1. Oscillations and Waves

3. What is a wave?

4. How do the particles move?

5. Some definitions… 1) Amplitude – this is “how high” the wave is: 2) Wavelength ( ) – this is the distance between two corresponding points on the wave and is measured in metres: 3) Frequency – this is how many waves pass by every second and is measured in Hertz (Hz)Frequency Define the terms displacement, amplitude, frequency and period.

6. Describing waves T T Time Displacement ω = 2 π /T f = 1/T ω = 2 π f ω = Angular frequency f = FrequencyFrequency T = Time period 2π2π 1 cycle is described by 2π radians of phase phase Hyperlink What are radians? Hyperlink and scroll down

7. Phase and angle

8. Examples of phase difference

9. Oscillations http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm#motion

11. Simple harmonic motion Any motion that repeats itself after a certain period is known as a periodic motion, and since such a motion can be represented in terms of sines and cosines it is called a harmonic motion. The following are examples of simple harmonic motion: a test-tube bobbing up and down in water (Figure 1) a simple pendulum a compound pendulum a vibrating spring atoms vibrating in a crystal lattice a vibrating cantilever a trolley fixed between two springs a marble on a concave surface a torsional pendulum liquid oscillating in a U-tube a small magnet suspended over a horseshoe magnet an inertia balance

12. Data loggers Use the data loggers to find the variation of displacement with time for an oscillating mass on a spring. Process this data to find velocity and acceleration with time. Now use the data to obtain a graph of Acceleration versus displacement.

13. Analysing your graphs From the graph, find the Time period Angular frequency Amplitude The value of the velocity at maximum displacement The value of the acceleration at zero displacement The relationship between displacement and acceleration.

14. SHM definition Find the gradient of your graph of acceleration and displacement. Use this to calculate ω Calculate T from the graph. How the graph fit in with the definition of SHM?

15. SHM

16. Free body diagram for SHM Draw the free body diagram for the mass when it is in the centre of the motion At the top of the motion Between the bottom and the middle, down Between the bottom and the middle, heading upwards. http://www.acoustics.salford.ac.uk/feschools/waves/shm2.htm Hyperlink

17. Spring pendulum Spring Pendulum Hyperlink

18. Oscillations to wave motion

19. Restoring forces

20. Simple Harmonic Motion Consider a pendulum bob: Let’s draw a graph of displacement against time: Displacement Time Equilibrium position “Sinusoidal”

21. Pendulum Simple Pendulum Hyperlink

22. SHM Graphs Time Displacement Velocity Acceleration Time T

23. Definition of SHM Acceleration Displacement Now write your OWN definition of SHM

24. The Maths of SHM Time Displacement Therefore we can describe the motion mathematically as: x = x 0 cos ω t v = -x 0 ω sin ω t a = -x 0 ω 2 cos ω t a = - ω 2 x

25. Students are expected to understand the significance of the negative sign in the equation and to recall the connection between ω and T. ω = 2 π /T

26. SHM questions a x 5 2 1)Calculate the gradient of this graph 2)Use it to work out the value of ω 3)Use this to work out the time period for the oscillations 4)Ewan sets up a pendulum and lets it swing 10 times. He records a time of 20 seconds for the 10 oscillations. Calculate the period and the angular speed ω. 5)The maximum displacement of the pendulum is 3cm. Sketch a graph of a against x and indicate the maximum acceleration. a x

27. Questions Q’s 1 – 9 from the worksheet Using the equations V = V 0 cosωt V = V 0 sinωt x = x 0 cos ωt x = x 0 sin ωt V = ± ω√( x 0 2 -x 2 ) When do you use cos or sin?

29. 4.2 Energy changes during simple harmonic motion (SHM) Energy Displacement (x) -x 0 x0x0 At which points are -max displacement? -max velocity? -max acceleration? - max E k -max E p -max total energy? Total energy

30. SHM: Energy change Equilibrium position Energy Time GPE K.E.

31. Energy formulae Energy Displacement (x) -x 0 x0x0 Total energy E k = ½ mω 2 (x 0 2 – x 2 ) E p = ½ mω 2 x 2 E total = ½ mω 2 x 0 2

32. Questions

33. Answers

34. 4.3 Forced oscillations and resonance 4.3.1 State what is meant by damping. “It is sufficient for students to know that damping involves a force that is always in the opposite direction to the direction of motion of the oscillating particle and that the force is a dissipative force.”

