Published on

1 Comment
No Downloads
Total views
On SlideShare
From Embeds
Number of Embeds
Embeds 0
No embeds

No notes for slide


  1. 1. Waves Topic 4.4 Wave characteristics
  2. 2. WAVES <ul><li>Energy can be transferred in a number of ways. </li></ul><ul><li>A moving car is an example of energy in motion </li></ul><ul><ul><li>kinetic energy </li></ul></ul><ul><li>Not only does the energy move; </li></ul><ul><ul><li>the car moves as well. </li></ul></ul><ul><li>Energy can move; </li></ul><ul><ul><li>without the object/particle moving with it. </li></ul></ul><ul><li>This occurs in waves. </li></ul>
  3. 3. Travelling Wave Characteristics <ul><li>A surfer; </li></ul><ul><ul><li>sitting on their board, </li></ul></ul><ul><ul><li>waiting for the right wave. </li></ul></ul><ul><li>While waiting; </li></ul><ul><ul><li>ocean waves pass under him, </li></ul></ul><ul><ul><li>while he bobs up and down. </li></ul></ul><ul><li>Flick a slinky spring; </li></ul><ul><ul><li>wave passes along the slinky while, </li></ul></ul><ul><ul><li>particles move up and down. </li></ul></ul>
  4. 4. Travelling Wave Characteristics <ul><li>Drop a stone in a still pond; </li></ul><ul><ul><li>you produce a wave that moves out from the centre, </li></ul></ul><ul><ul><li>in ever increasing circles. </li></ul></ul><ul><li>Check the water before and after the wave passes, </li></ul><ul><ul><li>You find that the water, </li></ul></ul><ul><ul><li>remained where it was. </li></ul></ul>
  5. 5. Travelling Wave Characteristics <ul><li>In these examples; </li></ul><ul><ul><li>the particles vibrate or oscillate. </li></ul></ul><ul><li>The wave has been transferred; </li></ul><ul><ul><li>without a transfer of matter. </li></ul></ul><ul><li>The signals from radio and T.V.’s; </li></ul><ul><ul><li>are waves. </li></ul></ul><ul><li>Sound and light travel as waves. </li></ul>
  6. 6. Transverse Waves <ul><li>If you create a wave; </li></ul><ul><ul><li>by shaking a slinky up and down, </li></ul></ul><ul><ul><li>the motion of the medium is at </li></ul></ul><ul><ul><li>right angles to the </li></ul></ul><ul><ul><li>motion of the wave. </li></ul></ul><ul><li>This type of wave is called; </li></ul><ul><ul><li>a transverse wave . </li></ul></ul>
  7. 7. Transverse Wave Direction of travel of the wave Direction of oscillation of the particles
  8. 8. Transverse Waves <ul><li>Stretched strings in a musical instrument </li></ul><ul><li>ocean waves, </li></ul><ul><li>radio and light </li></ul><ul><li>are all examples of transverse waves. </li></ul>
  9. 9. Transverse Waves Transverse Wave
  10. 10. Longitudinal <ul><li>In these waves the source that produces the wave oscillates in the same direction as the direction of travel of the wave </li></ul><ul><li>It means that the particle of the medium through which the wave travels also oscillates in the same direction as the direction of travel of the wave. </li></ul>
  11. 11. Longitudinal Waves <ul><li>When the particles of the medium; </li></ul><ul><ul><li>move in the same direction as the wave, </li></ul></ul><ul><ul><li>it is known as a longitudinal wave . </li></ul></ul><ul><li>They are less common. </li></ul><ul><li>Sound travels as a longitudinal wave. </li></ul>
  12. 12. Longitudinal Waves <ul><li>In both forms, the energy can be transferred; </li></ul><ul><ul><li>as a single pulse, </li></ul></ul><ul><ul><li>a number of pulses, </li></ul></ul><ul><ul><li>or a continuous wave. </li></ul></ul><ul><li>Particles may be set in motion by a wave; </li></ul><ul><ul><li>no particle travels far from its initial position. </li></ul></ul>
  13. 13. Longitudinal Waves <ul><li>As the wave particles set neighbouring particles into motion; </li></ul><ul><ul><li>the wave is propagated through the medium, </li></ul></ul><ul><ul><li>energy is transferred in the medium. </li></ul></ul>
  14. 14. Longitudinal Wave Direction of travel of the wave Direction of oscillations of the particles
  15. 15. Definitions <ul><li>The following definitions are given in terms of the particles that make up the medium through which the wave travels. </li></ul><ul><li>For the slinky spring a particle would be a single turn of the spring </li></ul><ul><li>For the water waves a particle would be a very small part of the water. </li></ul>
  16. 16. What is a Wave? <ul><li>A wave is a means by which energy is transferred between two points in a medium without any net transfer of the medium itself. </li></ul>
  17. 17. The Medium <ul><li>The substance or object in which the wave is travelling. </li></ul><ul><li>When a wave travels in a medium parts of the medium do not end up at different places </li></ul><ul><li>The energy of the source of the wave is carried to different parts of the medium by the wave . </li></ul>
  18. 18. <ul><li>Water waves however, can be a bit disconcerting. </li></ul><ul><li>Waves at sea do not transport water but the tides do. </li></ul><ul><li>Similarly, a wave on a lake does not transport water but water can actually be blown along by the wind . </li></ul>
  19. 19. <ul><li>However, if you set up a ripple tank you will see that water is not transported by the wave set up by the vibrating dipper. </li></ul>
  20. 20. Displacement <ul><li>(s) is the distance that any particle is away from its equilibrium position at an instance </li></ul><ul><li>Measured in metres </li></ul>
  21. 21. Crest <ul><li>This is a term coined from water waves and refers to the points at the maximum height of the wave. </li></ul><ul><li>It is the positive displacement </li></ul><ul><li>from equilibrium </li></ul>
  22. 22. Trough <ul><li>A term coined from water waves referring to the points at the lowest part of the wave. </li></ul><ul><li>The negative displacement from equilibrium. </li></ul>
  23. 23. Compression <ul><li>This is a term used in connection with longitudinal wave and refers to the region where the particles of the medium are &quot;bunched up&quot;. </li></ul><ul><li>High density </li></ul><ul><li>High pressure </li></ul>
  24. 24. Rarefaction <ul><li>A term used in connection with longitudinal waves referring to the regions where the particles are &quot;stretched out&quot;. </li></ul><ul><li>Low density </li></ul><ul><li>Low pressure </li></ul>
  25. 26. Defining Terms
  26. 27. Defining Terms
