My World: Lissajous

Introduction

Before we find out how the lissajous curve is formed, we should find out what that curve lissajous. In mathematics, a Lissajous curve is an lɪsəʒu//, also known as Lissajous figure or Bowditch curve/baʊdɪtʃ/, is a region of the graph of a parametric equation system

describing a complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch also in 1815, and later in more detail by Jules Antoine Lissajous in 1857

Lissajous figures (or Lissajous curves) are produced in two dimensions when the x and y coordinates are given by two sine waves, which may have any amplitude, frequency and phase. This is a support page to the multimedia chapter Interference and Consonance in the volume Waves and Sound, which introduces interactions between sine waves. Other pages support the topics beats, Tartini tones, linear and nonlinear superposition and phasor addition.

Lissajous curves can also be generated using an oscilloscope(as illustrated). An octopus circuit can be used to demonstrate the waveform images on an oscilloscope. Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure.

For more details you can read this task

Theorical in Physic

In mathematics, a Lissajous curve /ˈ l ɪ s ə ʒ uː /, also known as Lissajous figure or Bowditch curve /ˈ b aʊ d ɪ tʃ /, is the graph of a system of parametric equations

which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857.

The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/2). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.

Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.

Lissajous figures where a = 1, b = N (N is a natural number) and

are Chebyshev polynomials of the first kind of degree N. (Wikipedia.org)

Lissajous figures are patterns generated by the junction of a pair of sinusoidal waves with axes that are perpendicular to one another. The figures, sometimes called Bowditch curves, were first described in the early 1800s by American mathematician Nathaniel Bowditch, who experimented with a pendulum undulating around perpendicular axes at the same time. Later that century, French mathematician Jules-Antoine Lissajous began his own study of the figures, which he produced in multiple ways. One of his methods involved creating vibrations in a mirror with sounds of different frequencies, then reflecting light from that mirror to produce various patterns. More commonly today, Lissajous figures are generated with an oscilloscope, a type of cathode ray tube that provides a picture of electrical signals in the form of a graph. (Magnet Lab)

This Lissajous figure has 400 Hz on the horziontal (y₁ or x) axis and 500.1 Hz on the vertical (y₂ or y) axis. (This nomenclature is for consistency with the other web pages involving combinations of sine waves.) The slightly incommensurate ratio is chosen so that the phase changes gradually, as can be heard by the constructive and destructive interference in one of the sound files below.

By tracing the curve mentally, one can observe that, in four horizontal oscillations, there are five vertical oscillations. One can also determine this ratio by counting the number of intersections with the two axes (UNSW)

pict of lissajous curve

phase shift of lissajuous curve

Conclusion

In mathematics, a Lissajous curve /ˈ l ɪ s ə ʒ uː /, also known as Lissajous figure or Bowditch curve /ˈ b aʊ d ɪ tʃ /, is the graph of a system of parametric equations

which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857.

Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.

Lissajous figures where a = 1, b = N (N is a natural number) and

are Chebyshev polynomials of the first kind of degree N.

the Oscilloscope needs to be ' triggered ' to start a trace using the first channel (y1).

If not, use the trigger ' line ': new trail triggered in phase with a nominal 50 Hz power supply. This often varies slightly from the 50 Hz, so it's not exactly sync with for example 400 Hz signal from the computer.

Reference

http://en.wikipedia.org/wiki/Lissajous_curve

http://www.magnet.fsu.edu/education/tutorials/java/lissajous/

http://www.animations.physics.unsw.edu.au/jw/Lissajous.htm