36. Free and Forced oscillations

37. Forcing frequency too slow

38. Forcing frequency too fast

39. Forcing frequency equals natural frequency

40. Resonance

41. Resonance and frequency Physics Applets Hyperlink

42. Resonance and frequency The width of the curve (Q value) is determined by the damping in the system. The value of the resonant frequency depends factors such as the size of the object…..

43. Tacoma Narrows

44. Useful resonance Musical instruments Microwave ovens Electrical resonance when tuning a radio

45. Damping

46. Damped oscillations

47. Damping Amplitude of driven system Driver frequency Low damping High damping

48. Damping How much damping is best?

49. Critical damping

50. Wave characteristics The wave pulse transfers energy If the source continues to oscillate, then a continuous progressive wave is produced. Students should be able to distinguish between oscillations and wave motion, and appreciate that in many examples, the oscillations of the particles are simple harmonic.

51. Travelling Waves Definition: A travelling wave (or “progressive wave”) is one which travels out from the source that made it and transfers energy from one point to another. Energy dissipation Clearly, a wave will get weaker the further it travels. Assuming the wave comes from a point source and travels out equally in all directions we can say: Energy flux = (in Wm -2 ) Power (in W) Area (in m 2 ) φ = P 4 π r 2 An “inverse square law”

52. Example questions 1)Darryl likes doing his homework. His work is 2m from a 100W light bulb. Calculate the energy flux arriving at his book. 2)If his book has a surface area of 0.1m 2 calculate the total amount of energy on it per second (what assumption did you make?). 3)Matti doesn’t like the dark. He switches on a light and stands 3m away from it. If he is receiving a flux of 2.2Wm -2 what was the power of the bulb? 4)Matti walks 3m further away. What affect does this have on the amount of flux on him?

53. State that progressive (travelling) waves transfer energy. Students should understand that there is no net motion of the medium through which the wave travels.

54. Transverse vs. longitudinal waves Transverse waves are when the displacement is at right angles to the direction of the wave… Longitudinal waves are when the displacement is parallel to the direction of the wave… Displacement Direction Displacement

55. Transverse wave

56. Transverse waves Students should describe the waves in terms of the direction of oscillation of particles in the wave relative to the direction of transfer of energy by the wave. Students should know that light waves and water waves are transverse and that water waves cannot be propagated in gases or liquids.

58. Longitudinal waves Sound waves and earthquake P-waves are longitudinal

59. Longitudinal slinky

60. Loudspeaker

61. Describe waves in two dimensions, including the concepts of wavefronts and of rays. Energy is transferred in 2 dimensions Watch the wavefront(s) propagate

62. Wavefronts and rays.

63. Rays show the direction of travel of the energy. The wavefronts are where the crests of the waves are. The rays are always at 90 deg to the wavefronts. rays Wavefronts

65. Longitudinal waves Compressions and rarefactions

66. Transverse waves Crests Troughs

67. Displacement graphs

68. Define the terms displacement, amplitude, frequency, period, wavelength, wave speed and intensity WAVELENGTH - the distance from one crest to another or one trough to another. (In fact generally from any point on the wave to the next exactly similar point i.e. 2 consecutive points in phase) FREQUENCY - the number of vibrations of any part of the wave per second. The bigger the frequency the higher the pitch of the note or the bluer the light AMPLITUDE - the maximum distance that any point on the wave moves from its mean position. The bigger the amplitude the louder the sound, the rougher the sea, or the brighter the light

69. Period (T) The time it takes for one complete cycle of the wave. Displacement (x) How far the “particle” has travelled from its mean position. Wave speed (v) The speed at which the wavefronts pass a stationary observer Intensity (I) The power per unit area that is received by an observer. Students should know that intensity α amplitude 2

