So we need only two dimensions to describe the wave. In addition, we can swap the dimension in which the wave is at light speed with the oscillating dimension and the wave equation remains the same. We will differentiate between space and time dimensions, and we can reduce our descriptions to three unique orthogonal pairs,
\((t,x)\), \((t,t)\) and \((x,x)\). All three are sinusoidal with speed \(c\) along an orthogonal dimension.
And thanks to Fourier Transforms, the sum of any combinations, of any numbers of these three can be described analytically using integrable functions.
This is provided we describe waves as,
\(\cfrac{\partial^2\psi}{\partial\,t^2}=c^2\cfrac{\partial^2\psi}{\partial\,x^2}\) --- (*)
It is not wavelets but good old Fourier, that fully describe physics.
However, as we have noted previously, as we swap a time dimension for a space dimension, the wave becomes a standing wave wrap around a center; a spherical wave. A wave wrapped into a particle. This information is lost when we think space and time dimensions are interchangeable as we develop the equivalent view above. When a space dimension is swapped for a time dimension, the wave presents itself differently in space. And since each time dimensions has characteristic energy, electric, temperature or gravity, all time dimensions must be differentiated to retained their characteristics.
As such the wave equation (*) by itself is inadequate. The description of a particle is supplemented with information of which dimension carries oscillating energy, which time dimension the particle exist in and, which dimension it is at light speed.
Have a nice day.
