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Principle of Huygens' Wavefronts

Every point on a wavefront can be considered as a source of secondary spherical wavelets, and the new wavefront is the envelope of these wavelets.
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The statement of the theorem

Let x∈R3\mathbf{x} \in \mathbb{R}^3 be the spatial coordinates and t∈Rt \in \mathbb{R} be time. Assume the wave propagation satisfies the homogeneous wave equation: (∂2∂t2−c2∇2)Φ(x,t)=0 \left( \frac{\partial^2}{\partial t^2} - c^2 \nabla^2 \right) \Phi(\mathbf{x}, t) = 0 where Φ\Phi is the phase function and cc is the wave speed. Define the initial wavefront S(t0)S(t_0) as the level set Φ(x,t0)=C0\Phi(\mathbf{x}, t_0) = C_0. The Principle of Huygens' Wavefronts asserts that the subsequent wavefront S(t)S(t) is the envelope of the secondary spherical wavelets emanating from every point x0∈S(t0)\mathbf{x}_0 \in S(t_0). Formally, the phase Φ(x,t)\Phi(\mathbf{x}, t) must satisfy the Eikonal equation in the limit of high frequency ω\omega: ∣∇Φ∣2=(ωc)2 \left| \nabla \Phi \right|^2 = \left( \frac{\omega}{c} \right)^2 The new wavefront S(t)S(t) is defined by the locus of points x\mathbf{x} such that the phase Φ(x,t)\Phi(\mathbf{x}, t) is constant, and this phase must be determined by the integral representation: Φ(x,t)=min⁡x0∈S(t0)(Φ0(x0)+1v∫t0tc2+∣∇Φ∣2dt′) \Phi(\mathbf{x}, t) = \min_{\mathbf{x}_0 \in S(t_0)} \left( \Phi_0(\mathbf{x}_0) + \frac{1}{v} \int_{t_0}^{t} \sqrt{c^2 + \left| \nabla \Phi \right|^2} dt' \right) where Φ0(x0)\Phi_0(\mathbf{x}_0) is the initial phase, and the minimum is taken over all paths connecting x0\mathbf{x}_0 to x\mathbf{x} on S(t0)S(t_0). The wavefront S(t)S(t) is thus the level set Φ(x,t)=C(t)\Phi(\mathbf{x}, t) = C(t).