Here is the math on the Chandler wobble. We start with the seasonally-modulated draconic lunar forcing. This has an envelope of a full-wave rectified signal as the moon and sun will show the greatest gravitational pull on the poles during the full northern and southern nodal excursions (i.e. the two solstices). This creates a full period of a 1/2 year.

> ![nodes](http://imageshack.com/a/img924/1315/A00Y47.png)

The effective lunisolar pull is the multiplication of that envelope with the complete cycle draconic month of $2\pi / \omega_0$ =27.2122 days. Because the full-wave rectified signal will create a large number of harmonics, the convolution in the frequency domain of the draconic period with the biannually modulated signal generates spikes at intervals of :

$ 2\omega_0, 2\omega_0-4\pi, 2\omega_0-8\pi, ... 2\omega_0-52\pi $

According to the Fourier series expansion in the figure above, the intensity of the terms will decrease left to right as $1/n^2$, that is with *decreasing* frequency. The last term shown correlates to the Chandler wobble period of 1.185 years = 432.77 days.

One would think this decrease in intensity is quite rapid, but because of the resonance condition of the Chandler wobble nutation, a compensating amplification occurs. Here is the frequency response curve of a 2nd-order resonant DiffEq, written in terms of an equivalent electrical RLC circuit.

> ![rlc](http://imageshack.com/a/img922/5107/wwEHLX.png)

So, if we choose values for RLC to give a resonance close to 433 days and with a high enough Q-value, then the diminishing amplitude of the Fourier series is amplified by the peak of the nutation response. Note that it doesn't have to match exactly to the peak, but somewhere within the halfwidth, where Q = $\frac{\omega}{\Delta\omega}$

![rlc_full](http://imageshack.com/a/img922/5184/SazoGh.png)

So we see that the original fortnightly period of 13.606 days is retained, but what also emerges is the 13th harmonic of that signal located right at the Chandler wobble period.

That's how a resonance works in the presence of a driving signal. It's not the characteristic frequency that emerges, but the forcing harmonic closest to resonance frequency. And that's how we get the value of 432.77 days for the Chandler wobble. It may not be entirely intuitive but that's the way that the math of the steady-state dynamics works out.

Alas, you won't find this explanation anywhere in the research literature, even though the value of the Chandler wobble has been known since 1891! Apparently no geophysicist will admit that a lunisolar torque can stimulate the wobble in the earth's rotation. I find that mystifying, but maybe I am missing something.

> ![nodes](http://imageshack.com/a/img924/1315/A00Y47.png)

The effective lunisolar pull is the multiplication of that envelope with the complete cycle draconic month of $2\pi / \omega_0$ =27.2122 days. Because the full-wave rectified signal will create a large number of harmonics, the convolution in the frequency domain of the draconic period with the biannually modulated signal generates spikes at intervals of :

$ 2\omega_0, 2\omega_0-4\pi, 2\omega_0-8\pi, ... 2\omega_0-52\pi $

According to the Fourier series expansion in the figure above, the intensity of the terms will decrease left to right as $1/n^2$, that is with *decreasing* frequency. The last term shown correlates to the Chandler wobble period of 1.185 years = 432.77 days.

One would think this decrease in intensity is quite rapid, but because of the resonance condition of the Chandler wobble nutation, a compensating amplification occurs. Here is the frequency response curve of a 2nd-order resonant DiffEq, written in terms of an equivalent electrical RLC circuit.

> ![rlc](http://imageshack.com/a/img922/5107/wwEHLX.png)

So, if we choose values for RLC to give a resonance close to 433 days and with a high enough Q-value, then the diminishing amplitude of the Fourier series is amplified by the peak of the nutation response. Note that it doesn't have to match exactly to the peak, but somewhere within the halfwidth, where Q = $\frac{\omega}{\Delta\omega}$

![rlc_full](http://imageshack.com/a/img922/5184/SazoGh.png)

So we see that the original fortnightly period of 13.606 days is retained, but what also emerges is the 13th harmonic of that signal located right at the Chandler wobble period.

That's how a resonance works in the presence of a driving signal. It's not the characteristic frequency that emerges, but the forcing harmonic closest to resonance frequency. And that's how we get the value of 432.77 days for the Chandler wobble. It may not be entirely intuitive but that's the way that the math of the steady-state dynamics works out.

Alas, you won't find this explanation anywhere in the research literature, even though the value of the Chandler wobble has been known since 1891! Apparently no geophysicist will admit that a lunisolar torque can stimulate the wobble in the earth's rotation. I find that mystifying, but maybe I am missing something.