The discussion revolves around the application of the Fourier transform in physics, specifically focusing on the physical variables that can be transformed between different domains, such as momentum-space, position-space, frequency space, and wavenumber. Participants explore the conditions and implications of these transformations.
Participants express varying views on the appropriateness and usefulness of transforming different pairs of physical variables. The discussion remains unresolved regarding which specific pairs are allowed and under what conditions.
Participants highlight the need for certain mathematical properties, such as dimensional consistency, but do not reach a consensus on the broader implications of these transformations.
Andy Resnick said:Usually, one transforms between a time-varying signal and a temporal frequency, or a spatially-varying signal and 'wavenumber' (equivalently, spatial frequency or angle).
Does that help?
A Fourier transform is a way of writing a given function as a sum of sinusoids, so I can Fourier transform just about any function that meets some minimal standards for well-behavedness. The interesting question is whether that's useful: do the sinusoids correspond to any physically interesting function? For example, Fourier transforming a sound signal tells me what frequencies have been superimposed to produce that signal... but Fourier transforming the elevation above sea level along a path is unlikely to tell me anything interesting (unless the topography happens to include some unusually evenly spaced and symmetrical hills, which would show up as a spike in the transform).Higgsono said:But why? What pair of variables are allowed and why?
Higgsono said:But why? What pair of variables are allowed and why?
Andy Resnick said:I'm not sure what you are getting at. For one thing, the product of whatever conjugate variables you choose (call them 'K' and 'L') must be dimensionless.
Higgsono said:What about the dimensions? RHS should have the same dimension as the LHS?