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 /ˈlɪsəʒuː/, also known as Lissajous figure or Bowditch
curve /ˈbaʊdɪtʃ/, 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
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)
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.
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 /ˈlɪsəʒuː/, also known as Lissajous figure or Bowditch
curve /ˈbaʊdɪtʃ/, 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 (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.
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
