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According to huygen's principle, every point on a wavefront maybe considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave.

Why did he mention that the wavelets spread out in all directions? We don't get a new wave front by constructing a surface tangent to the 'Spheres', rather we get a new wave front by constructing a surface tangent to the 'Hemispheres'.

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Huygens introduced his Principle in Traité de la Lumière (1690), and the diagrams he used to illustrate the Principle show the wavelets as arcs of circles (representing caps of spheres) in the 'forward direction'. He doesn't seem to discuss the extent of the wavelets in the text.

There is a (much later) refinement of Huygens's principle that modulates the amplitudes of the wavelets by a factor of $(1+\text{cos} \ \theta)$ in which $\theta$ is the angle between the propagation direction of the wavelet and the propagation direction of the wave at the point from which the wavelet originates. This suppresses the parts of the wavelet more and more the further 'backwards' they are going.

Huygens does not seem to have considered the periodic nature of waves nor the effects of interference. But remember that he was was one of the earliest workers in the field of wave propagation!

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