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.