Until recently, I thought of Huygens' principle in the following way:
- Define a wave front as a continuous line in space along which a given wave has equal phase at any given point in time. For instance, a plane wave has equal phase at any point on a plane perpendicular to the direction of propagation, while a spherical wave has equal phase at any point on a sphere centered at the wave's origin point.
- Choose any one wave front of a given wave. Now the wave as a whole can be recovered as the superposition of spherical waves, each of those originating at one point of the wave front with its phase at the front being equal to that of the original wave.
But upon further inspection, this seems not to be true for at least two reasons, which I will explain using plane waves. In both scenarios, consider a plane wave that consists of only a single peak, after which the wave returns to the equilibrium position. Think something like a Gaußian, but where the wave isn't spread out infinitely, instead going to 0 after a finite time instead of at the limit to $\infty$.
- Now take a wave front. At some point in time, the peak will reach that front. Now according to Huygens, the wave should be recovered as a superposition of spherical waves emanating from this plane. But this superposition includes a wave propagating backwards from the plane, due to the symmetrical nature of the plane. This backwards wave does not exist in reality, though.
- Also consider this: at any point on the wave front, and at any point in time after the peak reaches said front, there will be peaks coming in from some spherical wave, resulting in a positive displacement, even though at some point the wave should be in equilibrium position. Essentially, the wave would be "smeared" in the sense that once the peak reaches a point, that point will never return to equilibrium.
My question: What is the correct way to state Huygens' principle, and how does it address the artifacts I mentioned above? Alternatively, can you point out where I went wrong in my arguments against the principle in the form stated at the beginning?
