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There have already been a lot of questions on this site on diffraction but I still believe this one might be slightly different. In electromagnetic waves, diffraction and any other phenomenon of wave propagation can be solved by Huygen's principle, a geometric construction that asks us to consider all points on a wavefront as secondary sources of wavefront.
A justification of this treatment is provided by Feynman:-
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I understood this completely. But then diffraction is a very general phenomenon. What if we talk about mechanical waves (sound waves) then this treatment by superposing the fields produced by opaque and a hypothetical plug is no longer valid, but the diffraction details are similar. Why then on an intuitive level is there diffraction, details of which can be worked out by considering secondary sources?

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I'd likely to further dmckee's answer by answering the OP's follow-up question:

Can you please explain why considering secondary sources as in Huygen's principle is justified, i.e., why do we get correct results by assuming secondary sources when there are actually no sources other than the original source? As Feynman explains in case of electromagnetic waves, it is because the diffracted wave is equivalent to the superposition of electric fields of a hypothetical plug containing several independent sources.

Huygen's principle is actually a fairly fundamental property of solutions of the Helmholtz equation $(\nabla^2 + k^2)\psi = 0$ or the D'Alembert wave equation $(c^2\nabla^2 - \partial_t^2)\psi = 0$. For these equations the Green's function is a spherical wave diverging from the source. All "physically reasonable" solutions (given reasonable physical assumptions such as the Sommerfeld Radiation Condition) in freespace regions away from the sources can be built up by linear superposition from a system of these sources outside the region under consideration. Already this is sounding like Huygen's principle, but one can go further and, with this prototypical solution and the linear superposition principle together with Gauss's divergence theorem, show that waves can be approximately thought of as arising from a distributed set of these "building block" spherical sources spread over the wavefront: this result leads to the Kirchoff Diffraction Integral, thence to various statements of Huygens's Principle.

This treatment is worked through in detail §8.3 and §8.4 of Born and Wolf, "Principles of Optics" or in Hecht, "Optics", which I don't have before me at the moment.

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Huygen's principle is valid for mechanical waves because the disturbed part of the medium is connected to the rest of the medium in every direction. In fluids the high pressure region pushes in every direction. In solids, the displaced region is connected by roughly elastic, intermolecular forces in all directions (in a isotropic/amorphous material or in all lattice directions in a ordered material).

So the disturbance can be relieved by doing work in every direction and is relieved in this way.

And just as in the electromagnetic case, it is coherence that causes the overall effect to take on long-range structure.

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Can you please explain why considering secondary sources as in Huygen's principle is justified, i.e., why do we get correct results by assuming secondary sources when there are actually no sources other than the original source? As Feynman explains in case of electromagnetic waves, it is because the diffracted wave is equivalent to the superposition of electric fields of a hypothetical plug containing several independent sources. –  Satwik Pasani Jan 17 at 17:00
    
@SatwikPasani It's a ffundamental property of the Wave equation or Helmholtz's equation: see my answer. –  WetSavannaAnimal aka Rod Vance Jan 19 at 3:07

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