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Huygens' principle states every point on the current wavefront acts as a source of secondary spherical waves. These secondary waves propagate outwards, in the forward direction, and a common tangent (an envelope) to all these waves constitute the new wavefront..but do these secondary wavefronts act as a source of another secondary waves?

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Sure, a wave $u(t,x,y,z)$ propagates along both time and space. For every time slice, there is a wavefront and it's the source for the next wavefront in the next time slice.

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