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Is there any law that says that the envelope coincides with the wavefront for any spatial dimension $n$?

If I am not wrong, Huygens principle is valid just for even spatial dimensions.

Is there some part of Huygens principle which is valid for all dimensions?

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    $\begingroup$ It's not clear what you're asking. Huygen's principle states that every point on a wavefront can be treated as a source of secondary wavelets. Are you asking for a justification of this idea? $\endgroup$
    – J. Murray
    Commented Aug 24, 2017 at 21:55
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    $\begingroup$ Not sure what you call envelope, but the wavefront at the next moment is a sum of wavelets from the current moment. This is what the principle says and it is correct for any number of dimensions. The difference is that in an odd number of dimensions this sum cancels in the reverse direction, so the wave only moves forward, but in an even number of dimensions this is not true. Shake a whip or a long rope and the wave runs away (1D). Turn off the lamp and it becomes dark (3D). Throw a rock in a lake and see that the wave expands, but also stays inside the circle and in the middle (2D). $\endgroup$
    – safesphere
    Commented Aug 25, 2017 at 15:20
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    $\begingroup$ PS. Also no sound echo in an empty space in 3D. You would always hear the echo of your voice in 2D or 4D. $\endgroup$
    – safesphere
    Commented Aug 25, 2017 at 15:27
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    $\begingroup$ My understanding is that the principle is valid for any wave anywhere, as long as the medium is linear (or, if in vacuum, the equations are linear) and therefore the waves obey the superposition principle of not interacting with each other. These include electromagnetic waves, sound, water, etc. The waves that do not obey the superposition principle include gravitational waves, turbulence on water, exceeding the speed of sound in air, saturation in transformers and tape, non linear springs or elastic materials. I am not sure if the Huygens principle fully applies to such interacting waves. $\endgroup$
    – safesphere
    Commented Aug 29, 2017 at 21:16
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    $\begingroup$ see researchgate.net/publication/… $\endgroup$
    – user45664
    Commented Jun 18, 2018 at 17:18

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