1. PHY2048, Lecture 23 – Wave Motion Chapter 14 – Tuesday November 16th Fifth and final mini exam on Thu. (Chs. 13-15) Fifth and final mini exam on Thu. (Chs. 13-15) Make up labs Nov. 29 at 12:30pm and 3:30pm in UPL107 Make up labs Nov. 29 at 12:30pm and 3:30pm in UPL107 Final exam – Wed. Dec. 8 th, 10am to noon, in MOR 104 Final exam – Wed. Dec. 8 th, 10am to noon, in MOR 104 (Moore Auditorium) Review of wave solutionsReview of wave solutions The wave equationThe wave equation Wave superposition and interferenceWave superposition and interference Spatial interference and standing wavesSpatial interference and standing waves Temporal beatingTemporal beating Sources of musical soundSources of musical sound Doppler effect (if time)Doppler effect (if time) Stephen Hill (covering for Prof. Wiedenhover)

2. Review - wavelength and frequency k is the angular wavenumber. Transverse wave  is the angular frequency. frequency velocity Amplitude Displacement Phase } shift angular wavenumber angular frequency

3. Traveling waves on a stretched string  is the string's linear density, or mass per unit length. Tension F T provides the restoring force (kg.m.s -2 ) in the string. Without tension, the wave could not propagate.Tension F T provides the restoring force (kg.m.s -2 ) in the string. Without tension, the wave could not propagate. The mass per unit length  (kg.m -1 ) determines the response of the string to the restoring force (tension), through Newtorn's 2nd law.The mass per unit length  (kg.m -1 ) determines the response of the string to the restoring force (tension), through Newtorn's 2nd law. Look for combinations of F T and  that give dimensions of speed (m.s -1 ).Look for combinations of F T and  that give dimensions of speed (m.s -1 ). FTFT FTFT F net a a a +ve curve  ve curve Zero curve

4. Traveling waves on a stretched string  is the string's linear density, or mass per unit length. The Wave Equation FTFT FTFT F net a a a +ve curve  ve curve Zero curve ll mass transverse acceleration Dimensionless parameter proportional to curvature

5. The wave equation General solution:General solution: v v

6. The principle of superposition for waves It often happens that waves travel simultaneously through the same region, e.g.It often happens that waves travel simultaneously through the same region, e.g.  Radio waves from many broadcasters  Sound waves from many musical instruments  Different colored light from many locations from your TV And have solutions:And have solutions: This is a result of the principle of superposition, which applies to all harmonic waves, i.e., waves that obey the linear wave equationThis is a result of the principle of superposition, which applies to all harmonic waves, i.e., waves that obey the linear wave equation Nature is such that all of these waves can exist without altering each others' motionNature is such that all of these waves can exist without altering each others' motion Their effects simply addTheir effects simply add

7. The principle of superposition for waves If two waves travel simultaneously along the same stretched string, the resultant displacement y' of the string is simply given by the summationIf two waves travel simultaneously along the same stretched string, the resultant displacement y' of the string is simply given by the summation where y 1 and y 2 would have been the displacements had the waves traveled alone. Overlapping waves algebraically add to produce a resultant wave (or net wave). Overlapping waves do not in any way alter the travel of each other This is the principle of superposition.This is the principle of superposition.

8. Interference of waves Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such thatSuppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such that The waves will add.The waves will add.

9. Interference of waves Noise canceling headphones

10. Interference of waves Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such thatSuppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such that The waves will add.The waves will add. If they are in phase (i.e.  = 0 ), they combine to double the displacement of either wave acting alone.If they are in phase (i.e.  = 0 ), they combine to double the displacement of either wave acting alone. If they are out of phase (i.e.  =  ), they combine to cancel everywhere, since sin(  ) =  sin(  ).If they are out of phase (i.e.  =  ), they combine to cancel everywhere, since sin(  ) =  sin(  ). This phenomenon is called interference.This phenomenon is called interference.

11. Interference of waves Mathematical proof:Mathematical proof:Then: But: So: Amplitude Wave part Phase shift

12. Interference of waves If two sinusoidal waves of the same amplitude and frequency travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in the same direction. If  = 0, the waves interfere constructively, cos½  = 1 and the wave amplitude is 2y m.If  = 0, the waves interfere constructively, cos½  = 1 and the wave amplitude is 2y m. If  = , the waves interfere destructively, cos(  /2) = 0 and the wave amplitude is 0, i.e. no wave at all.If  = , the waves interfere destructively, cos(  /2) = 0 and the wave amplitude is 0, i.e. no wave at all. All other cases are intermediate between an amplitude of 0 and 2y m.All other cases are intermediate between an amplitude of 0 and 2y m. Note that the phase of the resultant wave also depends on the phase difference.Note that the phase of the resultant wave also depends on the phase difference. Adding waves as vectors (phasors) described by amplitude and phase Adding waves as vectors (phasors) described by amplitude and phase

13. Interference - Standing Waves If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave. This is clearly not a traveling wave, because it does not have the form f(kx  t).This is clearly not a traveling wave, because it does not have the form f(kx  t). In fact, it is a stationary wave, with a sinusoidal varying amplitude 2y m cos(  t).In fact, it is a stationary wave, with a sinusoidal varying amplitude 2y m cos(  t). t dependencex dependence Link

14. Reflections at a boundary Waves reflect from boundaries.Waves reflect from boundaries. This is the reason for echoes - you hear sound reflecting back to you.This is the reason for echoes - you hear sound reflecting back to you. However, the nature of the reflection depends on the boundary condition.However, the nature of the reflection depends on the boundary condition. For the two examples on the left, the nature of the reflection depends on whether the end of the string is fixed or loose.For the two examples on the left, the nature of the reflection depends on whether the end of the string is fixed or loose.

15. Standing waves and resonance At ordinary frequencies, waves travel backwards and forwards along the string.At ordinary frequencies, waves travel backwards and forwards along the string. Each new reflected wave has a new phase.Each new reflected wave has a new phase. The interference is basically a mess, and no significant oscillations build up.The interference is basically a mess, and no significant oscillations build up.

16. Standing waves and resonance However, at certain special frequencies, the interference produces strong standing wave patterns.However, at certain special frequencies, the interference produces strong standing wave patterns. Such a standing wave is said to be produced at resonance.Such a standing wave is said to be produced at resonance. These certain frequencies are called resonant frequencies.These certain frequencies are called resonant frequencies.

17. Standing waves and resonance Standing waves occur whenever the phase of the wave returning to the oscillating end of the string is precisely in phase with the forced oscillations.Standing waves occur whenever the phase of the wave returning to the oscillating end of the string is precisely in phase with the forced oscillations. Thus, the trip along the string and back should be equal to an integral number of wavelengths, i.e.Thus, the trip along the string and back should be equal to an integral number of wavelengths, i.e. Each of the frequencies f 1, f 1, f 1, etc, are called harmonics, or a harmonic series; n is the harmonic number.Each of the frequencies f 1, f 1, f 1, etc, are called harmonics, or a harmonic series; n is the harmonic number. determined by geometry determined by geometry

18. Standing waves and resonance Here is an example of a two-dimensional vibrating diaphragm.Here is an example of a two-dimensional vibrating diaphragm. The dark powder shows the positions of the nodes in the vibration.The dark powder shows the positions of the nodes in the vibration.

19. Standing waves in air columns Simplest case:Simplest case: - 2 open ends - Antinode at each end - 1 node in the middle Although the wave is longitudinal, we can represent it schematically by the solid and dashed green curves.Although the wave is longitudinal, we can represent it schematically by the solid and dashed green curves.

20. Standing waves in air columns A harmonic series 2 3 4

21. Standing waves in air columns A different harmonic series 1 3 5 7

22. Musical instruments Flute Oboe Saxophone Link

23. Musical instruments n = even if B.C. same at both ends of pipe/string n = odd if B.C. different at the two ends

24. Wave interference - spatial

25. Interference - temporal (or beats) In order to obtain a spatial interference pattern, we placed two sources at different locations, i.e. we varied the first term in the phase of the waves.In order to obtain a spatial interference pattern, we placed two sources at different locations, i.e. we varied the first term in the phase of the waves. We can do the same in the time domain whereby, instead of placing sources at different locations, we give them different angular frequencies   and  . For simplicity, we analyze the sound at x = 0.We can do the same in the time domain whereby, instead of placing sources at different locations, we give them different angular frequencies   and  . For simplicity, we analyze the sound at x = 0.

26. Interference - temporal (or beats) A maximum amplitude occurs whenever  't has the value  or .A maximum amplitude occurs whenever  't has the value  or . This happens twice in each time period of the cosine function.This happens twice in each time period of the cosine function. Therefore, the beat frequency is twice the frequency  ', i.e.Therefore, the beat frequency is twice the frequency  ', i.e. Link

27. Doppler effect Consider a source of sound at the ‘proper frequency’, f ', moving relative to a stationary observer.Consider a source of sound at the ‘proper frequency’, f ', moving relative to a stationary observer. The observer will hear the sound with an apparent frequency, f, which is shifted from the proper frequency according to the following equation:The observer will hear the sound with an apparent frequency, f, which is shifted from the proper frequency according to the following equation: Here, v is the sound velocity (~330 m/s in air), and u is the relative speed between the source and detector.Here, v is the sound velocity (~330 m/s in air), and u is the relative speed between the source and detector. When the motion of the detector or source is towards the other, use the plus (+) sign so that the formula gives an upward shift in frequency. When the motion of the detector or source is away from the other, use the minus (  ) sign so that the formula gives a downward shift.

28. Mach cone angle:

29. Kinetic energy: dK = 1 / 2 dm v y 2 Divide both sides by dt, where dx/dt = v x Energy in traveling waves Similar expression for elastic potential energy Energy is pumped in an oscillatory fashion down the string Note: I dropped the subscript on v since it represents the wave speed dm =  dx