Fourier Transforms and Waves: In Four Lectures

When earth material properties are constant in any of the cartesian variables then it is useful to Fourier transform (FT) that variable.
In seismology, the earth does not change with time (the ocean does!) so for the earth, we can generally gain by Fourier transforming the time axis thereby converting time-dependent differential equations (hard) to algebraic equations (easier) in frequency (temporal frequency).
In seismology, the earth generally changes rather strongly with depth, so we cannot usefully Fourier transform the depth axis and we are stuck with differential equations in . On the other hand, we can model a layered earth where each layer has material properties that are constant in . Then we get analytic solutions in layers and we need to patch them together.
Thirty years ago, computers were so weak that we always Fourier transformed the coordinates. That meant that their analyses were limited to earth models in which velocity was horizontally layered. Today we still often Fourier transform but not, so we reduce the partial differential equations of physics to ordinary differential equations (ODEs). A big advantage of knowing FT theory is that it enables us to visualize physical behavior without us needing to use a computer.