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My book references this principle but does not prove it. Could someone give the proof? The context for the principle in in 3 Space - 1 Time dimensions for the wave equation.

$$\int_{B(0,c(T-s))} (u_t^2 + c^2|\bigtriangledown u|^2 )(x,s)dx \leq \int_{B(0,cT)} (u_t^2 + c^2|\bigtriangledown u|^2 )(x,0)dx$$

For $B(0,R)$ representing a ball centered at the origin with radius $R$. The book also talks about a physical interpretation where a 4-D divergence theorem is applied to some "energy" quantity over the frustum of a light cone subtended on the two 3-D balls at time 0 and s.

Any help would be greatly appreciated.

Masacroso
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joe smith
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1 Answers1

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For a geometric derivation see https://www.researchgate.net/publication/316994209

This paper proves Huygens Principle (excluding the backward wave) and shows geometrically why plane and spherical waves propagate without a wake. Basically an impulsive sphere and an infinite impulsive plane are examined. The wave field as a function of time at an external point in space is computed, taking into account 1/r type spherical spreading. Then the time derivative of the wave field is taken to get the wave fronts. The derived wave fronts consist of Dirac Deltas corresponding to the start (and end for the sphere) of the wave fields. (The wave field in between is constant so the derivative there is zero.) So for those sources there are no wakes and the waves propagate cleanly.

user45664
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