Plane waves in three dimensions

My view of the world

I start from the idea that the four-dimensional world $(t,x,y,z)$ is filled with expanding spherical waves and with quasispherical waves that result from reflection from quasiplanar objects and refraction through quasihomogeneous materials. We rarely, if ever see in an observational data cube, an entire expanding spherical wave, but we normally have a two- or three-dimensional slice with some wavefront curvature. We analyze data subcubes that I call bricks. In any brick we see only local patches of apparent plane waves. I call them platelets. From the microview of this brick, the platelets come from the ``great random-point-generator in the sky,'' which then somehow convolves the random points with a plate-like impulse response. If we could deconvolve these platelets back to their random source points, there would be nothing left inside the brick because the energy would have gone outside. We would have destroyed the energy inside the brick. If the platelets were coin shaped, then the gradient magnitude would convert each coin to its circular rim. The plate sizes and shapes are all different and they damp with distance from their centers, as do Gaussian beams. If we observed rays instead of wavefront platelets then we might think of the world as being filled with noodles, and then. . . .

How is it possible that in a small brick we can do something realistic about deconvolving a spheroidal impulse response that is much bigger than the brick? The same way as in one dimension, where in a small time interval we can estimate the correct deconvolution filter of a long resonant signal. A three-point filter destroys a sinusoid.

The inverse filter to the expanding spherical wave might be a huge cube. Good approximations to this inverse at the brick level might be two small planes. Their time extent would be chosen to encompass the slowest waves, and their spatial extent could be two or three points, representing the idea that normally we can listen to only one person at a time, occasionally we can listen to two, and we can never listen to three people talking at the same time.

Plane waves in three dimensions