spherical waves

Spherical harmonic waves

In our lecture — VIII, we worked out the form of plane harmonic traveling waves. Note that soon we will have to a. address the concept of wave profile and b. how to convert a wave profile into its corresponding time-dependent or traveling form.

But before we do that here is yet another general form of a traveling wave which we often meet in the physicists work-place. The traveling spherical wave fronts. Let us work out its details.

Spherical Waves
When a stone is dropped in water it sends out circular waves. Similarly a sphere or a glob of matter that oscillates inside of a water body would send out 3 – dimensional waves or ripples.

Sources of light wave, which we will study in great detail, in this course, to fulfill our insatiable hunger for understanding the nature of optical phenomena similarly send out oscillations which propagate radially and uniformly in all directions. These are the spherical waves and the points or regions that move out with equal phase are the wave fronts in this case, spherical in shape and known as spherical wave fronts.

We obviously need to describe the spherical wave fronts in spherical polar coordinate system, due to the spherical symmetry of problems of 3 – dimensional propagation of light waves.

Let us recall that the Laplacian in spherical polar coordinate system is given as: