Waves 2

• Phase is the position of a point on a wave cycle relative to a reference point at the same phase.
• Phase difference is the amount one wave lags behind another. They are measured in radians. (π=180 degree)
• If a part of the wave continually has the same displacement, it is in phase. It will have a phase difference of 0 or multiples of 2π radians.
• One complete cycle of a wave is 360 degrees. (2π radians)
• Antiphase waves will have a phase difference of π or odd multiples of π radians.
• Phase Difference = (distance / wavelenght) X 2 π Radians
• Superposition happens when two or more waves pass through each other. The principle of superposition states that when two or more waves cross, the resultant displacement equals the vector sum of the individual displacements.
• When waves superpose with each other, interference happens. They can be constructive or distructive.
• To get stable interference patterns, sources must be coherent. Coherent means the waves have the same wavelenght and frequency, and have a constant phase difference (E.g. lasers )
• Path difference is the difference in of the distance that the waves travel.
• A maxima occurs when the path difference is a whole number of wavelengths. It is a constructive interference, and the waves are in phase.

  
• A minima occurs when the path difference is a odd number of half wavelenghts ( n+ 1/2). It is a destructive interference, and the waves are anti-phase.

  
• Observing interference through sound waves:

  • Connect two speakers to the same oscillator, so they are coherent. Place them in a line with each other.
  • Walk along a line perpendicular to the speakers.
  • Loud = maxima = constructive interference Quiet = minima = destructive interference
• Young‘s double slit experiment, proved that Newton’s theory of light of ”corpuscular (particle) theory” was wrong.
• Laser light is coherent and monochromatic (it has only one wavelengh), so it can be used in Young’s double slit experiment.
• To do Young’s double slit experiment with a normal light source:

  • We need a coloured filter for it to be single wavelenght source (monochromatic)
  • We need a single slit so it will be coherent.
• Young’s double slit experiment:

  • Monochromatic and coherent source of light is directed towards a double slit.
  • They diffract and the diffracted waves interfere as they hit the screen and create patterns of light fridges (constructive interference) and dark fridges (destructive interference).
• Wavelenght = (Slit seperation x distance between fridges) / Distance to the screen

  $λ =$$ax / D$

  
• If we do do the double slit experiment with more than two slits, the interference pattern is sharper: The bright bands are brighter and narrower, dark areas are darker.
• 
  

  tan angle = Fringe width / Distance to the screen
• slit separation x sin θ = Order x Wavelenght

  $d sin θ =$$n λ$

  
• Shining white light through diffraction grating:

  The interference pattern would be a coloured spectrum along the screen, with white point directly in front of the slit.
• If the diffraction grating has N slits per meter, the the slit separation (d) is 1/N meters.
• A stationary wave is the superposition of two progressive waves with the same wavelengh, that are moving in the opposite direction.
• A node is where there are no vibrations and the amplitude is 0. Two adjacent nodes are half a wavelenght apart.
• Antinodes are when the amplitude is maximum.
• Producing stationary waves:

  • Use an oscillator to pass a wave along a string, which is fixed at one end.
  • The stationary wave will form when the progressive wave is reflected off the fixed end.
• Progressive waves transfer energy. Stationary waves doesn’t.
• Progressive- All parts of the wave have the same amplitude.

  Stationary- Maximum amplitude at antinodes, zero at nodes.
• Progressive- The phase changes across one complete cycle of the wave.

  Stationary- All parts of the wave between the nodes are in phase.
• Stringed instruments are stretched between two points. Those points act as nodes. When the string is plucked, it vibrates in its fundamental frequency (fo). Wavelenght of the progressive wave is double that of the length of the spring.
• Harmonics are points where the stationary wave form doesn’t change because the waves in each directions are reinforcing each other.
• Wavelenght = 2L / order of harmonic

  
• Coherent = constant phase difference