Editing Wave equation (section)

===Solution of a general initial-value problem===
The wave equation is linear in {{mvar|u}} and is left unaltered by translations in space and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let {{math|''φ''(''ξ'', ''η'', ''ζ'')}} be an arbitrary function of three independent variables, and let the spherical wave form {{mvar|F}} be a [[delta function]]. Let a family of spherical waves have center at {{math|(''ξ'', ''η'', ''ζ'')}}, and let {{mvar|r}} be the radial distance from that point. Thus

<math display="block">r^2 = (x - \xi)^2 + (y - \eta)^2 + (z - \zeta)^2.</math>

If {{mvar|u}} is a superposition of such waves with weighting function {{mvar|φ}}, then
<math display="block">u(t, x, y, z) = \frac{1}{4\pi c} \iiint \varphi(\xi, \eta, \zeta) \frac{\delta(r - ct)}{r} \, d\xi \, d\eta \, d\zeta;</math>
the denominator {{math|4''πc''}} is a convenience.

From the definition of the delta function, {{mvar|u}} may also be written as
<math display="block">u(t, x, y, z) = \frac{t}{4\pi} \iint_S \varphi(x + ct\alpha, y + ct\beta, z + ct\gamma) \, d\omega,</math>
where {{mvar|α}}, {{mvar|β}}, and {{mvar|γ}} are coordinates on the unit sphere {{mvar|S}}, and {{mvar|ω}} is the area element on {{mvar|S}}. This result has the interpretation that {{math|''u''(''t'', ''x'')}} is {{mvar|t}} times the mean value of {{mvar|φ}} on a sphere of radius {{math|''ct''}} centered at {{mvar|x}}:
<math display="block">u(t, x, y, z) = t M_{ct}[\varphi].</math>

It follows that
<math display="block">u(0, x, y, z) = 0, \quad u_t(0, x, y, z) = \varphi(x, y, z).</math>

The mean value is an even function of {{mvar|t}}, and hence if
<math display="block">v(t, x, y, z) = \frac{\partial}{\partial t} \big(t M_{ct}[\varphi]\big),</math>
then
<math display="block">v(0, x, y, z) = \varphi(x, y, z), \quad v_t(0, x, y, z) = 0.</math>

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point {{mvar|P}}, given {{math|(''t'', ''x'', ''y'', ''z'')}} depends only on the data on the sphere of radius {{math|''ct''}} that is intersected by the '''[[light cone]]''' drawn backwards from {{mvar|P}}. It does ''not'' depend upon data on the interior of this sphere. Thus the interior of the sphere is a [[Petrovsky lacuna|lacuna]] for the solution. This phenomenon is called '''[[Huygens' principle]]'''. It is only true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure.{{sfn | Atiyah | Bott | Gårding | 1970 | pp=109–189}}{{sfn | Atiyah | Bott | Gårding | 1973 | pp=145–206}}