Superposition

• The superposition principle states that the wave function of an object can be expressed as a sum of all possible states.
• If two waves meet in phase, they add together to give a wave with an amplitude that is the sum of the of the original waves. This is called constructive interference.
• If two waves are in complete anti phase the waves cancel each other out. This is called destructive interference.
• Constructive and destructive waves

  
• Standing waves are produced by the interference of two waves, of the same type and wavelength and speed but move in opposite direction. the most common occurrence is a wave traveling in one direction meeting its reflection moving in the opposite direction
• Waves meet the pulley end of the string and they are reflected back along the string. This causes two waves of the same wavelength to arise in opposite directions, resulting in a standing wave
• the lowest frequency that the string vibrates at is called the fundamental, f1, this results in a half wavelength anti-node.
• fundamental:

  frequency = f1 and wavelength = w1 = 2L

  v = f1w1

  f1 = v/w1

  v/2L
• 2nd mode of vibration:

  frequency = f2 and w2 = L

  v = f2w2

  f2= v/w2

  v/L=2f1
• 3rd mode of vibration:

  frequency = f3 and w3 = 2L/3

  v=f3w3

  f3=v/w3

  3v/2L = 3f1
• f(nth mode) = nv/2L
• when a standing wave is created some points along the wave are always at rest, these are known as nodes. between the nodes are points of maximum displacement these are known as anti-nodes
• for standing waves between fixed points:

  standing waves are formed when two waves of the same wavelength and frequency travel in opposite directions

  there are always nodes at fixed points

  the points between nodes are vibrating in phase

  the distance between neighboring nodes is 1/2 wavelength

  the frequency and wavelength are different for different modes of vibration but the speed is the same
• Using the first and second positions of resonance:

  one frequency is used

  the fundamental mode of vibration is found

  this is the shortest length at which a loud sound is heard at (Length 1, L1)is measured.

  Using the same frequency the next shortest length that a loud sound is heard at(Length 2, L2)is measured.

  L1 = 1/4 wavelength and L2 = 3/4 wavelength

  so L2 - L1 = 1/2 wavelength

  v= fw