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  • 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)

1857

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
  • Died on June 24, 1880

1

x

1

-1

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

-1

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

max ernst

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

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