  27. 28. Defining Terms <ul><li>Wavelength: </li></ul><ul><ul><li>The distance covered in a complete wave cycle. </li></ul></ul><ul><ul><li>The distance between two consecutive points in phase. </li></ul></ul><ul><ul><li>Symbol: </li></ul></ul><ul><ul><ul><li>Greek letter  </li></ul></ul></ul><ul><ul><li>Unit (SI) </li></ul></ul><ul><ul><ul><li>metre. </li></ul></ul></ul>
  28. 29. Defining Terms
  29. 30. Amplitude <ul><li>(A, a) This is the maximum displacement of a particle from its equilibrium position </li></ul><ul><li>(It is also equal to the maximum displacement of the source that produces the wave). </li></ul><ul><li>Normally measured in metres </li></ul>
  30. 31. Defining Terms
  31. 32. Period <ul><li>(T) This is the time that it takes a particle to make one complete oscillation </li></ul><ul><li>(It also equals the time for the source of the wave to make one complete oscillation). </li></ul><ul><li>Measured in seconds </li></ul>
  32. 33. Frequency <ul><li>(f) This is the number of oscillations made per second by a particle </li></ul><ul><li>(It is also equal to the number of oscillations made per second by the source of the wave) </li></ul><ul><li>The SI unit of frequency is the Hertz - Hz. (1 Hz is 1 oscillation per second) </li></ul><ul><li>Clearly then, f = 1/T </li></ul>
  33. 34. Defining Terms <ul><li>Wave Speed: </li></ul><ul><ul><li>Is the speed at which a given point on the wave, </li></ul></ul><ul><ul><ul><li>is travelling through the medium. </li></ul></ul></ul><ul><ul><li>The product of frequency and wavelength. Mathematically represented by: </li></ul></ul><ul><li>v =f  </li></ul><ul><li>Unit (SI) </li></ul><ul><ul><li>ms -1 . </li></ul></ul>
  34. 35. Eg <ul><li>For example, the speed of sound waves in air is typically 330 ms -1 to 350 ms -1 depending on the density of the air and is four times faster in water. </li></ul><ul><li>Velocity = displacement of crest/time taken </li></ul><ul><li>If the time taken is equal to the period T of the wave, the displacement of one crest in this time is equal to  and the equation can be rewritten as: </li></ul><ul><ul><ul><li>v =  /T </li></ul></ul></ul><ul><ul><ul><li>But f = 1/T </li></ul></ul></ul><ul><ul><ul><li>so v = f  </li></ul></ul></ul>
  35. 36. Eg 1 <ul><li>What will be the time delay in hearing the sound from a brass band for an observer 660 m away? Assume the light arrives instantaneously and the sound travels at 330 ms -1 . </li></ul>
  36. 37. Solution <ul><li>v = 330 ms -1 </li></ul><ul><li>s = 660 m </li></ul><ul><li>t = ? </li></ul><ul><li>v = s/t </li></ul><ul><li>and rearranging; </li></ul><ul><li>t = s/v </li></ul><ul><li>t = 660/330 </li></ul><ul><li>t = 2.0 s </li></ul>
  37. 38. Eg 2 <ul><li>Waves reaching the beach from an offshore storm arrives at 4 s intervals. Calculate the frequency of the waves </li></ul>
  38. 39. Solution <ul><li>T = 4 s </li></ul><ul><li>T = 1/f </li></ul><ul><li>f = ¼ </li></ul><ul><li>f = 0.25 Hz </li></ul>
  39. 40. Eg 3 <ul><li>Find the period of a 1 kHz sound wave. </li></ul>
  40. 41. Solution <ul><li>f = 1 kHz = 1000 Hz </li></ul><ul><li>T = ? </li></ul><ul><li>F = 1/T </li></ul><ul><li>rearranging; </li></ul><ul><li>T = 1/f </li></ul><ul><li>T = 1/1000 </li></ul><ul><li>T = 0.001 or 10 -3 s. </li></ul>
  41. 42. Eg 4 <ul><li>Calculate the speed of an earthquake wave with a wavelength of 2 km and a frequency of 3 Hz. </li></ul>
  42. 43. Solution <ul><li> = 2000m </li></ul><ul><li>f = 3 Hz </li></ul><ul><li>v = ? </li></ul><ul><li>v = f  </li></ul><ul><li>v = 3 x 2000 </li></ul><ul><li>v = 6000 m s -1 </li></ul>
  43. 44. Eg 5 <ul><li>Given that the speed of sound in air is 330 ms -1 , find the wavelength of (a) 20Hz and (b) 20000 Hz sounds. </li></ul>
  44. 45. Solution <ul><li>Part (a) </li></ul><ul><li>v = 330 m s -1 </li></ul><ul><li>f = 20 Hz </li></ul><ul><li> = ? </li></ul><ul><li>v = f  </li></ul><ul><li> = 330/20 </li></ul><ul><li> = 16.5 m </li></ul>Part (b) v = 330 m s -1 f = 20 000 Hz  = ? v = f   = 330/20 000  = 0.0165 m  = 1.65 x 10 -2 m
  45. 46. <ul><li>A very important property associated with all waves is their so-called periodicity. </li></ul><ul><li>Waves in fact are periodic both in time and space and this sometimes makes it difficult to appreciate what actually is going on in wave motion. </li></ul>Periodicity
  46. 47. <ul><li>If we drew a diagram that froze time on waves in water </li></ul><ul><li>We would have an instantaneous snapshot of the whole of the water surface </li></ul><ul><li>The next diagram shows the periodicity of the wave in space </li></ul>
  47. 48. Displacement / Distance p displacement distance
  48. 49. <ul><li>The y-axis shows the displacement of the water from its equilibrium position </li></ul><ul><li>The graph is a displacement-distance graph. </li></ul>
  49. 50. <ul><li>We now look at one part of the wave that is labeled p and &quot;unfreeze&quot; time </li></ul><ul><li>The next diagram shows how the position of p varies with time </li></ul><ul><li>This illustrates the periodicity of the wave in time </li></ul>
  50. 51. Displacement / Time displacement of point p from equilibrium position time
  51. 52. <ul><li>The y-axis now shows the displacement of the point p from equilibrium </li></ul><ul><li>The graph is a displacement-time graph. </li></ul>
  52. 53. <ul><li>The space diagram and the time diagram are both identical in shape </li></ul><ul><li>If we mentally combine them we have the whole wave moving both in space and time. </li></ul>
  53. 54. Graphical Representation of Waves <ul><li>Two ways of describing a periodic wave motion by means of a graph. </li></ul><ul><li>One way is to consider a particular particle of the medium and plot its displacement against time as shown below: </li></ul>
  54. 55. Graphical Representation of Waves
  55. 56. Graphical Representation of Waves <ul><li>Tracks the movement of a particle; </li></ul><ul><ul><li>as a wave moves through it. </li></ul></ul><ul><li>With displacement on the vertical axis; </li></ul><ul><ul><li>time on the horizontal, </li></ul></ul><ul><ul><li>the particle will move up and down, </li></ul></ul><ul><ul><li>In a straight line, but the graph will be </li></ul></ul><ul><ul><li>a sine curve type pattern </li></ul></ul>
  56. 57. Graphical Representation of Waves <ul><li>Allows us to find both frequency; </li></ul><ul><ul><li>which will be the number of crests in 1 sec, </li></ul></ul><ul><li>Period; </li></ul><ul><ul><li>which will be the time between crests but, </li></ul></ul><ul><li>tells us nothing about the wave speed or wavelength. </li></ul>
  57. 58. Graphical Representation of Waves <ul><li>Alternatively, the horizontal axis; </li></ul><ul><ul><li>Can be used to represent a distance in medium, </li></ul></ul><ul><ul><li>and plot the displacement of the particles, </li></ul></ul><ul><ul><li>at successive points along a line in the medium. </li></ul></ul>
  58. 59. Graphical Representation of Waves <ul><li>The graph obtained is a snapshot; </li></ul><ul><ul><li>of the wave at a particular instant, </li></ul></ul><ul><ul><li>known as a displacement vs distance graph. </li></ul></ul>
  59. 60. Graphical Representation of Waves
  60. 61. Graphical Representation of Waves <ul><li>The graphs above show the profile of a wave at two instants. </li></ul><ul><li>The graphs could represent a transverse or longitudinal wave motion. </li></ul>
  61. 62. Graphical Representation of Waves <ul><li>The distance between peaks represents the wavelength. </li></ul><ul><li>The wave speed can not be calculated directly from this graph, </li></ul><ul><ul><li>only by combining the information from this and the previous one. </li></ul></ul>
  62. 63. Wavefront <ul><li>All the points that started from a source at one time make up the whole of that wavefront, </li></ul><ul><li>If it was a single point, it will be a circular wavefront </li></ul><ul><li>If it is a straight line, it will be a straight wave front </li></ul>
  63. 64. Wavelength again! <ul><li>Wavelength will therefore be equal to the distance between successive crests and successive troughs. </li></ul>
  64. 65. Deriving v = f  <ul><li>Imagine a wave with velocity v </li></ul><ul><li>Being produced from a source of frequency f </li></ul><ul><li>In 1 second the 1 st wavefront would have travelled a distance of f  </li></ul><ul><li>As speed = distance / time </li></ul><ul><li>v = f  / 1 </li></ul><ul><li> v = f  </li></ul>
  65. 66. 2 Important Points <ul><li>1. The frequency of a wave depends only on the source producing the wave </li></ul><ul><ul><li>It will therefore not change if the wave enters a different medium or the properties of the medium change </li></ul></ul>
  66. 67. <ul><li>2. The Speed of waves only depends on the nature and the properties of the medium </li></ul><ul><ul><li>Water waves do travel faster in deeper water </li></ul></ul><ul><ul><li>Light travels slower in more optically dense material </li></ul></ul>
  67. 68. Electromagnetic Waves <ul><li>Light is energy that is emitted by; </li></ul><ul><ul><li>vibrating electric charges in atoms. </li></ul></ul><ul><li>Energy travels in a wave that is; </li></ul><ul><ul><li>partly electric and partly magnetic. </li></ul></ul><ul><li>Such a wave is called an; </li></ul><ul><ul><li>electromagnetic wave . </li></ul></ul>
  68. 69. Electromagnetic Waves <ul><li>Light is only one part of a; </li></ul><ul><ul><li>broad family of e/m waves. </li></ul></ul><ul><li>They are all radiated by; </li></ul><ul><ul><li>vibrating electrons within the atom. </li></ul></ul><ul><li>The range of e/m waves; </li></ul><ul><ul><li>or em spectrum is shown below </li></ul></ul>
  69. 70. Electromagnetic Waves
  70. 71. Electromagnetic Waves <ul><li>EM waves are transverse; </li></ul><ul><ul><li>travel at the speed of light, </li></ul></ul><ul><ul><ul><li>3 x 10 8 ms -1 </li></ul></ul></ul><ul><ul><li>through a vacuum. </li></ul></ul>
  71. 72. Electromagnetic Waves <ul><li>E/M waves are usually defined by; </li></ul><ul><ul><li>their wavelength, </li></ul></ul><ul><ul><li>assumed to be in a vacuum </li></ul></ul><ul><ul><ul><li>which is rather silly, </li></ul></ul></ul><ul><ul><ul><li>frequency never changes, </li></ul></ul></ul><ul><ul><ul><li>is what defines the characteristics </li></ul></ul></ul><ul><ul><ul><ul><li>i.e. colour. </li></ul></ul></ul></ul>
  72. 73. The EM Spectrum Itself Short  Long  High f Low f V I S I B L E Radio Waves Micro Waves Infra red Gamma rays Ultra Violet X rays
  73. 74. Frequencies of Regions (Hz) <ul><li>Gamma Rays >10 21 </li></ul><ul><li>X-rays 10 17 - 10 21 </li></ul><ul><li>Ultraviolet 10 1 4 - 10 17 </li></ul><ul><li>Violet 7.5 x 10 14 > Visible > Red 4.3 x 10 14 </li></ul><ul><li>Infrared 10 11 - 10 14 </li></ul><ul><li>Microwaves 10 9 - 10 11 </li></ul><ul><li>Radio and TV < 10 9 </li></ul>
  74. 75. Sources of Regions <ul><li>Gamma – certain radioactive material’s nuclei </li></ul><ul><li>X-rays – by firing an electron stream at a tungsten metal target in a highly evacuated tube. </li></ul><ul><li>Ultraviolet – the Sun, ultraviolet lamp </li></ul><ul><li>Visible – hot bodies </li></ul><ul><li>Infrared – the Sun (heat), hot bodies </li></ul><ul><li>Microwaves – Ovens, communication systems </li></ul><ul><li>Radio and TV – transmitter stations, Azteca TV </li></ul>
  75. 76. The Different Regions <ul><li>In the context of wave motion, common properties of all parts of the electromagnetic spectrum are </li></ul><ul><ul><li>all transverse waves </li></ul></ul><ul><ul><li>all travel at the speed of light in vacuo </li></ul></ul><ul><ul><li>(3.0 x 10 8 ms -1 ) </li></ul></ul><ul><ul><li>all can travel in a vacuum </li></ul></ul>
  76. 77. Discrete Pulses and Continuous Waves <ul><li>A single shake of a slinky will send a discrete pulse down it </li></ul><ul><li>Shake the slinky up and down and a continuous travelling wave travels down it </li></ul><ul><li>This applies to longitudinal waves too </li></ul>
  77. 79. Reflections in one Dimension <ul><li>When a wave reaches a boundary between two media; </li></ul><ul><ul><li>some or all of the wave bounces back, </li></ul></ul><ul><ul><li>into the first medium. </li></ul></ul>
  78. 80. Reflections in one Dimension <ul><li>A pulse is sent along a slinky spring; </li></ul><ul><ul><li>which is attached at one end to a wall. </li></ul></ul><ul><li>All the energy is reflected back; </li></ul><ul><ul><li>along the spring, </li></ul></ul><ul><ul><li>rather than into the wall. </li></ul></ul>
  79. 81. Reflections in one Dimension Reflection from a boundary
  80. 82. Reflection From a Fixed End
  81. 83. Reflections in one Dimension <ul><li>The pulse becomes inverted; </li></ul><ul><ul><li>as it is reflected. </li></ul></ul><ul><li>This is called phase reversal . </li></ul>
  82. 84. Reflections in one Dimension <ul><li>This is why metals are so shiny. </li></ul><ul><li>A Metal surface is rigid; </li></ul><ul><ul><li>to the light waves that shine upon it. </li></ul></ul>
  83. 85. Reflections in one Dimension <ul><li>Most of the light is reflected; </li></ul><ul><ul><li>apart from a small energy loss, </li></ul></ul><ul><ul><li>due to the friction of, </li></ul></ul><ul><ul><li>the vibrating electrons in the surface. </li></ul></ul><ul><li>Metals can be used as mirrors for this reason. </li></ul>
  84. 86. Reflection From a Free End
  85. 87. Reflections in one Dimension <ul><li>The part of the spring; </li></ul><ul><ul><li>adjacent to the boundary is free to be displaced, </li></ul></ul><ul><ul><li>and no phase change occurs on reflection. </li></ul></ul>
  86. 88. Reflections in one Dimension <ul><li>If the wall is replaced; </li></ul><ul><ul><li>with a heavy spring as a new medium, </li></ul></ul><ul><ul><li>some energy is transmitted, </li></ul></ul><ul><ul><li>some energy is reflected. </li></ul></ul>Reflection from a boundary
  87. 89. Reflections in one Dimension
  88. 90. Partial Reflection from a Heavier Spring
  89. 91. Reflections in one Dimension <ul><li>The heavy spring acts; </li></ul><ul><ul><li>as an imperfect ‘rigid’ boundary, </li></ul></ul><ul><ul><li>partially reflecting the pulse, </li></ul></ul><ul><ul><li>with a change of phase but, </li></ul></ul><ul><ul><li>also partially transmitting it. </li></ul></ul>
  90. 92. Reflections in one Dimension <ul><li>Two pulses of reduced amplitude </li></ul><ul><ul><li>move at speeds characteristic of the media </li></ul></ul><ul><li>result. </li></ul>
  91. 93. Partial Reflection From a Lighter Spring
  92. 94. Reflections in one Dimension <ul><li>The lighter spring acts; </li></ul><ul><ul><li>as an imperfect ‘free end’, </li></ul></ul><ul><ul><li>partially reflecting the pulse, </li></ul></ul><ul><ul><li>without change of phase and, </li></ul></ul><ul><ul><li>partially transmitting it. </li></ul></ul><ul><li>Two pulses with reduced amplitude are produced. </li></ul>
  93. 95. Huygen’s Principle <ul><li>In the late 1600s; </li></ul><ul><ul><li>Dutch mathematician </li></ul></ul><ul><ul><li>Christian Huygens </li></ul></ul><ul><li>studied waves.  </li></ul>
  94. 96. Huygen’s Principle <ul><li>Using light waves; </li></ul><ul><ul><li>he suggested that the waves spread out from a point source, </li></ul></ul><ul><ul><li>may be regarded as, </li></ul></ul><ul><ul><li>overlapping tiny secondary wavelets, </li></ul></ul><ul><ul><li>every point on any wave front may be regarded as, </li></ul></ul><ul><ul><li>a new point source of secondary waves. </li></ul></ul>
  95. 97. Huygen’s Principle <ul><li>In simpler terms, wave fronts are made up of tinier wave fronts. </li></ul>
  96. 98. Huygen’s Principle <ul><li>The drawing and page is from </li></ul><ul><ul><li>Huygen’s book </li></ul></ul><ul><ul><li>Treatise on Light </li></ul></ul><ul><li>Light expands in wave-fronts, </li></ul><ul><ul><li>every point behaves </li></ul></ul><ul><ul><li>as if it were a new source of waves. </li></ul></ul>
  97. 99. Huygen’s Principle <ul><li>Secondary wavelets starting @ b,b,b,b </li></ul><ul><ul><li>form a new wave front d, d, d, d </li></ul></ul><ul><ul><li>form still another new wave front DCEF. </li></ul></ul>
  98. 100. Huygen’s Principle Defined <ul><li>Huygen’s principle states that every point on a wavefront may be regarded as a point source of secondary circular wavelets. </li></ul><ul><li>The new wavefront is formed along the common tangent to these secondary wavelets. </li></ul>
  99. 101. Huygen’s Principle Explained <ul><li>Throw a stone into water; </li></ul><ul><ul><li>the wave produced is circular. </li></ul></ul>
  100. 102. Huygen’s Principle Explained <ul><li>In a wave from a straight edge; </li></ul><ul><ul><li>such as a beach wave, </li></ul></ul><ul><ul><li>wavefronts from the straight edge, </li></ul></ul><ul><ul><li>move in the same direction. </li></ul></ul>
  101. 103. Huygen’s Principle Explained <ul><li>In both circular and straight wavefronts; </li></ul><ul><ul><li>the direction of propagation of the wave </li></ul></ul><ul><ul><li>is perpendicular to the wavefronts. </li></ul></ul>
  102. 104. Huygen’s Principle Explained <ul><li>Consider a water wave; </li></ul><ul><ul><li>moving in a ripple tank. </li></ul></ul><ul><li>The wavefront is made up of many points </li></ul><ul><ul><li>generating its own secondary circular wave. </li></ul></ul>
  103. 105. Huygen’s Principle Explained <ul><li>Draw a common tangent to the secondary wavefronts; </li></ul><ul><ul><li>you then have created the new wavefront. </li></ul></ul><ul><li>The original point wave front; </li></ul><ul><ul><li>creates a new wavefront at bbbb, </li></ul></ul><ul><ul><li>which creates a new wavefront at dddd, </li></ul></ul><ul><ul><li>and so on as shown below. </li></ul></ul>
  104. 106. Huygen’s Principle Explained
  105. 107. Huygen’s Principle Explained
  106. 108. Reflections in Two Dimensions <ul><li>In one dimension; </li></ul><ul><ul><li>the reflected wave simply travels back, </li></ul></ul><ul><ul><li>in the direction from which it came. </li></ul></ul><ul><li>In two dimensions, </li></ul><ul><ul><li>the situation is a little different. </li></ul></ul>
  107. 109. Reflections in Two Dimensions <ul><li>Direction of incident & reflected waves; </li></ul><ul><ul><li>described by straight lines called rays. </li></ul></ul><ul><li>The incoming ray </li></ul><ul><ul><ul><li>incident ray </li></ul></ul></ul><ul><ul><li>and the reflected ray makes, </li></ul></ul><ul><ul><li>equal angles with the normal. </li></ul></ul>
  108. 110. Reflections in Two Dimensions <ul><li>Angle between incident ray & normal; </li></ul><ul><ul><li>called the angle of incidence </li></ul></ul><ul><li>Angle between the reflected ray & normal; </li></ul><ul><ul><li>called the angle of reflection . </li></ul></ul>
  109. 111. Reflections in Two Dimensions
  110. 112. Reflections in Two Dimensions <ul><li>Relationship is called; </li></ul><ul><li>Law of reflection . </li></ul><ul><li>Law applies equally to both; </li></ul><ul><ul><li>partially reflected and, </li></ul></ul><ul><ul><li>totally reflected waves. </li></ul></ul><ul><li>Stated mathematically: </li></ul><ul><li> i =  r </li></ul>Reflection of light
  111. 113. Reflection <ul><li>If a lit candle is placed; </li></ul><ul><ul><li>in front of a plane mirror, </li></ul></ul><ul><ul><li>rays of light are reflected in all directions. </li></ul></ul><ul><li>There are an infinite number; </li></ul><ul><ul><li>all obey the law of reflection. </li></ul></ul>
  112. 114. Reflection <ul><li>The rays diverge from the tip of the flame and; </li></ul><ul><ul><li>continue to diverge upon reflection. </li></ul></ul><ul><li>These rays appear to originate from; </li></ul><ul><ul><li>a point located behind the mirror. </li></ul></ul>
  113. 115. Reflection <ul><li>This is called a virtual image ; </li></ul><ul><ul><li>the light does not actually pass through the image, </li></ul></ul><ul><ul><li>but behaves as though it virtually did. </li></ul></ul><ul><li>The image appears as far behind the mirror; </li></ul><ul><ul><li>as the object is in front of it and, </li></ul></ul><ul><ul><li>the object and the image is the same. </li></ul></ul>
  114. 116. Reflection
  115. 117. Reflection <ul><li>When the mirror is curved; </li></ul><ul><ul><li>sizes & distances of the object and image, </li></ul></ul><ul><ul><li>are no longer equal, </li></ul></ul><ul><ul><li>but the law of reflection still holds. </li></ul></ul>
  116. 118. Reflection
  117. 119. Reflection
  118. 120. Reflection <ul><li>Light incident on a rough surface; </li></ul><ul><ul><li>it is reflected in many directions. </li></ul></ul>
  119. 121. Reflection <ul><li>Although each individual ray obeys the law of reflection; </li></ul><ul><ul><li>many different angles light rays encounter; </li></ul></ul><ul><ul><li>in striking a rough surface cause, </li></ul></ul><ul><ul><li>reflection in many directions. </li></ul></ul><ul><li>This is called diffuse reflection . </li></ul>
  120. 122. Reflection Reflection of Light
  121. 123. Reflection <ul><li>What is a rough surface? </li></ul><ul><li>If the elevations in the surface; </li></ul><ul><ul><li>less than one eighth of the wavelength, </li></ul></ul><ul><ul><li>of the wave that falls upon it, </li></ul></ul><ul><ul><li>it is called polished. </li></ul></ul>
  122. 124. Reflection <ul><li>Sound waves also obey the laws of reflection. </li></ul><ul><li>Reflected sound is called an echo . </li></ul><ul><li>If the reflecting surface is large; </li></ul><ul><ul><li>rigid and smooth, </li></ul></ul><ul><ul><li>a large echo is heard. </li></ul></ul>
  123. 125. Reflection <ul><li>If the reflecting surface is small; </li></ul><ul><ul><li>soft and irregular, </li></ul></ul><ul><ul><li>little echo is heard. </li></ul></ul><ul><li>Sound energy not reflected; </li></ul><ul><ul><li>is transmitted or absorbed. </li></ul></ul>
  124. 126. Reflection <ul><li>If a room or hall is too reflective; </li></ul><ul><ul><li>the sound becomes garbled, </li></ul></ul><ul><ul><li>due to multiple reflections, </li></ul></ul><ul><ul><li>called reverberations . </li></ul></ul>
  125. 127. Reflection
  126. 128. Reflection <ul><li>If reflective surface is too absorbent; </li></ul><ul><ul><li>sound level would be low, </li></ul></ul><ul><ul><li>room or hall would sound dull and lifeless. </li></ul></ul><ul><li>A good hall has a balance between; </li></ul><ul><ul><li>absorption and reverberation. </li></ul></ul>
  127. 129. Reflection
  128. 130. Refraction of Waves in 1 & 2 Dimensions <ul><li>Place a pencil in a glass of water; </li></ul><ul><ul><li>it appears bent, </li></ul></ul><ul><ul><li>at the air/water interface. </li></ul></ul><ul><li>Bending or change in direction; </li></ul><ul><ul><li>that occurs at the boundary, </li></ul></ul><ul><ul><li>of two different media is called refraction . </li></ul></ul>
  129. 131. Refraction of Waves in 1 & 2 Dimensions <ul><li>Place coin on bottom of empty coffee mug. </li></ul><ul><li>Position yourself so the coin is just out of view; </li></ul><ul><ul><li>the coin becomes visible as water is added. </li></ul></ul><ul><li>The coin still appears to be on the bottom; </li></ul><ul><ul><li>the image of the coin and the bottom of the mug, </li></ul></ul><ul><ul><li>must have moved up. </li></ul></ul>
  130. 132. Refraction of Waves in 1 & 2 Dimensions
  131. 133. Refraction of Waves in 1 & 2 Dimensions
  132. 134. Refraction of Waves in 1 & 2 Dimensions <ul><li>Water in a pond appears to be; </li></ul><ul><ul><li>only ¾ its true depth. </li></ul></ul><ul><li>The depth an object appears to be; </li></ul><ul><ul><li>is called the apparent depth </li></ul></ul><ul><ul><li>while its true depth is called, </li></ul></ul><ul><ul><li>the real depth . </li></ul></ul>
  133. 135. Refraction of Waves in 1 & 2 Dimensions
  134. 136. Refraction of Waves in 1 & 2 Dimensions <ul><li>This phenomenon can be generalised for any two media. </li></ul>
  135. 137. Refraction of Waves in 1 & 2 Dimensions <ul><li>i = angle of incidence </li></ul><ul><li>R = angle of refraction </li></ul><ul><li>D = angle of deviation </li></ul>
  136. 138. Refraction of Waves in 1 & 2 Dimensions <ul><li>Angle of refraction is less than angle of incidence; </li></ul><ul><ul><li>when the 2 nd medium is more optically dense than the first medium, </li></ul></ul><ul><ul><li>such as when light travels from air to glass. </li></ul></ul><ul><li>This is reversed when light travels from glass to air. </li></ul>
  137. 139. Refraction of Waves in 1 & 2 Dimensions
  138. 140. Refraction of Waves in 1 & 2 Dimensions <ul><li>Light bends towards the normal when; </li></ul><ul><ul><li>it enters a more optically dense medium. </li></ul></ul><ul><li>Light bends away from the normal; </li></ul><ul><ul><li>when it enters a less optically dense medium. </li></ul></ul><ul><li>The amount the incident ray is deviated; </li></ul><ul><ul><li>depends on the nature of the transparent material </li></ul></ul>
  139. 141. Huygen’s Principle and Reflection <ul><li>Each point on the wavefront ABCD; </li></ul><ul><ul><li>acts as a source of semicircular wavelets. </li></ul></ul><ul><li>As each point on the wavefronts hits the reflecting surface; </li></ul><ul><ul><li>the wavelets change direction. </li></ul></ul>
  140. 142. Huygen’s Principle and Reflection
  141. 143. Huygen’s Principle and Reflection
  142. 144. Huygen’s Principle and Reflection
  143. 145. Huygen’s Principle and Reflection
  144. 146. Huygen’s Principle and Reflection
  145. 147. Refraction & Huygen’s Principal <ul><li>Consider a wavefront advancing through medium 1; </li></ul><ul><ul><li>travelling at velocity v 1 </li></ul></ul><ul><ul><li>falling at an angle on to a medium 2, </li></ul></ul><ul><ul><li>travelling at velocity is v 2 . </li></ul></ul>
  146. 148. Refraction & Huygen’s Principal
  147. 149. Refraction & Huygen’s Principal <ul><li>The wavefront with the points; </li></ul><ul><ul><li>A, B, C, D are a source of secondary wavelets. </li></ul></ul><ul><li>After a time t , the secondary source wavelet from D; </li></ul><ul><ul><li>has moved a distance s 1 = v 1 t while, </li></ul></ul><ul><ul><li>wavelet from A has moved, </li></ul></ul><ul><ul><li>smaller distance s 2 = v 2 t, </li></ul></ul><ul><ul><li>in the denser medium 2, </li></ul></ul><ul><ul><li>where the velocity is less. </li></ul></ul>
  148. 150. Refraction & Huygen’s Principal <ul><li>The time for the wavelet to travel; </li></ul><ul><ul><li>from B to B 2 is identical, </li></ul></ul><ul><ul><li>for the wavelet to travel from C to C 2 . </li></ul></ul><ul><li>The wavelet from C; </li></ul><ul><ul><li>spends a longer time in, </li></ul></ul><ul><ul><li>less dense medium 1 in travelling, </li></ul></ul><ul><ul><li>from C to C 1 than for, </li></ul></ul><ul><ul><li>B to travel from B to B 1 . </li></ul></ul>
  149. 151. Refraction & Huygen’s Principal <ul><li>Thus there is less time for the wave; </li></ul><ul><ul><li>to travel C 1 to C 2 in the denser medium 2, </li></ul></ul><ul><ul><li>than for the wavelet to travel from B 1 to B 2 . </li></ul></ul><ul><li>The distance CC 1 is thus less than BB 1 . </li></ul>
  150. 152. Refraction & Huygen’s Principal <ul><li>The new wavefront at the end of this time; </li></ul><ul><ul><li>envelope of tangents to, </li></ul></ul><ul><ul><li>the wavelet wavefronts at A 1 B 2 C 2 D 1 . </li></ul></ul><ul><li>The direction of movement of the wavefront has changed; </li></ul><ul><ul><li>refracted towards the normal at point A. </li></ul></ul>
  151. 153. Deriving Snell’s Law <ul><li>This can be stated as a mathematical formula. </li></ul><ul><li>It was first discovered; </li></ul><ul><ul><li>in 1621 </li></ul></ul><ul><ul><li>Dutch physicist, </li></ul></ul><ul><ul><li>Snell. </li></ul></ul>
  152. 154. Deriving Snell’s Law <ul><li>Since the wave; </li></ul><ul><ul><li>travels in a direction, </li></ul></ul><ul><ul><li>perpendicular to its wavefront, </li></ul></ul><ul><li> IAD = 90 o and </li></ul><ul><li> AA 1 D 1 = 90 o . </li></ul>
  153. 155. Deriving Snell’s Law <ul><li>From  IAD, </li></ul><ul><ul><li>i +  = 90 o , </li></ul></ul><ul><ul><li>so i = 90 o -  . </li></ul></ul><ul><li>From  N’AD 1 , </li></ul><ul><ul><li>R +  = 90 o , </li></ul></ul><ul><ul><li>so R = 90 o -  . </li></ul></ul><ul><li>But from  D 1 AN, </li></ul><ul><ul><li> +  = 90 o , </li></ul></ul>
  154. 156. Deriving Snell’s Law <ul><li>so  = 90 o -  , </li></ul><ul><ul><li>so  = i . </li></ul></ul><ul><li>In triangle D 1 AA 1 , </li></ul><ul><ul><li> +  + 90 o = 180 o . </li></ul></ul><ul><li> = 90 o -  , </li></ul><ul><ul><li>so  = R </li></ul></ul>
  155. 157. Deriving Snell’s Law <ul><li>In triangle ADD 1 , </li></ul><ul><ul><li>sin i = v 1 t /AD 1 </li></ul></ul><ul><li>In triangle AA 1 D 1 , </li></ul><ul><ul><li>sin R = v 2 t /AD 1 </li></ul></ul><ul><li>Dividing </li></ul>
  156. 158. Deriving Snell’s Law <ul><li>1 n 2 = a constant </li></ul>
  157. 159. Snell’s Law <ul><li>Snell’s law states that the ratio of the sine of the angle of incidence to the sine of the refraction is constant and equals the ratio of the velocity of the wave in the incident medium to the velocity of the refracting medium. </li></ul>
  158. 160. Snell’s Law
  159. 161. Snell’s Law <ul><li>The constant is a number without units; </li></ul><ul><ul><li>as it is a ratio and is called the refractive index. </li></ul></ul>
  160. 162. Snell’s Law <ul><li>When a wave is incident from a vacuum; </li></ul><ul><ul><li>on to a medium M, </li></ul></ul><ul><ul><li>the refractive index is written n M </li></ul></ul><ul><ul><li>is called the absolute refractive index </li></ul></ul><ul><ul><li>of medium M. </li></ul></ul>
  161. 163. Example 5 <ul><li>Light strikes a glass block at an angle of 60 o to the surface. If air n glass = 1.5, calculate: </li></ul><ul><li>(a) the angle of refraction (R) and </li></ul><ul><li>(b) the angle of deviation (D). </li></ul>
  162. 164. Solution
  163. 165. Solution (a) <ul><li>i = 30 0 </li></ul><ul><li>air n glass = 1.5 </li></ul>
  164. 166. Solution (a) <ul><li>air n glass </li></ul>sin R = sin R = R = 19.5 o R = 20 o (2 sig digits)
  165. 167. Solution (b)
  166. 168. Solution (b) <ul><li>i = 30 0 </li></ul><ul><li>R = 19.5 o </li></ul><ul><li>D = i - R </li></ul><ul><li>D = 30 – 19.5 </li></ul><ul><li>D = 10.5 o </li></ul><ul><li>D = 11 o (2 sig digits) </li></ul>
  167. 169. Snell’s Law <ul><li>When the refractive index 1 n 2; </li></ul><ul><ul><li>does not involve a vacuum, </li></ul></ul><ul><ul><li>as the incident medium, </li></ul></ul><ul><ul><li>the value is determined by, </li></ul></ul><ul><ul><li>the absolute refractive indices of each medium. </li></ul></ul>
  168. 170. Snell’s Law
  169. 171. Snell’s Law <ul><li>If the wave direction is reversed then; </li></ul><ul><li>We know that v = f  , </li></ul><ul><ul><li>frequency cannot be changed, </li></ul></ul><ul><ul><ul><li>this is determined by the wave source, </li></ul></ul></ul><ul><ul><li>Snell’s law can be expanded to; </li></ul></ul>
  170. 172. Snell’s Law <ul><li>Equation that summarises all of the above is; </li></ul>
  171. 173. Example 6 <ul><li>Light of wavelength 500 nm is incident on a block of glass ( air n glass ) at 60 o to the glass surface. </li></ul><ul><li>Calculate (a) the velocity, and (b) the wavelength of the light waves in the glass. </li></ul>
  172. 174. Solution (a) <ul><li>v air = 3.0 x 10 8 m s -1 </li></ul><ul><li>air n glass = 1.5 </li></ul>
  173. 175. Snell’s Law <ul><li>v glass = 2.0 x 10 8 m s -1 </li></ul>
  174. 176. Solution (b)
  175. 177. Snell’s Law (b) <ul><li>air n glass = 1.5 </li></ul><ul><li>i = 30 o </li></ul><ul><li> air = 500 nm = 5.0 x 10 -7 m </li></ul><ul><li>v air = 3.0 x 10 8 m s -1 </li></ul>
  176. 178. Snell’s Law (b) <ul><li> glass = 3.3 x 10 -7 m </li></ul>
