Page History

The figures commonly called Lissajous Figures were investigated by Jules Antoine Lissajous in the 1850s, but had earlier been generated and studied by Nathaniel Bowditch of Salem, Massachusetts in 1815, who built a special pendulum to generate them. Later, a Scottish Professor named Hugh Blackborn would create a similar device, which would develop into the "Harmonograph". The figures it generated would become a 19th century craze.

The basic Lissajous figure is made up of two sinusoidal oscillations along directions at right angles to each other, and when the periods of the oscillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The above figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose strings form a "Y" shape. The Lissajous figures are the paths taken by the pendulum bob during its swings, if viewed from directly above or below.

Solution

First, consider the Y support for the pendulum:

The pendulum effectively has length L₂ when swinging in the horizontal plane in and out of the page, but length L₁ along the horizontal direction in the plane of the page.

We ignore the distribution of tensions in the upper cables, and simply view the pendulum as a simple pendulum along either the plane of the drawing or perpendicular to it. In the plane perpendicular to the drawing (where the mass oscillates toward and away from the reader) the pendulum length is L₂ and the angular frequency of oscillation is given by the formula for the Simple Pendulum (see Simple Harmonic Motion.

\begin{large}\[ \omega_{2} = \sqrt{\frac{g}{L_{\rm 2}}} \]\end{large}

Along the plane lying in the page, where the mass moves left and right, the pendulum length is the shorter L₁ and the angular frequency is

\begin{large}\[ \omega_{1} = \sqrt{\frac{g}{L_{\rm 1}}} \]\end{large}

the ratio of frequencies is thus:

\begin{large}\[ \frac{\omega_{2}}{\omega_{1}}= \frac{\sqrt{\frac{g}{L_{2}}}}{\sqrt{\frac{g}{L_{1}}}} = \sqrt{\frac{L_{1}}{L_{2}}} \]\end{large}

in order to have a ratio of 1:2, one thus needs pendulum lengths of ratio 1:4. In order to get a ratio of 3:4 (as in the figure at the top of the page), the lengths must be in the ration 9:16.