illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions. For physical examples of solutions to the 3D wave equation that possess angular dependence, see dipole radiation. Monochromatic spherical wave Although the word "monochromatic" is not exactly accurate, since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on plane-wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined constant angular frequency , then the transformed function has simply plane-wave solutions: or From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude drops at the rate proportional to , an example of the inverse-square law. Solution of a general initial-value problem The wave equation is linear in 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 be an arbitrary function of three independent variables, and let the spherical wave form be a delta function: that is, let be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at , and let be the radial distance from that point. Thus If is a superposition of such waves with weighting function , then the denominator is a convenience. From the definition of the delta function, may also be written as where , , and are coordinates on the unit sphere , and is the area element on . This result has the interpretation that is times the mean value of on a sphere of radius centered at : It follows that The mean value is an even function of , and hence if then These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point , given depends only on the data on the sphere of radius that is intersected by the light cone drawn backwards from . 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