1:23
  • $\begingroup$ Yes, we know the Green's function propagates the wave from all points. It still is not explicitly shown the usual picture of the adding and cancellations that form the wavefront a small time later. $\endgroup$
    – Bob Bee
    Apr 9, 2017 at 19:13

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