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

Post on 21-Dec-2015

216 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

• 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