Waves in 2 Dimensions


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Waves in 2 Dimensions

  1. 1. Unit 1 - Waves <ul><li>Many waves have the ability to travel in more than one dimension. Two dimensional waves have the ability to travel around corners. </li></ul><ul><li>Sound waves are a good example of 2 dimensional waves. </li></ul><ul><li>Water waves are visible 2 dimensional waves, and as a result we will study the behavior of water waves. </li></ul>Waves in 2 Dimensions
  2. 2. Unit 1 - Waves <ul><li>The study of waves in two dimensions is often done using a ripple tank. A ripple tank is a large glass-bottomed tank of water that is used to study the behavior of water waves. A light shines upon the water from above and illuminates a white sheet of paper placed below the tank. </li></ul><ul><li>Because rays of light undergo bending as they pass through the troughs and crests, there is a pattern of light and dark spots on the white sheet of paper. The dark spots represent wave troughs and the bright spots represent wave crests . As the water waves move through the ripple tank, the dark and bright spots move as well. As the waves encounter obstacles in their path, their behavior can be observed by watching the movement of the dark and bright spots on the sheet of paper. </li></ul>Waves in 2 Dimensions
  3. 3. Unit 1 - Waves Ripple Tank
  4. 4. Unit 1 - Waves <ul><li>A wave coming from a point source is circular, whereas a wave originating from a linear source is straight. The wave equation and its properties of velocity, wavelength, and frequency hold true for two dimensional waves as well. </li></ul><ul><li>When examining a wave, the wavefront is referred to as a continuous crest or trough. The waveray represents the direction of motion of a point on the wavefront. The direction of motion of the waveray is perpendicular to the wavefront at that point. </li></ul>Waves in 2 Dimensions
  5. 5. Unit 1 - Waves Wave Front’s & Wave Rays
  6. 6. Unit 1 - Waves Wave Properties Group Assignment
  7. 7. Unit 1 - Waves Reflection of Two Dimensional Waves
  8. 8. Unit 1 - Waves <ul><li>The incident ray refers to the incoming wave ray. </li></ul><ul><li>The reflected wave ray refers to the outgoing wave ray. </li></ul><ul><li>The normal refers to the imaginary line that is perpendicular to the barrier. </li></ul><ul><li>The angle of incidence refers to the angle between the incident ray and the normal. The angle of reflection is the angle between the reflected ray and the normal. </li></ul><ul><li>A straight wave will be reflected back along its path if it hits a straight barrier at right angles (wave ray perpendicular to barrier). The angle of incidence equals the angle of reflection. </li></ul><ul><li>A straight wave reflecting off a barrier at an angle will reflect such that the angle of incidence ( i ) equals the angle of reflection ( r ). </li></ul>Reflection of Two Dimensional Waves
  9. 9. Unit 1 - Waves Reflection of Two Dimensional Waves
  10. 10. Unit 1 - Waves <ul><li>A circular wave reflecting off a straight barrier results in i = r . </li></ul><ul><li>A straight wave striking a parabolic reflector also obeys the laws of reflection. The waves are reflected to a focal point. </li></ul><ul><li>If a wave is generated from the focal point towards a parabolic reflector, the reflected wave will be a straight wave. </li></ul>Reflection of Two Dimensional Waves
  11. 11. Unit 1 - Waves Reflection of Two Dimensional Waves
  12. 12. Unit 1 - Waves <ul><li>Refraction of waves involves a change in the direction of waves as they pass from one medium to another. </li></ul><ul><li>Refraction, or bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. The frequency remains unchanged. </li></ul><ul><li>A wave traveling in deep water has a speed expressed by v 1 = f 1  1 . In shallow water the equation is v 2 = f 2  2 . </li></ul><ul><li>Therefore: </li></ul><ul><li>v 1 =  1 </li></ul><ul><li>v 2 =  2 </li></ul>Refraction of Two Dimensional Waves
  13. 13. Unit 1 - Waves <ul><li>Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down. </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>The change in direction will only occur if the wave meets the boundary between the two mediums at an angle, not straight on. </li></ul><ul><li>When a wave enters a medium at an angle and its speed decreases, the refracted ray is bent towards the normal. </li></ul><ul><li>When the wave enters a medium in which the speed increases, the wave is refracted away from the normal. </li></ul>Refraction of Two Dimensional Waves
