I am rationalizing why solving the wave equation is so difficult. This is a 2nd-order differential equation, with a few forcing factors applied. It shouldn't be hard to solve via a multiple linear regression or by applying a Fourier or Laplace transform and solving in reciprocal space. I am sure that researchers have tried this, but perhaps became frustrated. I think the reason for the difficulty is that any Hill/Mathieu modulation on the characteristic frequency invalidates the traditional linear approaches of solving such a model. The best way is to brute force a solution by solving numerically. That's why I have leaned so heavily on using Mathematica.

I mentioned a couple of electrical engineering analogies in the last comment -- that of carrier scattering representing statistical noise and Maxwell's equation modeling a dipole. Another one is solving Schroedinger's equation in the context of a semiconductor lattice and how this relates. Fundamentally, the math behind semiconductor band formation is no different than the Hill or Mathieu equation applied to the wave equation.

But no one harbors any illusions that solving the band structure of a semiconductor lattice is a walk in the park. The periodic modulation of a lattice potential as applied to the electron will cause all sorts of odd spatial wiggles in the allowed states. This is the basis of the infamous [Bloch wave](http://en.wikipedia.org/wiki/Bloch_wave). I recognized this in the [first blog post](http://contextearth.com/2014/02/10/the-southern-oscillation-index-model/) I wrote when I started looking at ENSO.

The first iteration of finding the Hill/Mathieu modulation is reviewed below.

The equation I am solving in Mathematica is

NDSolve[{y''[t]+(CF+Hill[t])*y[t] == RHS[t]

where CF is the characteristic frequency term squared, Hill is the modulation, and RHS is the forcing.

The factors contributing to the Hill and RHS modulation are based on models of QBO, Chandler Wobble, and TSI as I described earlier. The shape of these profiles are captured as a Fourier series of sine waves with a fidelity that achieves at least 70% of a unity correlation coefficient. Those numbers are shown in panels 3 through 5 below with the correlation coefficient percentage shown to upper left of each graph. The last two panels show the modulation. Interestingly, the Hill modulation appears dominated by the TSI factor while the RHS shows the more rapid wiggles of the QBO.


Isn't applied mathematical physics fun?