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Waves on a string

THIS LECTURE. Waves on a string. Standing waves. Dispersive and non-dispersive waves. Travelling waves. No boundaries. x. With boundaries. Standing waves. Two ends fixed. One end fixed. Standing waves. Two ends fixed. Standing waves. Two ends fixed. x. x. Travelling waves.

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Waves on a string

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  1. THIS LECTURE Waves on a string • Standing waves • Dispersive and non-dispersive waves

  2. Travelling waves No boundaries x With boundaries Standing waves Two ends fixed One end fixed

  3. Standing waves Two ends fixed

  4. Standing waves Two ends fixed

  5. x x Travelling waves Standing waves Boundaries No boundaries 2 2 Each section of the string vibrates with same frequency w Each section of the string vibrates with same amplitude A Each section of the string vibrates with different phase f = kx

  6. x x Travelling waves Standing waves Boundaries No boundaries 2 2 Each section of the string vibrates with same frequency w Each section of the string vibrates with same frequency w Each section of the string vibrates with same amplitude A Each section of the string vibrates with different amplitude 2Asin(knx) Each section of the string vibrates with different phase f = kx Each section of the string vibrates with phase 0 or out of phase by p

  7. Standing waves One end fixed

  8. Superposition of standing waves

  9. Relative intensities of the harmonics for different instruments

  10. Playing different instruments x x

  11. Dispersive wave: it changes shape t = 0 t > 0 Dispersive and non-dispersive waves Non-dispersive wave: it does not change shape t = 0 t > 0

  12. Two velocities to describe the wave Group velocity, Vg Velocity at which the envelope of wave peaks moves Phase velocity, Vp Velocity at which successive peaks move For non-dispersive waves Vg =Vp For dispersive waves VgVp http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-further-dispersive.htm

  13. Group velocity If VpVg  dispersive wave If Vp=Vg  non-dispersive wave Group and phase velocity Phase velocity Relation between Vg and Vp

  14. Superposition Wave-packet Superposition of sinusoidal waves Sinusoidal waves w1, k1 w2, k2 w3, k3

  15. Dispersive wave Sinusoidal waves have different speed w1/ k1= c1 w2/ k2= c2 w3/ k3= c3 Non-dispersive wave Sinusoidal waves have the same speed Wave propagates with speed c maintaining its shape w1/ k1= c w2/ k2= c t = 0 w3/ k3= c t > 0 Wave changes its shape t = 0 t > 0

  16. Dispersion relation c= slope Real string (e.g. a piano string) c2 c1 Waves on a string Ideal string Vp=w/k=c does not depend on k Non-dispersive wave Vp=w/k=c depends on k Dispersive wave

  17. Dispersion relation Group velocity Phase velocity Real string Ideal string

  18. Problem Determine phase and group velocity for waves whose dispersion relation is described by :

  19. The resulting wave is given by Group velocity 2p/Dk Phase velocity 2p/k Superposition of sinusoidal waves

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