  14. 14. Unit 1 - Waves <ul><li>Snell’s Law allows scientists to predict the direction a wave ray would take in various media. </li></ul>Refraction of Two Dimensional Waves
  15. 15. Unit 1 - Waves <ul><li>A mathematical equation relating the two angles (angles of incidence and refraction) and speed of the two waves on each side of the boundary is known as the Snell's Law equation and is expressed as follows. </li></ul><ul><li>sin  i /sin  r = v 1 /v 2 </li></ul><ul><li>  </li></ul><ul><li>The ratio of the speeds is equal to the index of refraction. </li></ul><ul><li>1 n 2 = v 1 /v 2 </li></ul><ul><li>  </li></ul><ul><li>therefore </li></ul><ul><li>  </li></ul><ul><li>sin  i /sin  r = 1 n 2 </li></ul>Refraction of Two Dimensional Waves
  16. 16. Unit 1 - Waves Refraction of Two Dimensional Waves
  17. 17. Unit 1 - Waves <ul><li>Diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path.   </li></ul><ul><li>Water waves have the ability to travel around corners, around obstacles and through openings. This ability is most obvious for water waves with longer wavelengths. </li></ul><ul><li>The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength. </li></ul><ul><li>When the wavelength of the waves are smaller than the obstacle, no noticeable diffraction occurs. </li></ul>Diffraction of Two Dimensional Waves
  18. 18. Unit 1 - Waves <ul><li>The amount of diffraction is increased when the relative size of the opening is decreased. </li></ul>Diffraction of Two Dimensional Waves
  19. 19. Unit 1 - Waves Diffraction of Two Dimensional Waves
  20. 20. Unit 1 - Waves Diffraction of Two Dimensional Waves
  21. 21. Unit 1 - Waves <ul><li>Interference in two dimensional water waves may be both constructive or destructive. </li></ul><ul><li>As in 1-dimensional waves, interfering waves that have the same wavelength, amplitude, and frequency will result in nodes and antinodes. </li></ul>Interference of Two Dimensional Waves
  22. 22. Unit 1 - Waves <ul><li>The interference pattern formed from two identical point sources results in a symmetrical pattern of nodal lines </li></ul>Interference of Two Dimensional Waves
  23. 23. Unit 1 - Waves <ul><li>The central region of constructive interference is known as the central maximum. </li></ul><ul><li>Destructive interference occurs when the crest of one wave meets the trough of another wave. </li></ul><ul><li>On either side of the central maximum are the first order nodes, N 1 . These are regions of destructive interference. </li></ul><ul><li>On either side of N 1 are the next antinodes, A 1 . </li></ul>Interference of Two Dimensional Waves
  24. 24. Unit 1 - Waves <ul><li>This alternating pattern of nodes and antinodes continues throughout the construction. </li></ul><ul><li>Constructive interference occurs when the crests or troughs of two circular waves meet. </li></ul>Interference of Two Dimensional Waves
  25. 25. Unit 1 - Waves Interference of Two Dimensional Waves
  26. 26. Unit 1 - Waves <ul><li>The two point source interference pattern is useful because it allows for direct measurement of the wavelength while the interference pattern remains relatively stationary. </li></ul><ul><li>The relationship between the differences in path length between the two point sources and a nodal point can be used to determine the wavelength. </li></ul>The Path Difference Relationship
  27. 27. Unit 1 - Waves <ul><li>The equation below shows the relationship between path length and wavelength. </li></ul><ul><li> P n S 1 – P n S 2  = (n – ½)  </li></ul>The Path Difference Relationship
  28. 28. Unit 1 - Waves <ul><li>The relationship can be used in the ripple tank to find the wavelength of interfering waves. A wavelength can be calculated simply by locating a point on a specific nodal line, measuring the path lengths, and substituting in the above equation. </li></ul>The Path Difference Relationship
  29. 29. Unit 1 - Waves <ul><li>Sample Problem: </li></ul><ul><li>Two point sources generate identical waves that interfere in a ripple tank. The sources are located 5.0 cm apart, and the frequency of the waves is 8 Hz. A point on the first nodal line is 10 cm from one source and 11 cm from the other source. What is the speed of the waves? </li></ul>The Path Difference Relationship