Here is another analysis in Fourier (frequency) space

Consider the sloshing differential equation

$ f''(t) + (a + q(t)) f(t) = h(t) $

in frequency space this would be

$ (\omega^2 + a) F(\omega)+ Q(\omega) \ast F(\omega) = H(\omega) $

The tricky part is in the second term, which is a convolution.

The Hill or Mathieu factor $q(t)$ is generally a sinusoid. The Fourier transform of a sinusoid is:

$ \mathcal{F}[q(t)] = \mathcal{F}[b \cos(\omega_o t)] = b \frac{\delta(\omega-\omega_0) + \delta(\omega+\omega_0) }{2} $

doing the convolution

$ Q(\omega) \ast F(\omega) = b \frac{\delta(\omega-\omega_0) + \delta(\omega+\omega_0) }{2} \ast F(\omega) = b \frac{F(\omega-\omega_0) + F(\omega+\omega_0) }{2} $

What this does is bifurcate or splits the spectral peaks.

$ (\omega^2+ a)F(\omega) + b \frac{F(\omega-\omega_0)+F(\omega+\omega_0)}{2} = H(\omega) $

This is an unusual looking equation because of what look like "shifted" frequency terms. Ordinarily this is is evaluated recursively, and for a delta forcing, a Mathieu function results for *F* for a *q(t)* that is a single sinusoid.

Otherwise, one can picture in you mind that for a RHS of a QBO frequency corresponding to $2\pi/2.33$ rads/year and a $\omega_0$ corresponding to the Chandler wobble of $2\pi/6.43$, then two sidebands will appear as the sum and difference of these two values.

![modulation](http://imageshack.com/a/img537/7186/mTFAd5.gif)

Note the two new spectral components at approximately 44 months and 21 months. Well, it seems that, in particular, 44 months appears in the power spectra of the ENSO metrics such as SOI and NINO34.

The physicist Lubos Motl just rediscovered this fact last week : http://motls.blogspot.co.uk/2015/04/geomagnetic-44-month-cycle-seen-in.html

And so did somebody named "willis" on the WUWT blog, as Lubos pointed out.

It is also found in the [examples documentation for Matlab](http://www.mathworks.com/help/curvefit/custom-nonlinear-enso-data-analysis.html) where they isolate the spectral components of ENSO :

> "In conclusion, Fourier analysis of the data reveals three significant cycles. The annual cycle is the strongest, but cycles with periods of approximately 44 and 22 months are also present. These cycles correspond to El Nino and the Southern Oscillation (ENSO)."

What do you know -- the spectral decomposition agrees with the model fit shown in comment #80 and explanations for the emergent significant frequencies are to be had. Now you understand why the 2.333 year = 28 month period is not observed directly in the ENSO power spectrum -- it becomes bifurcated. No wonder everyone was mystified by this missing link, as only a sloshing formulation will reveal this subtle transformation from a forcing frequency to a seemingly unrelated response frequency.

I think all the moons are in phase and the flowers are [blooming parsimoniously](http://books.google.com/books?id=iVkugqNG9dAC&pg=PA277&lpg=PA277&dq=blooming+parsimoniously).

Consider the sloshing differential equation

$ f''(t) + (a + q(t)) f(t) = h(t) $

in frequency space this would be

$ (\omega^2 + a) F(\omega)+ Q(\omega) \ast F(\omega) = H(\omega) $

The tricky part is in the second term, which is a convolution.

The Hill or Mathieu factor $q(t)$ is generally a sinusoid. The Fourier transform of a sinusoid is:

$ \mathcal{F}[q(t)] = \mathcal{F}[b \cos(\omega_o t)] = b \frac{\delta(\omega-\omega_0) + \delta(\omega+\omega_0) }{2} $

doing the convolution

$ Q(\omega) \ast F(\omega) = b \frac{\delta(\omega-\omega_0) + \delta(\omega+\omega_0) }{2} \ast F(\omega) = b \frac{F(\omega-\omega_0) + F(\omega+\omega_0) }{2} $

What this does is bifurcate or splits the spectral peaks.

$ (\omega^2+ a)F(\omega) + b \frac{F(\omega-\omega_0)+F(\omega+\omega_0)}{2} = H(\omega) $

This is an unusual looking equation because of what look like "shifted" frequency terms. Ordinarily this is is evaluated recursively, and for a delta forcing, a Mathieu function results for *F* for a *q(t)* that is a single sinusoid.

Otherwise, one can picture in you mind that for a RHS of a QBO frequency corresponding to $2\pi/2.33$ rads/year and a $\omega_0$ corresponding to the Chandler wobble of $2\pi/6.43$, then two sidebands will appear as the sum and difference of these two values.

![modulation](http://imageshack.com/a/img537/7186/mTFAd5.gif)

Note the two new spectral components at approximately 44 months and 21 months. Well, it seems that, in particular, 44 months appears in the power spectra of the ENSO metrics such as SOI and NINO34.

The physicist Lubos Motl just rediscovered this fact last week : http://motls.blogspot.co.uk/2015/04/geomagnetic-44-month-cycle-seen-in.html

And so did somebody named "willis" on the WUWT blog, as Lubos pointed out.

It is also found in the [examples documentation for Matlab](http://www.mathworks.com/help/curvefit/custom-nonlinear-enso-data-analysis.html) where they isolate the spectral components of ENSO :

> "In conclusion, Fourier analysis of the data reveals three significant cycles. The annual cycle is the strongest, but cycles with periods of approximately 44 and 22 months are also present. These cycles correspond to El Nino and the Southern Oscillation (ENSO)."

What do you know -- the spectral decomposition agrees with the model fit shown in comment #80 and explanations for the emergent significant frequencies are to be had. Now you understand why the 2.333 year = 28 month period is not observed directly in the ENSO power spectrum -- it becomes bifurcated. No wonder everyone was mystified by this missing link, as only a sloshing formulation will reveal this subtle transformation from a forcing frequency to a seemingly unrelated response frequency.

I think all the moons are in phase and the flowers are [blooming parsimoniously](http://books.google.com/books?id=iVkugqNG9dAC&pg=PA277&lpg=PA277&dq=blooming+parsimoniously).