Waves 2

• Stationary wave —> formed when two progressive waves, each with the same frequency and wavelength and moving in opposite directions, interfere with one another
• Node —> points on an object where the displacement is zero
• Antinodes —> points on a stationary wave with maximum amplitude. They oscillate from the furthest point upwards to the furthest point downwards.
• Superposition --> At the point where 2 or more waves meet the resultant displacement is equal to the sum of the individual displacement (of the waves)
• • As displacement is a vector quantity, when the displacement of two waves are added together the resultant can be greater or smaller that the individual displacements of each wave
• Interference --> when 2 progressive waves continually pass through each other they superpose & produce a resultant wave with a displacement equal to the sum of the individual displacements from the 2 waves
• Constructive interference --> If the 2 waves are in phase then the maximum positive displacements from each wave line up, creating a resultant displacement with increased amplitude
• As intensity is directly proportional to (amplitude)^2, the increase in amplitude resulting from constructive interference increases the intensity: sound waves are louder, and light is brighter
• Destructive interference --> If 2 progressive waves are in antiphase, then the maximum positive displacement from one wave lines up with the maximum negative displacement from the other, and the resultant displacement is smaller than for each individual wave.
• • If the waves have the same amplitude the resultant wave will have zero amplitude - it is cancelled out completely
• Coherence --> waves emitted from 2 sources having a constant phase difference. In order to be coherent the two waves must have the same frequency
• • Filament lamps emit light of a range of different frequencies and ever changing phase difference between different waves. Therefore not possible to produce stable interference patterns using 2 filament lamps
• • At a maximum the waves interfere constructively
  • At a minimum they interfere destructively
• Path difference --> the difference in distance that 2 waves must travel from their sources to a given point
• wavelength = ( source/slit separation x fringe separation) / distance between the source & screen
• Bright fringe --> complete number of wavelength path difference, constructive interference
• Dark fringe --> odd number of half wavelengths path difference, destructive interference
• Coherent light source:

  Works by the interference of 2 light sources which must be coherent meaning:

  • Constant phase difference (which could be zero)
  • Same frequency
  • Approximately the same amplitude to show interference
  • Same frequency & monochromatic light
• Young's double slit experiment:

  • 2 coherent waves are needed to form an interference pattern
  • Used a monochromatic source of light & a narrow single slit to diffract the light
  • Light diffracting from the single slit arrives at the double slit in phase
  • Then diffracts again from the double slit
  • Each slit acts s a source of coherent waves, which spread from each slit, overlapping & forming an interference pattern that can be seen on a screen as alternating bright & dark regions called fringes
• Diffraction gratings:

  For there to be constructive interference the path difference:-

  • d x Sin theta = n x wavelength
  • Where n is the order of diffraction
• Diffraction gratings:

  • As the wavelength gets bigger, sin theta gets bigger, meaning theta gets bigger and the bright fringes spread out
  • As d gets smaller, sin theta gets bigger, meaning theta gets bigger and the bright fringes spread out
  • Sin theta can't be greater than 1 and so if you calculate that it is that order doesn't exist
• • The separation between 2 adjacent nodes (or antinodes) is equal to half the wavelength of the original progressive wave & the frequency is the same as that of the original wave
  • As the progressive waves are travelling in opposite directions, there is no net energy transfer by a stationary wave, unlike a single progressive wave
• Harmonics:

  • Each string has a fundamental mode of vibration
  • The frequency of this vibration is the fundamental frequency, and depends on the string's mass, tension and length
• Stationary waves on strings:

  • If a string is stretched between two fixed points, these points act as nodes
  • When the string is plucked a progressive wave travels along the string & reflects off its end
  • This creates 2 progressive waves travelling in opposite directions that then form a stationary wave
• Harmonics and wavelength:

  • The fundamental frequency, f0, is the minimum frequency of a stationary wave for a string
  • Along with this fundamental mode of vibration, the string can form other stationary waves called harmonics at higher frequencies
  • For a given string at a fixed tension, the speed of progressive waves along the string is constant
  • At a frequency of 2f0, the wavelength is half the wavelength at f0
• Stationary waves in sound:

  • Sound waves reflected off a surface can form a stationary wave
  • The original wave & the reflected wave travel in opposite directions and superpose
  • Stationary sound waves can also be made in tubes by making the air column inside the tube vibrate at frequencies related to the length of the tube
  • The stationary wave formed depends on whether the ends of the tube are open or closed
• Stationary waves in a tube closed at one end:

  • In order for a stationary wave to form in a tube closed at one end there must be an antinode at the open end & a node at the closed end.
  • The air at the closed end can't move & so must form a node
  • At the open end, the oscillations of the air are at their greatest amplitude, so it must be an antinode
  • The fundamental mode of vibration simply has a node at the base and an antinode at the open end
• Stationary waves in open tubes:

  • A tube open at both ends must have an antinode at each end in order to form a stationary wave
  • Unlike a tube closed at one end, harmonics at all integer multiples of the fundamental frequency (f0, 2f0, 3f0 ...) are possible in an open tube