Partial differential equations Hyperbolic equations Theorem

Huygens principle

Statement

Let $n \ge 3$ be odd and let $g \in C^{m+1}(\R^n)$ , $h \in C^{m}(\R^n)$ with $m = \frac{n+1}{2}$ . Let $u \in C^2(\R^n \times [0, \infty))$ be the solution of the wave equation $u_{tt} = \Delta u \ \text{ on } \R^n \times (0, \infty), \qquad u(\cdot, 0) = g, \quad u_t(\cdot, 0) = h.$ Let $x \in \R^n$ and $t > 0$ . Then $u(x, t)$ depends only on the values of $g$ , $h$ and their derivatives of order at most $m$ on the sphere $\set{y : \abs{y - x} = t}$ . In particular, if $g$ and $h$ vanish outside a ball $B(0, R)$ , then $u(x, t) = 0$ whenever $t > \abs{x} + R$ or $t < \abs{x} - R$ .

Remarks

For $n = 3$ , Kirchhoff's formula makes this explicit: $u(x, t) = \frac{1}{4\pi t^2}\int_{\abs{y - x} = t} \big( t\, h(y) + g(y) + \nabla g(y) \cdot (y - x) \big)\, dS(y).$ The proof in general odd dimension uses spherical means, which reduce the equation to a one-dimensional wave equation solved by d'Alembert's formula. The hypothesis that $n$ is odd and at least $3$ is necessary. For $n = 2$ , Poisson's formula gives $u(x, t)$ as an integral over the whole disc $\abs{y - x} \le t$ , and a pulse leaves a tail: $u(x, t)$ need not vanish for $t > \abs{x} + R$ . For $n = 1$ , the integral term of d'Alembert's formula also depends on $h$ in the whole interval $[x - t, x + t]$ . This sharp form is also called the strong Huygens principle.

Going deeper

For a student

The statement follows from the explicit solution formulas for the wave equation obtained by spherical means and the method of descent. In odd dimensions the formula contains only surface integrals over the sphere of radius $t$ , while in even dimensions it contains integrals over the whole ball. The weaker statement that $u(x, t)$ depends only on data in the closed ball of radius $t$ holds in all dimensions; it is finite speed of propagation. The name refers to the physical Huygens principle of wave optics, where each point of a wave front acts as a source of secondary waves.

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Wave equation concept
Huygens principle statement

Tag	`1SH2WZVZ66`
URI	`m2://1SH2WZVZ66`
Kind	statement, theorem
Writing	W0 stub
MSC	35L05, 35L15
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About	Wave equation
Uses	Wave equation