Chapter 12 - Waves

Chapter 12 - Waves

1. Standing waves

   • This forms when two waves with the same frequency travelling in opposite directions are superposed
     • An interference pattern is set up by superposition of waves reflected from end to end of a wave
       • A node is a position on a standing wave where there is no movement
         • An antinode has maximum movement (peak and trough)
         • The separation between to adjacent nodes (or antinodes) is equal to half a wavelength of the original wave
       • As two progressive waves are travelling in opposite directions there is no net energy transfer
   • Formation of nodes/antinodes
     • A node is formed where the displacement is zero
       • So the displacements cancel out
     • An antinode is formed when the two waves are in phase and the displacements add up to make more

2. Superposition of waves

   • When two waves of the same type meet they pass through each other
     • Where the waves superpose (overlap) they produce a single wave
       • This displacement can be found by adding the displacements of each wave
   • When two progressive waves continuously pass through each other they superpose and produce a resultant wave with displacement of waves added together
     • If two waves are in phase then max. positive displacements add together to make increased amplitude
       • This is constructive interference
         • Increase in amplitude results in an increase in intensity
           • intensity is proportional to amplitude^2
     • If two waves are anti-phase then peak of one wave and the trough of another superpose to create zero amplitude
       • This is destructive interference
         • Drop in amplitude means a drop in intensity

3. Interference

   • Thin film interface
     • The film must be 1/2 a wavelength thick
   • Coherent waves have a constant phase difference and must have the same frequency and wavelength
   • Path difference
     • Maxima is where the waves interfere constructively
       • Minima is where they interfere destructively
     • Maxima and minima are a result of two waves having travelled different distances from their sources
       • The difference of the distance travelled is the path difference

4. Young double-slit experiment

   • Two coherent waves are needed to form an interference pattern
     • At the double slit the light is coherent, as the wave passes through the slit it diffracts and overlaps the waves coming from the other slit
       • When the waves overlap they create interference patterns
         • i.e. light and dark patterns
   • Wavelength = ax/D
     • a is distance of slits
     • x is separation between fringes
     • D is distance

5. Harmonics

   • Notes and fundamental frequencies
     • So the fundamental has 2 nodes 1 antinode
       • 2nd harmonic has 3 nodes 2 antinodes etc.
     • The fundamental frequency changes how the wave behaves
       • e.g. if a string has a fundamental frequency of 200Hz
         • If the string is vibrated at 700Hz then the line will be flat and straight because it is not a multiple
       • If the string length is doubled then you half the frequency
         • If the length of the string is halved then you double its frequency
       • V = k Root T
         • T is tension in N
           • k is constant
       • Mass per unit length also affects the fundamental frequency
   • Standing waves in pipes
     • Open ended pipes
       • Anti-nodes exist at open ends
         • As air particles are free to move
       • <-- L*2 = wavelength
     • Closed ended pipes
       • Nodes exist at the closed end
         • Air particles can't move
         • Only odd numbered harmonics can be produced
           • This is impossible because you can't have a node at an open end
         • Antinode at closed end
     • Resonance is making something vibrate at its natural frequency

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