- http://mathforum.org/mathimages/index.php/Lissajous_Curve
- http://ibiblio.org/e-notes/Lis/Lissa.htm
- flashback...1815 - Nathaniel Bowditch observed vibrations years earlier using a compound pendulum...
- 1867 - Lissajous' tuning fork experiment was exhibited at the Paris Universal Exposition (with assistance from Jules Duboscq)
- 1873 - he won the Lacaze Prize for his optical observation of vibration and "for his beautiful experiments"
- superposition of two harmonic vibrations in perpendicular directions
- a dynamic pattern
- parametric equation
- x(t) = Asin(mt+c) y(t) = Bsin(nt)
- appearance of the figure is highly sensitive to the ratio m/n
- when the ratio is rational, a simple, repeating, closed pattern is formed
- when irrational, the pattern cannot close and continues forever, eventually filling the whole axis
- c determines the phase shift
some more tidbits...
About the Curve
- Interested in developing an optical method for studying vibrations
- Studied waves created by...
- tuning fork in contact water
- Sur la position des noeuds dans les lames qui vibrent transversalement (1850)
- reflecting a light beam from a mirror attached to a vibrating object
- Sur un cas particulier de stéréoscopie fourni par l'étude optique des mouvements vibratoires (1856)
- Mémoire sur l'étude optique des mouvements vibratoires (1857)
- Sur les vibrations transversales des lames élastique (1858)
Lissajous' studies
Nathaniel Bowditch
- Born on March 26, 1773 in Salem, Massachusetts
- stopped going to school at age 10 because of family financial troubles
- educated himself by reading Richard Kirwan's scientific works
- learned algebra, calculus, Latin and French
- 1795-1802: continued to study while on sea voyages
- 1804: became president of the Essex Fire and Marine Insurance Company
- developed a good reputation in academics
Jules Antoine Lissajous
Lissajous Curve
- Born on March 4, 1822 in Versailles, France
- Entered the École Normale Supérieure in 1841
- Became professor of mathematics at the Lycée Saint-Louis in 1847
- Awarded a doctorate in 1850 for thesis on vibrating bars using Chladni's sand pattern method to determine nodal positions
x(t) = Asin(mt+ c)
y(t) = Bsin(nt)
- Became rector of Academy at Chambéry and Academy at Besançon in 1874 and 1875
x
http://plumedeplombe.blogspot.com/2012/04/max-ernst-levity-and-gravity-in-his.html
http://physics.kenyon.edu/EarlyApparatus/Oscillations_and_Waves/Lissajous_Figures/Lissajous_Figures.html
http://www-history.mcs.st-andrews.ac.uk/Biographies/Lissajous.html
http://www-history.mcs.st-and.ac.uk/Biographies/Bowditch.html
http://ibiblio.org/e-notes/Lis/Lissa.htm
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2003.fall/Shiver/assignment10/assignment10.htm
http://mathforum.org/mathimages/index.php/Lissajous_Curve
http://plumedeplombe.blogspot.com/2012/04/max-ernst-levity-and-gravity-in-his.html
y
x(t) = sin(3t)
y(t) = sin(t)
bibliography
- offered chair of math/physics at Harvard and at the University of Virginia, but turned them down
- interested in the mathematics involved in celestial navigation and astronomy
- New American Practical Navigator (1802)
- paper on a meteor explosion (1807)
- three papers on orbits of comets
- 1815 - studied Lissajous curves....or Bowditch curves!
- examined the motion of a pendulum suspended from two points
- elected to the American Academy of Arts and Sciences in 1799
more on Bowditch
art
more applications
- music
- Lissajous tuning fork experiment used to standardize pitch
- art
- harmonograph
- draw beautiful figures
- laser shows
Applications of Lissajous Curves
- calculus class!
- parametric equations
- two superimposed sine functions
- find area between two curves
- physics
- electronics, audio
- waves and nodes
- oscilloscopes - analyze curve
- astronomy
- Lissajous orbit - very stable
varying ratios with fixed c = 0
the relationship between the frequency ratio and phase shift