Jules Antoine Lissajous (1822–1880) may not rank among the most prominent figures in science history, but his name is familiar to physics students thanks to the "Lissajous figures"—patterns formed when two perpendicular vibrations combine. Lissajous, a student at the École Normale Supérieure and later a physics professor at the Lycée Saint-Louis in Paris, made significant contributions to studying vibrations and sound.


Image source: https://en.wikipedia.org/wiki/Jules_Antoine_Lissajous

In 1855, he devised a groundbreaking optical method to visualize compound vibrations: by attaching mirrors to vibrating objects (like tuning forks) and reflecting light beams onto a screen, he created two-dimensional patterns. This innovation allowed sound to be "seen," marking a significant departure from traditional reliance on hearing alone.

                               Figure 1: Lissjous figures from adventures with Lissjous figures

Lissajous figures are created when two simple harmonic motions, represented as sinusoidal waves, interact. Their shapes depend on parameters such as amplitudes, angular frequencies, and phase differences. For equal frequencies and zero phase difference, the result might be a straight line; a phase difference of π/2 creates an ellipse, or even a circle if amplitudes are equal. Unequal frequencies generate more intricate shapes, with periodic or non-periodic motion depending on whether the frequency ratio is rational or irrational.

                                                  Figure 2: Lissajous Figure on Oscilloscope or CRO

Lissajous’ work gained recognition among contemporaries like John Tyndall and Lord Rayleigh and earned him the Lacaze Prize in 1873. His method was showcased at the 1867 Paris Universal Exposition, solidifying his contributions to acoustics and physics. Interestingly, Lissajous figures were independently discovered earlier in 1815 by American scientist Nathaniel Bowditch, who used a compound pendulum to create similar patterns. This concept inspired popular 19th-century science demonstrations, such as the harmonograph, which traced these mesmerizing figures on paper.

In a Lissajous figure, the ratio of the frequencies (ω1/ω2) can be determined visually by counting how many times the curve touches the sides (left and right) and the top and bottom of the bounding box (the rectangle formed by the maximum and minimum values of the X and Y axes). Here's how you can find the frequency ratio using this method:

Steps to Find the Ratio in the Lissajous Curve or Figure from the Number of Contacts:

1. Examine the Lissajous curve: First, check the Lissajous figure based on two perpendicular oscillations with known frequencies ω1 and ω2. You can either create this using a physical setup or a simulation software.
   
   
2. Identify the bounding box: The bounding box is formed by the maximum and minimum points along the X-axis (left-right) and the Y-axis (top-bottom).
   
   
3. Count the number of contacts with the sides:
   Left and Right sides: Count how many times the curve touches or crosses the vertical sides of the rectangle (X-axis boundaries). This tells you how many times the X-component completes one full cycle.
   
   
4. Count the number of contacts with the top and bottom:
   Top and Bottom sides: Count how many times the curve touches or crosses the horizontal sides of the rectangle (Y-axis boundaries). This tells you how many times the Y-component completes one full cycle.
   
   
5. Determine the ratio:

1. The number of contacts on the X-axis (left and right sides) will be the numerator.
2. The number of contacts on the Y-axis (top and bottom sides) will be the denominator.
   
   The ratio of these two counts gives you the ratio of the frequencies ω1/ω2.

Example:

Let’s assume we have a Lissajous figure where:

• The curve touches the left and right sides of the rectangle 3 times (this is the number of times the X-component completes a cycle).
• The curve touches the top and bottom sides of the rectangle 2 times (this is the number of times the Y-component completes a cycle).

Step 1: Count the contacts

• Left and Right (X-axis): 3 contacts
  
  
• Top and Bottom (Y-axis): 2 contacts

Step 2: Calculate the ratio

The ratio of the frequencies is:$$\frac{\text{Contacts on X-axis}}{\text{Contacts on Y-axis}} = \frac{3}{2}$$

This means the frequency ratio is 3:2.

Lissajous curve in short

In short we can say that:

A Lissajous curve forms when two simultaneous motions occur—one along the X-axis and the other along the Y-axis. These motions combine to create intricate patterns, influenced by the frequency of each movement and their phase difference (the delay between them). For instance, when both motions have the same speed, the result could be a straight diagonal line or a circle, depending on the phase. If one movement is faster than the other, it can create a figure-eight or more complex shapes.

Learn such patterns which can be created on our portable oscilloscope ZOOLARK 

Oscilloscope Function Generator - Digital Storage Oscilloscope DSO

which can be connected to a computer and can be used as a display waveforms as well.