  177. 179. Refraction <ul><li>Refraction occurs because the wave; </li></ul><ul><ul><li>takes the path that takes the least time, </li></ul></ul><ul><ul><li>Fermat’s Principle. </li></ul></ul>
  178. 180. Refraction <ul><li>Consider a lifesaver. </li></ul><ul><li>If a person was in trouble; </li></ul><ul><ul><li>and the lifesaver could move at the same speed, </li></ul></ul><ul><ul><li>in water as on sand, </li></ul></ul><ul><ul><li>the fastest route would be a straight line. </li></ul></ul>
  179. 181. Refraction <ul><li>As he can move faster on the sand; </li></ul><ul><ul><li>the fastest route would be the one that, </li></ul></ul><ul><ul><li>spends more time on sand. </li></ul></ul>
  180. 182. Refraction <ul><li>Refraction also takes place; </li></ul><ul><ul><li>when a wave moves through water. </li></ul></ul><ul><li>Waves travel faster in deep water; </li></ul><ul><ul><li>than shallow water. </li></ul></ul>
  181. 183. Refraction
  182. 184. Refraction <ul><li>As the waves move more slowly in shallow water; </li></ul><ul><ul><li>the crests are closer together. </li></ul></ul><ul><li>Diagram above; </li></ul><ul><ul><li>each line represents a crest, </li></ul></ul><ul><ul><li>called a wavefront. </li></ul></ul>
  183. 185. Refraction <ul><li>Waves can also be refracted in air. </li></ul><ul><li>This can happen when; </li></ul><ul><ul><li>winds are uneven or, </li></ul></ul><ul><ul><li>when sound travels through air, </li></ul></ul><ul><ul><li>of uneven temperature. </li></ul></ul>
  184. 186. Refraction <ul><li>On a warm day; </li></ul><ul><ul><li>the air near the ground may be warmer, </li></ul></ul><ul><ul><li>the sound will travel faster. </li></ul></ul><ul><li>The refraction is gradual; </li></ul><ul><ul><li>the sound tends to bend away, </li></ul></ul><ul><ul><li>from the ground. </li></ul></ul><ul><li>This makes the sound appear; </li></ul><ul><ul><li>not to travel well. </li></ul></ul>
  185. 187. Refraction
  186. 188. Refraction <ul><li>On a cold day or night; </li></ul><ul><ul><li>when the air near the ground is colder than above, </li></ul></ul><ul><ul><li>the speed of sound near the ground is reduced. </li></ul></ul>
  187. 189. Refraction <ul><li>The higher speed of the wave fronts; </li></ul><ul><ul><li>cause bending of the sound towards Earth. </li></ul></ul><ul><li>This causes sound to be heard; </li></ul><ul><ul><li>over considerably longer distances. </li></ul></ul><ul><li>Occurs with light. </li></ul>
  188. 190. Diffraction of Water, Sound & Light. <ul><li>The bending of waves by means other; </li></ul><ul><ul><li>than reflection or refraction, </li></ul></ul><ul><ul><li>is called diffraction. </li></ul></ul>
  189. 191. Diffraction of Water, Sound & Light. <ul><li>In water; </li></ul><ul><ul><li>when the opening is wide compared to the wavelength, </li></ul></ul><ul><ul><li>the effect is small. </li></ul></ul><ul><li>As the opening becomes narrower, </li></ul><ul><ul><li>the effect is more pronounced. </li></ul></ul>
  190. 192. Diffraction of Water, Sound & Light.
  191. 193. Diffraction of Water, Sound & Light. <ul><li>This also happens when light; </li></ul><ul><ul><li>passes through an opening. </li></ul></ul><ul><li>If the opening is small; </li></ul><ul><ul><li>such as a thin razor blade, </li></ul></ul><ul><ul><li>the shadow becomes fuzzy. </li></ul></ul>
  192. 194. Diffraction of Water, Sound & Light.
  193. 195. Diffraction of Water, Sound & Light. <ul><li>Diffraction can occur to some degree; </li></ul><ul><ul><li>for all shadows. </li></ul></ul><ul><li>The amount of diffraction; </li></ul><ul><ul><li>depends on the wavelength. </li></ul></ul><ul><li>Long waves ‘fill in’ shadows; </li></ul><ul><ul><li>better than short ones. </li></ul></ul><ul><li>Fog Horns emit low frequency sound; </li></ul><ul><ul><li>to ‘fill in’ blind spots. </li></ul></ul>
  194. 196. Diffraction of Water, Sound & Light. <ul><li>AM radio waves are long; </li></ul><ul><ul><li>compared to the objects in their path, </li></ul></ul><ul><ul><li>bend around small buildings. </li></ul></ul><ul><li>FM waves are shorter; </li></ul><ul><ul><li>don’t diffract around buildings as well. </li></ul></ul><ul><li>This is why some areas have; </li></ul><ul><ul><li>poor FM reception compared to AM. </li></ul></ul>
  195. 197. Diffraction of Water, Sound & Light. <ul><li>When objects are about the same size; </li></ul><ul><ul><li>as the wavelength of light (10 -7 m), </li></ul></ul><ul><ul><li>the image through a microscope is very blurred. </li></ul></ul><ul><li>If it is smaller than light; </li></ul><ul><ul><li>no image can be seen. </li></ul></ul><ul><li>To overcome this; </li></ul><ul><ul><li>illuminate objects with shorter wavelengths. </li></ul></ul>
  196. 198. Diffraction of Water, Sound & Light. <ul><li>Electrons have a wavelength. </li></ul><ul><li>Electron microscopes are used; </li></ul><ul><ul><li>to illuminate objects that are very small, </li></ul></ul><ul><ul><li>as the diffraction effect is less. </li></ul></ul>
  197. 199. Diffraction of Water, Sound & Light. <ul><li>This concept is also used by dolphins. </li></ul><ul><li>They use high frequency sound - ultrasound. </li></ul><ul><li>Echoes of long wavelength sound; </li></ul><ul><ul><li>gives the dolphin an overall image of objects. </li></ul></ul><ul><li>Shorter wavelength sound is used; </li></ul><ul><ul><li>to examine their surroundings in more detail. </li></ul></ul>
  198. 200. Diffraction of Water, Sound & Light. <ul><li>This concept is also used by dolphins. </li></ul><ul><li>They use high frequency sound - ultrasound. </li></ul><ul><li>Echoes of long wavelength sound; </li></ul><ul><ul><li>gives the dolphin an overall image of objects. </li></ul></ul>
  199. 201. Diffraction of Water, Sound & Light. <ul><li>Shorter wavelength sound is used; </li></ul><ul><ul><li>to examine their surroundings in more detail. </li></ul></ul><ul><li>They can ‘see’ bones, teeth, gas cavities; </li></ul><ul><ul><li>as well as cancers and tumours. </li></ul></ul><ul><li>Medical science can now reproduce; </li></ul><ul><ul><li>what dolphins can do naturally. </li></ul></ul>
  200. 202. What Happens When Two Waves Meet? <ul><li>As the velocity of all pulses; </li></ul><ul><ul><li>in a given medium is the same, </li></ul></ul><ul><ul><li>no matter what shape or amplitude, </li></ul></ul><ul><ul><li>each pulse may have. </li></ul></ul><ul><li>Two pulses can only meet therefore; </li></ul><ul><ul><li>only if they are travelling, </li></ul></ul><ul><ul><li>in opposite directions. </li></ul></ul>
