Lissajous Figures and Cathode Ray Oscilloscope
Introduction
Curves formed by the superposition of two simple harmonic motions at right angles are known as Lissajous figures. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail in 1857 by Jules Antoine Lissajous (for whom it has been named).
Two SHMs acting at right angles given by
X = A sin ( ω1 t + δ ), Y = B sin ( ω2 t )
The appearance of the figure is dependent on the ratio ω1/ω2. If ω1/ω2 is rational then a closed curve is formed. Visually, the ratio ω1/ω2 determines the number of "lobes" of the figure. For example, a ratio of 3/2 produces a figure with three vertical lobes and two horizontal lobes. Similarly, a ratio of 4/5 produces a figure with four vertical lobes and five horizontal lobes.
The ratio A/B determines the relative width-to-height ratio of the curve and δ determines the apparent rotation angle of the figure.
Superposition of two SHMs acting at right angles
- Frequency ratio 1:1
X = A Sin ( ω t + δ ), Y = B Sin ( ω t )
The resultant motion can be obtained by eliminating t. Then resultant motion is given by the equation
It represents the general equation of an ellipse bounded within a rectangle of sides 2A and 2B. Thus the resultant motion is in general elliptic.
- Frequency ratio 1:2
X = A Sin ( ω t + δ ), Y = B Sin ( 2ω t )
Then the resultant motion is given by the equation
This is an equation of fourth degree in X and in general represents a closed curve having two lobes.
- Frequency ratio 1:3
X = A Sin ( ω t + δ ), Y = B Sin ( 3ω t )
Then the resultant motion is given by
This is an equation of sixth degree in X.
Different frequency ratio
Facebook's rebrand into Meta Platforms is also a Lissajous Curve, shaped like M.
( ω1 = 1, ω2 = -2, δ = π/20 )
■ Roses
The family of curves related to the Lissajous curves is the Roses. In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves were named by the Italian mathematician Guido Grandi.
- Polar equation : r = A Cos( kθ )
Roses can also be specified using Sine function.
- Cartesian equation :
X = r Cos( θ ) = r Cos( kθ ) Cos( θ )
Y = r Sin ( θ ) = r Cos( kθ ) Sin( θ )
Roses specified by the sinusoid r = A Cos(kθ) for various rational numbered values of the angular frequency k=n/d.
■ Petals
Graphs of roses are composed of petals. A petal is the shape formed by the graph of a half-cycle of the sinusoid that specifies the rose. The shape of each petal is the same. If k is an even number, the rose has 2k petals and if k is an odd number, the rose has k petals.
■ Cathode Ray Oscilloscope
For demonstrating Lissajous figures the cathode ray oscilloscope is most suitable. It has three main parts
- Electron gun
- Deflecting plates
- Fluorescent screen
The electron gun projects a narrow beam of high-speed electrons that passes through two pairs of parallel metal plates (deflecting plates) arranged at right angles to each other. When an electric field is set up one pair of plates deflects the electrons in horizontal direction (X-plates) and the other pair of plates deflect them in vertical direction (Y-plates). The electrons finally impinge on a fluorescent screen and produce visible spots.
To display a Lissajous figure on the CRO screen two simple harmonic vibrations are at first converted into sinusoidal voltages. These voltages after proper magnification are applied to the two pairs of deflecting plates. As the electrons move under the simultaneous action of two sinusoidal electric fields at right angles to each other they trace out the Lissajous figure on the CRO screen.
■ Medical Oscilloscope
A medical oscilloscope is used to monitor brain activity with an electroencephalogram (EEG), and muscle activity with an electromyogram (EMG), in addition to monitoring the heart through an electrocardiogram (ECG).
For the experiment with Lissajous figures visit
https://www.desmos.com/calculator/hf7iwjhhup
https://datagenetics.com/blog/april22015/index.html