2 INTRODUCTION is the usual Laplacian in Euclidean three dimensional space. In terms of polar coordinates we have Aw = r2 —1—— ' 0— 1 9 2 M r * 3 r r 9r r 2 sin0MSm 3* r2sin20 3/2 so that if u is spherically symmetric, u = w(r, *), the wave equation becomes 32w 1 9 2 3w i r j « 92w1 1 92 , x Thus v = ru satisfies the one dimensional wave equation The general solution of this equation is given by v(rj) =f(r + t) + g(r-t) and so the general solution of the symmetric wave equation is given by t x f{r + t) g(r - t) \ / r r Here the first term represents an incoming wave and the second term represents an outgoing wave. In particular, if we take/ = 0 and g(s) = elks then wk(t,r) = e Hr-t) represents an outgoing (sinusoidal) wave of frequency k. Indeed, up to normal- izing constants, it is easy to check that eikr is the fundamental solution to the reduced wave operator A 4- k , i.e., (A + k2)Ek = CS for a suitable constant C, in fact C = —477. Thus, let y be a point in R3. The function c(y)^k(t,\x - y\)y x ¥= y, then describes a steady emission of radiation from y of frequency k. Here the complex number c(y) gives the amplitude and phase of the emitted radiation.
Previous Page Next Page