  201. 203. What Happens When Two Waves Meet? Superposition
  202. 204. What Happens When Two Waves Meet? <ul><li>The pulses retain their shape; </li></ul><ul><ul><li>velocity and amplitude, </li></ul></ul><ul><ul><li>as they emerge from the interaction. </li></ul></ul>
  203. 205. What Happens When Two Waves Meet? <ul><li>When the waves meet; </li></ul><ul><ul><li>the particles suffer displacements, </li></ul></ul><ul><ul><li>which at any instant are the vector sums, </li></ul></ul><ul><ul><li>of the displacements due to, </li></ul></ul><ul><ul><li>each separate wave. </li></ul></ul>
  204. 206. What Happens When Two Waves Meet? <ul><li>Forces acting on particles of the spring; </li></ul><ul><ul><li>given at any instant by the vector sums, </li></ul></ul><ul><ul><li>of the forces due to each separate wave. </li></ul></ul>
  205. 207. What Happens When Two Waves Meet? <ul><li>The statements regarding; </li></ul><ul><ul><li>the resultant displacements, </li></ul></ul><ul><ul><li>and forces are aspects, </li></ul></ul><ul><ul><li>of the principle of superposition . </li></ul></ul>
  206. 208. Standing Waves in 1 Dimension <ul><li>When two waves of equal wavelength; </li></ul><ul><ul><li>frequency and amplitude meet, </li></ul></ul><ul><ul><li>they produce a wave whose shape, </li></ul></ul><ul><ul><li>is determined by the principle of superposition. </li></ul></ul><ul><li>Under the right conditions; </li></ul><ul><ul><li>a standing wave may result. </li></ul></ul>
  207. 209. Standing Waves in 1 Dimension <ul><li>The standing wave pattern produces; </li></ul><ul><ul><li>some points that do not oscillate </li></ul></ul><ul><ul><ul><li>Nodes </li></ul></ul></ul><ul><ul><li>others that have a maximum oscillation </li></ul></ul><ul><ul><ul><li>Antinodes. </li></ul></ul></ul>
  208. 210. Standing Waves in 1 Dimension <ul><li>The process by which waves interact; </li></ul><ul><ul><li>to produce a standing wave, </li></ul></ul><ul><ul><li>is called interference. </li></ul></ul><ul><li>Identical waves that are travelling; </li></ul><ul><ul><li>in opposite directions, </li></ul></ul><ul><ul><li>can easily be produced by reflection. </li></ul></ul>
  209. 211. Standing Waves in 1 Dimension <ul><li>Two wave sources can be used; </li></ul><ul><ul><ul><li>eg 2 signal generators, </li></ul></ul></ul><ul><ul><li>but it is more difficult. </li></ul></ul>
  210. 212. Standing Waves in 1 Dimension
  211. 213. Standing Waves in 1 Dimension
  212. 214. Standing Waves in 1 Dimension
  213. 215. Standing Waves in 1 Dimension
  214. 216. Standing Waves in 1 Dimension
  215. 217. Criteria for Interference in 2 D <ul><li>Consider a ripple tank; </li></ul><ul><ul><li>with two dippers producing waves, </li></ul></ul><ul><ul><li>of the same frequency and in phase. </li></ul></ul><ul><li>A two dimensional standing wave; </li></ul><ul><ul><li>would be seen. </li></ul></ul>
  216. 218. Criteria for Interference in 2 D
  217. 219. Criteria for Interference in 2 D <ul><li>Even if the dippers were out of phase; </li></ul><ul><ul><li>by  radians (  /2), </li></ul></ul><ul><ul><li>the 2D standing wave pattern would still be seen. </li></ul></ul><ul><li>In both cases, the dippers maintain; </li></ul><ul><ul><li>a constant phase relationship, </li></ul></ul><ul><ul><li>referred to as mutually coherent sources. </li></ul></ul>
  218. 220. Criteria for Interference in 2 D <ul><li>Mutually coherent wave sources maintain a constant phase relationship. </li></ul>
  219. 221. Criteria for Interference in 2 D
  220. 222. Criteria for Interference in 2 D <ul><li>For a point to be on a nodal line; </li></ul><ul><ul><li>difference between its distance, </li></ul></ul><ul><ul><li>from one source and the other source, </li></ul></ul><ul><ul><ul><li>called the geometric path difference, G.P.D. </li></ul></ul></ul><ul><ul><li>must be an odd number of half wavelengths. </li></ul></ul><ul><ul><li>In the diagram above; </li></ul></ul>
  221. 223. Criteria for Interference in 2 D <ul><li>For P 1 </li></ul><ul><li>G.P.D. is given by: </li></ul><ul><li>S 1 P 1 | - |S 2 P 1 | = </li></ul>
  222. 224. Criteria for Interference in 2 D <ul><li>For P 2 </li></ul><ul><li>The G.P.D. is given by: </li></ul>
  223. 225. Criteria for Interference in 2 D <ul><li>In general, </li></ul><ul><li>annulment occurs when </li></ul><ul><li>G.P.D. = (2m+1)  /2, </li></ul><ul><ul><li>m = 0,1,2,.......... </li></ul></ul>
  224. 226. Criteria for Interference in 2 D <ul><li>P 4 is a point on an antinodal line </li></ul><ul><li>The G.P.D. is given by: </li></ul>
  225. 227. Criteria for Interference in 2 D <ul><li>For any point on an antinodal line; </li></ul><ul><ul><li>G.P.D. must be an even number of  /2. </li></ul></ul><ul><li>This means that reinforcement occurs; </li></ul><ul><ul><li>when G.P.D. = m  , </li></ul></ul><ul><li>m = 0,1,2,........ </li></ul>
  226. 228. Criteria for Interference in 2 D <ul><li>If one source reversed its phase; </li></ul><ul><ul><li>it would be  radians out of phase. </li></ul></ul><ul><li>For the wave to reinforce the wave from the other source; </li></ul><ul><ul><li>G.P.D. must = (2m+1)  /2, </li></ul></ul><ul><ul><li>in order to get back in step. </li></ul></ul><ul><li>For annulment; </li></ul><ul><ul><li>G.P.D. = m  . </li></ul></ul>
  227. 229. Criteria for Interference in 2 D <ul><li>If both sources suffer a phase reversal; </li></ul><ul><ul><li>the conditions for reinforcement remains the same, </li></ul></ul><ul><ul><li>as though neither source had undergone phase reversal. </li></ul></ul><ul><li>If wavelength decreased; </li></ul><ul><ul><li>nodal & antinodal lines would be closer together. </li></ul></ul>
  228. 230. Criteria for Interference in 2 D <ul><li>This would mean the bandwidth; </li></ul><ul><ul><ul><li>distance between consecutive nodal and antinodal lines </li></ul></ul></ul><ul><ul><li>would decrease. </li></ul></ul><ul><li>The bandwidth would also decrease if; </li></ul><ul><ul><li>distance between the sources is increased. </li></ul></ul>
  229. 231. Criteria for Interference in 2 D Phase relationship Annulment Reinforcement in phase G.P.D. = (2m+1)  /2 G.P.D. = m  phase reversal of one wave G.P.D. = m  G.P.D. = (2m+1)  /2 phase reversal of both waves G.P.D. = (2m+1)  /2 G.P.D. = m 