Wave equation
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 ω, then the transformed function ru (r, t) has simply plane-wave solutions: r u (r, t) = A e i (ω t ± k r), {\displaystyle ru(r,t)=Ae^{i(\omega t\pm kr)},} or u (r, t) = A r e i (ω t ± k r). {\displaystyle u(r,t)={\frac {A}{r}}e^{i(\omega t\pm kr)}.}

From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude I = | u (r, t) | 2 = | A | 2 r 2, {\displaystyle I=|u(r,t)|^{2}={\frac {|A|^{2}}{r^{2}}},} drops at the rate proportional to 1/ r, an example of the inverse-square law.
keywords:letter_frequency