One of the tricky parts of dealing with an uncommon mathematical formulation is in finding appropriate analysis techniques. The Laplace's Tidal Equation solution is essentially a "sine of sine" formulation, which is rare to come across. Yet it is quite important in the realm of [Mach-Zehnder modulation](https://en.wikipedia.org/wiki/Electro-optic_modulator) (MZM), which is applied in interferometry-based measurement applications and in fiber optic communication devices.
This is from a book called Fiber Optics: Physics and Technology by Mitschke, where the sine of sine modulation is defined and often applied as a means to finely control the phase of a signal.
In a specific optical application, MZM can be used to encrypt comms such that enough phase shifts are applied as to make the signal nearly impossible to decode. This paper describes a method operating in the Fresnel domain where \\( 2 \pi\\) phase shifts are the norm.
> Abstract: A method for optical image hiding and for optical image encryption and hiding in the Fresnel domain via completely optical means is proposed, which encodes original object image into the encrypted image and then embeds it into host image in our modified Mach-Zehnder interferometer architecture. The modified Mach-Zehnder interferometer not only provides phase shifts to record complex amplitude of final encrypted object image on CCD plane but also introduces host image into reference path of the interferometer to hide it. The final encrypted object image is registered as interference patterns, which resemble a Fresnel diffraction pattern of the host image, and thus the secure information is imperceptible to unauthorized receivers. The method can simultaneously realize image encryption and image hiding at a high speed in pure optical system. The validity of the method and its robustness against some common attacks are investigated by numerical simulations and experiments.
So this is essentially what we are up against when trying to invert an LTE modulation that follows a \\( sin(A sin(t)) \\) behavior. If the value of \\(A\\) is small, there's no issue with doing a Taylor's series expansion, but if \\(A\\) gets too large, we have no way of extracting the number of \\( 2 \pi\\) phase shifts that are occurring, particularly if the internal \\( sin(t) \\) is an unknown Fourier series representation of the embedded signal. That's part of the reason it works effectively as an encryption approach, as only the sender and receiver know the key = number of phase shifts involved.
The best I have been able to do so far with an \\(arcsin()\\) inversion is to start with a candidate solution for LTE (e.g. a tropical lunar sinewave mixed with an annual impulse) and then estimate the \\( 2 \pi\\) phase shifts required to match the ENSO signal. This works fairly effectively but suffers from a biased coupling so as to render any correlation coefficient meaningless. The approach is essentially working from both ends of the following figure towards the middle.
The other aspect of MZM that I need to ponder is whether there is some commonality in the topological physics between the LTE formulation for constrained fluid dynamics and the wave modulation created via a MZ device. They are both waves, but there is likely something more fundamental involving how the "modulation of a modulation" arises.
I have been toying with the idea that it may have something to do with the physics of topological insulators from the cite below. This group is already trying to understand equatorial indices such as ENSO by analogizing from the fractional quantum Hall effect world, so perhaps there is a happy medium.
[Topological Origin of Equatorial Waves : Pierre Delplace, J. B. Marston, Antoine Venaille](https://arxiv.org/pdf/1702.07583.pdf)
> "The first Chern number is an integer that quantifies the number of phase singularities in a bundle of eigenmodes parameterized on a closed manifold. These singularities are somewhat analogous to amphidromic points (±2π phase vortices of tidal modes), but they occur in parameter space rather than in physical space"
Famous last words *"defined only up to a phase"*