waves on a string this lecture standing waves standing waves dispersive and non-dispersive waves...

Download Waves on a string THIS LECTURE Standing waves Standing waves Dispersive and non-dispersive waves Dispersive and non-dispersive waves

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  • Slide 1
  • Waves on a string THIS LECTURE Standing waves Standing waves Dispersive and non-dispersive waves Dispersive and non-dispersive waves
  • Slide 2
  • Travelling waves x Standing waves No boundaries With boundaries Two ends fixed One end fixed
  • Slide 3
  • Standing waves Two ends fixed
  • Slide 4
  • Standing waves Two ends fixed
  • Slide 5
  • Travelling waves Each section of the string vibrates with same frequency Each section of the string vibrates with different phase = kx Each section of the string vibrates with same amplitude A No boundaries x x Standing waves Boundaries 2 2
  • Slide 6
  • Travelling waves Each section of the string vibrates with same frequency Each section of the string vibrates with different phase = kx Each section of the string vibrates with same amplitude A No boundaries x x Standing waves Boundaries Each section of the string vibrates with phase 0 or out of phase by Each section of the string vibrates with different amplitude 2Asin(k n x) Each section of the string vibrates with same frequency 2 2
  • Slide 7
  • One end fixed Standing waves
  • Slide 8
  • Superposition of standing waves
  • Slide 9
  • Relative intensities of the harmonics for different instruments
  • Slide 10
  • Playing different instruments x x
  • Slide 11
  • Dispersive and non-dispersive waves Non-dispersive wave Non-dispersive wave: it does not change shape t = 0 t > 0 Dispersive wave Dispersive wave: it changes shape t = 0 t > 0
  • Slide 12
  • Two velocities to describe the wave Group velocity, V g Velocity at which the envelope of wave peaks moves Phase velocity, V p Velocity at which successive peaks move For non-dispersive waves V g = V p For dispersive waves V g V p http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-further-dispersive.htm
  • Slide 13
  • Group velocity Phase velocity Group and phase velocity Relation between V g and V p dispersive wave If V p V g dispersive wave dispersive wave If V p = V g non-dispersive wave
  • Slide 14
  • Superposition of sinusoidal waves Sinusoidal waves 1, k 1 2, k 2 3, k 3 Superposition Wave-packet
  • Slide 15
  • Wave propagates with speed c maintaining its shape t = 0 t > 0 Wave changes its shape t = 0 t > 0 Sinusoidal waves have the same speed 1 / k 1 = c 2 / k 2 = c 3 / k 3 = c Non-dispersive wave Sinusoidal waves have different speed 1 / k 1 = c 1 2 / k 2 = c 2 3 / k 3 = c 3 Dispersive wave
  • Slide 16
  • Ideal string Real string (e.g. a piano string) V p = /k=c does not depend on k V p = /k=c depends on k c= slope Dispersion relation c1c1 c2c2 Non-dispersive wave Dispersive wave Waves on a string
  • Slide 17
  • Ideal string Dispersion relation Real string Group velocity Phase velocity
  • Slide 18
  • Problem Determine phase and group velocity for waves whose dispersion relation is described by :
  • Slide 19
  • Group velocity Phase velocity The resulting wave is given by Superposition of sinusoidal waves k k