3.1 progressive and stationary waves

• Radians

  Phase difference (radians) = 2x pi x d / wavelength  

  Phase difference = pi / 180

  1 wave = 2 x pi

  Radian-degree conversions

  • angle in radians x 360 / 2 x pi = angle in degrees
  • angle in degrees x 2 x pi / 360 = angle in radians

  
• Wave speed equations

  hc / wavelength = hf   wavelength =hc / hf = cf   c = wavelength x f

  • Wavespeed = wavelength x frequency

  f=1 / T

  Progressive waves - transfer energy from one point to another

  
• Transverse waves

  • All electromagnetic waves are transverse
  • All travel at the same speed in a vacuum
  • Travel as vibrating magnetic and electric fields
  • Vibrations perpendicular to the direction of the energy transfer
  • E.g. water ripples, waves on strings

  
• Longitudinal waves

  • Vibrations propagate parallel to the direction of energy transfer and consist of compressions and rarefactions of the medium travelling through
  • Cannot travel through a vacuum as there isn't a medium to compress/rarefaction
  • Difficult to be represented graphically - plotted as displacement-time graphs
  • Appear as transverse waves
  • E.g. sound, primary seismic waves, slinky

  
• Polarisation

  • This only happens for transverse waves
  • Polarised waves only oscillate in one direction
  • Evidence of electromagnetic waves being transverse
  • 1808, Etienne-Louis Malus discovered light was polarised by reflection
  • 1817, Young suggested light was a transverse wave consisting of vibrating electric and magnetic fields perpendicular to the transfer of energy - explained why light could be polarised

  
• • Polarising filters only transmit vibrations in one direction
  • Light waves are a mixture of different directions of vibrations and can be polarised through a polarising filter
  • Two perpendicular polarising filters stop all light from passing through
  • Light is partially polarised when reflected from some surfaces - some waves vibrate in the same direction 
  • Reflecting partially polarised light through a polarising filter at the correct angle blocks out unwanted glare - e.g. polaroid sunglasses

  
• • Television and radio signals are polarised 
  • Rods on aerials are horizontal - TV/radio signals are polarised by the orientation of rods
  • To receive a strong signal, line up rods on the receiving aerial with rods on the transmitting aerial
  • Tuning the radio and moving aerial will result in fluctuating signal strength
• The principle of superposition - when two or more waves meet at a point, the resultant displacement at the point is equal to the sum of the displacements of the individual waves at that point
• Stationary waves

  • The superposition of two progressive waves with the same frequency and wavelength, and are moving in opposite directions
  • No energy is transmitted
  • At the resonant frequency, a stationary wave is formed where the wave pattern doesn't move

  
• • When a string is connected to a driving oscillator and is fixed at the other end, a wave is generated by the oscillator and is reflected back and forth
  • The pattern appears jumbled at most frequencies, but when the exact number of waves are produced within the time it takes for a wave to travel to the end and back, the original and reflected waves reinforce each other
• Nodes and antinodes

  Nodes - point of no displacement, have no energy

  Antinodes - point of maximum displacement, have the most energy

  
• Microwaves

  • Reflect off a metal plate set up a stationary wave
  • Find nodes and antinodes by moving the probe between the transmitter and the reflecting plate

  Sound waves 

  • Powder shows stationary waves in a tube of air
  • Stationary sound waves produced in a glass tube
  • Lycopodium powder laid on the bottom of the tube shook away from antinodes but was undisturbed at nodes

  
• First harmonic (fundamental frequency)

  • f = 1/2L x (square root of T / mass per unit length) 
  • Stationary wave vibrating at the lowest possible resonant frequency
  • It has two nodes and an antinode
  • Half a wavelength - 1/2 x wavelength
  • Length of string - L = wavelength/2
  • Frequency - f = c / wavelength    f1=c / 2L

  
• Second harmonic 

  • Twice the frequency of the first harmonic
  • Three nodes and two antinodes
  • One wavelength - wavelength
  • Length of string - L = wavelength

  Frequency - f2 = c / L   

  Third harmonic 

  • Thrice the frequency of the first harmonic
  • Four nodes and three antinodes
  • One and a half wavelengths - 3/2 x wavelength
  • Length of string - L = 3/2 x wavelength
  • Frequency - f3 = c / (2/3 x L)  
• Core practical (1) - investigating stationary waves

  Aims

  • Measure how the fundamental frequency is affected by changing variables
  • Length of string, tension, mass per unit length

  Variables

  • Independent variables - length, tension, or mass per unit length
  • Dependent variable - frequency of the first harmonic
  • Control variables - tension, mass per unit length, or length

  Resolution

  • Metre ruler = 1mm
  • Signal generator 10nHz
  • Top-pan balance =0.005g

  
• Method

  1. Measure the mass and length of different types of string using a mass balance and ruler, to find the mass per unit length of each - mass per unit length = m / L   (kgm-1)     
  2. Set up apparatus with one string and record the mass per unit length and length to work out tension T=mg   (kg)
  3. Turn on the signal generator and vary the frequency until you find the first harmonic - this is the fundamental frequency
  4. Repeat the test with different values to obtain multiple results

  
• Investigate how length, tension, or mass per unit length affects the resonant frequency:

  1. Keep the string and tension constant, adjust length by moving vibration transducer towards/away from pulley to find first harmonic; record frequency against the length.
  2. Keep the string and length the same, adding or removing mass to change the tension, finding the first harmonic and recording frequency against tension
  3. Keep the length and tension the same but use a different string to change the mass per unit length, finding the first harmonic and recording the frequency against the mass per unit length