Wave equation
Solution of a general initial-value problem

The wave equation is linear in 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 φ (ξ, η, ζ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta function. Let a family of spherical waves have center at (ξ, η, ζ), and let r be the radial distance from that point. Thus

r 2 = (x − ξ) 2 + (y − η) 2 + (z − ζ) 2. {\displaystyle r^{2}=(x-\xi)^{2}+(y-\eta)^{2}+(z-\zeta)^{2}.}

If u is a superposition of such waves with weighting function φ, then u (t, x, y, z) = 1 4 π c ∭ φ (ξ, η, ζ) δ (r − c t) r d ξ d η d ζ; {\displaystyle 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 ;} the denominator 4 πc is a convenience.

From the definition of the delta function, u may also be written as u (t, x, y, z) = t 4 π ∬ S φ (x + c t α, y + c t β, z + c t γ) d ω, {\displaystyle u(t,x,y,z)={\frac {t}{4\pi }}\iint _{S}\varphi (x+ct\alpha,y+ct\beta,z+ct\gamma)\,d\omega,} where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u (t, x) is t times the mean value of φ on a sphere of radius ct centered at x: u (t, x, y, z) = t M c t. {\displaystyle u(t,x,y,z)=tM_{ct}.}

It follows that u (0, x, y, z) = 0, u t (0, x, y, z) = ϕ (x, y, z). {\displaystyle u(0,x,y,z)=0,\quad u_{t}(0,x,y,z)=\phi (x,y,z).}

The mean value is an even function of t, and hence if v (t, x, y, z) = ∂ ∂ t (t M c t), {\displaystyle v(t,x,y,z)={\frac {\partial }{\partial t}}{\big (}tM_{ct}{\big)},} then v (0, x, y, z) = ψ (x, y, z), v t (0, x, y, z) = 0. {\displaystyle v(0,x,y,z)=\psi (x,y,z),\quad v_{t}(0,x,y,z)=0.}

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t, x, y, z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a 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.
combination:two_responses