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Why is the direction of propagation of a wave perpendicular to the wavefront at each point? I know this sounds intuitively true, but a simple mathematical proof would be very helpful.

If I have an arbitrary point on a wavefront, then as time passes the wavefront expands in space, and it would appear that that point is moving in space. Now the velocity of that point at any instant of time, is said to be in the direction of normal vector of the wavefront at that point. But why?

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  • $\begingroup$ Imagine if it were not perpendicular! $\endgroup$ Feb 15, 2020 at 6:55
  • $\begingroup$ Well I'm trying to think about it, but honestly i'm not able to think of the consequences of imagining if velocity were not perpendicular, maybe that would result in intersection of two wavefronts having different phasewhich is of course impossible (but that's just an intuition popping up in my mind) $\endgroup$
    – Shivansh J
    Feb 15, 2020 at 7:05

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Take a point source

point source

it is a tautology, since the wavefront is radial. One could call it a definition.

Look at the plane wave before the point source. Can there be another defintion of the direction of propagation of the wavefront, except perpendicular? (the image is from the description of the Huygens' principle)

I would consider it a matter of definition. Plane waves can mathematically build any shape.

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    $\begingroup$ One can add that for a macroscopic wave, the energy distribution follows this pattern. If you follow a specific block of energy within the wave, it propagates at right angles to the wavefront. $\endgroup$ Feb 15, 2020 at 11:52

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