In Chapter 12 of the book, we describe in detail the solution to Laplace’s Tidal Equations (LTE), which were introduced in Chapter 11. Like the solution to the linear wave equation, where there are even (cosine) and odd (sine) natural responses, there are also even and odd responses for nonlinear wave equations such as the Mathieu equation, where the natural response solutions are identified as MathieuC and MathieuS. So we find that in general the mix of even and odd solutions for any modeled problem is governed by the initial conditions of the behavior along with any continuing forcing. We will describe how that applies to the LTE system next:

The same even/odd formulation holds for the simplified LTE solution of Navier-Stokes along the equator, where there are two constrained natural responses, *A sin*(*k f*(*t*)) and *B cos*(*k f*(*t*)), per standing wave mode, where *f*(*t*) is a multi-factored tidal forcing waveform. Crucially, *A* and *B* can evolve with time, intuitively guided by varying initial conditions due to changes in the forcing.

This is a highly refined solution fitted to the ENSO time series, where the *A* and *B* terms are gradually modified (while *k* is held constant) for the main low-*f* dipole and the secondary high-*f* wavenumber (likely corresponding to a tropical instability wavetrain). The lower panel is the amplitude (not power) frequency spectrum

**(click to enlarge)**

Below is the essential phase modulation applied to the terms, starting in 1880 corresponding to the light-gray-colored waveform and then transitioning in 20-year increments to fitting the current data applying the darkest-gray curve. In mathematical terms, this is modeling a convolution of the natural response with a gradual change in forced initial (or boundary) conditions.

This is not a particularly strong perturbation, yet it does improve the model fit significantly. The *A+B* phase shifting is still a fraction of both the dipole and TIW modulation cycles.