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

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

Standing waves Two ends fixed

Standing waves Two ends fixed

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

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

One end fixed Standing waves

Superposition of standing waves

Relative intensities of the harmonics for different instruments

Playing different instruments x x

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

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

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

Superposition of sinusoidal waves Sinusoidal waves  1, k 1  2, k 2  3, k 3 Superposition Wave-packet

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

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

Ideal string Dispersion relation Real string Group velocity Phase velocity

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

Group velocity Phase velocity The resulting wave is given by Superposition of sinusoidal waves  k  k