70. Derive and apply the relationship between wave speed, wavelength and frequency. Speed = Dist / time For 1 cycle of the wave, dist = λ and time =T Speed = λ / Tf = 1/T Therefore V=f x λ

71. The Wave Equation The wave equation relates the speed of the wave to its frequency and wavelength: Wave speed (v) = frequency (f) x wavelength ( ) in m/s in Hz in m V f

72. 1)A water wave has a frequency of 2Hz and a wavelength of 0.3m. How fast is it moving? 2)A water wave travels through a pond with a speed of 1m/s and a frequency of 5Hz. What is the wavelength of the waves? 3)The speed of sound is 330m/s (in air). When Dave hears this sound his ear vibrates 660 times a second. What was the wavelength of the sound? 4)Purple light has a wavelength of around 6x10 -7 m and a frequency of 5x10 14 Hz. What is the speed of purple light? Some example wave equation questions 0.2m 0.5m 0.6m/s 3x10 8 m/s

73. Electromagnetic waves Click to play

77. 4.5 Wave properties

78. Wave diagrams 1) Reflection 4) Diffraction3) Refraction 2) Refraction

79. Describe the reflection and transmission of waves at a boundary between two media. This should include the sketching of incident, reflected and transmitted waves.

80. The amount of transmission and reflection depends upon the difference in the “density” of the 2 media. i.e the bigger the difference, the greater the amount of reflection.

81. Refraction through a glass block: Wave slows down and bends towards the normal due to entering a more dense medium Wave speeds up and bends away from the normal due to entering a less dense medium Wave slows down but is not bent, due to entering along the normal

83. Refraction of Light applet Hyperlink

84. Finding the Critical Angle… 1) Ray gets refracted 4) Ray gets internally reflected 3) Ray still gets refracted (just!) 2) Ray still gets refracted THE CRITICAL ANGLE

88. Optical fibres

89. Uses of Total Internal Reflection Optical fibres: An optical fibre is a long, thin, _______ rod made of glass or plastic. Light is _______ reflected from one end to the other, making it possible to send ____ chunks of information Optical fibres can be used for _________ by sending electrical signals through the cable. The main advantage of this is a reduced ______ loss. Words – communications, internally, large, transparent, signal

90. Other uses of total internal reflection 1) Endoscopes (a medical device used to see inside the body): 2) Binoculars and periscopes (using “reflecting prisms”)

91. Huygen’s principle 1.Velocity decreases 2.Wavelength decreases 3.Frequency same

92. Snell’s law

94. Questions 10,11,12 Practice Q 2 n air = 1.00 n water = 1.33 n diamond = 2.42 n glass = 1.50

95. Diffraction More diffraction if the size of the gap is similar to the wavelength More diffraction if wavelength is increased (or frequency decreased)

98. Diffraction Hyperlink

99. Sound can also be diffracted… The explosion can’t be seen over the hill, but it can be heard. We know sound travels as waves because sound can be refracted, reflected (echo) and diffracted.

100. Diffraction depends on frequency… A high frequency (short wavelength) wave doesn’t get diffracted much – the house won’t be able to receive it…

101. Diffraction depends on frequency… A low frequency (long wavelength) wave will get diffracted more, so the house can receive it…

102. i) Diffraction by a "large" object ii) Diffraction at a "large" aperture iii) Diffraction by a "small" object iv) Diffraction by a "narrow" aperture

103. Superposition Superposition is seen when two waves of the same type cross. It is defined as “the vector sum of the two displacements of each wave”:

104. Superposition

105. Interference of 2 pulses Click to play

106. Constructive interference i.e. Loud or bright. Waves are in phase Destructive interference i.e. dark or quiet. Waves are π rads out of phase.

107. Interference of sound waves Where are the positions of constructive and destructive interference?

108. Interference of 2 point sources Click to play

109. Hyperlink

110. Superposition patterns Consider two point sources (e.g. two dippers or a barrier with two holes):

111. Superposition of Sound Waves

112. Path Difference Constructive interference Destructive interference Max 1 st Max Min 2 nd Max