4. Tuesday 25th March 2014 Mr Spence
IB Physics: Waves
Tuesday 25th March 2014
Mr Spence
Present/copy up to slide 60, SHM.
Copy 6 slides per page, double sided = 4 sides of A4 each.

5. WAVES Wave Characteristics – “what do they look like?”
Longitudinal and Transverse
3 Representations
Defining the wave
The wave equation
2. Wave Properties – “how do they act?”
Reflection and transmission
Snell’s Law
Diffraction
3. Standing Waves
4. The Doppler Effect

6. Waves
Waves can transfer energy and information without a net (total) motion of the medium through which they travel. They involve vibrations (oscillations) in either the x or y directions.

7. Transverse waves
The oscillations are perpendicular to the direction of energy transfer.
peak
Direction of energy transfer
oscillation
trough

8. Transverse waves Water ripples EM Radiation On a rope Earthquake:
S waves

9. Longitudinal waves
The oscillations are parallel to the direction of energy transfer.
Direction of energy transfer
oscillation

10. Longitudinal waves
compression
rarefraction

11. Longitudinal waves Sound Slinky: back and forth Earthquake: P waves
The most common example of longitudinal waves. Frequency in Hz, amplitude in Decibels.
Slinky: back and forth
Earthquake: P waves

12. Wave fronts
Wave fronts highlight the part of a wave that is moving together (in phase).
= wavefront
Ripples formed by a stone falling in water

13. Rays
Rays highlight the direction of energy transfer.

14. Wave Characteristics: Displacement - x
This measures the change that has taken place as a result of a wave passing a particular point. Zero displacement refers to the average position.
= displacement

15. Amplitude - A
The maximum displacement from the mean (equilibrium) position. Often measured in dB or metres.
amplitude
Examples: 5dB is a whisper, 90dB is painful. Large sea waves may be around 5 metres.

16. Period - T
The time taken (in seconds) for one complete oscillation. It is also the time taken for a complete wave to pass a given point.
One complete wave

17. Frequency - f
The number of oscillations in one second. Measured in Hertz.
50 Hz = 50 vibrations/waves/oscillations in one second.
Period and frequency are reciprocals of each other
f = 1/T T = 1/f

18. Wavelength - λ
The shortest distance between points that are in phase (points moving together or “in step”).
wavelength

19. Wave speed - v
The speed at which the wave fronts pass a stationary observer.
330 m.s-1

20. v = λ/T = fλ The Wave Equation
The time taken for one complete oscillation is the period T. In this time, the wave will have moved one wavelength λ.
The speed of the wave therefore is distance/time
v = λ/T = fλ
Example: Purple light has a wavelength of around 6x10-7m and a frequency of 5x1014Hz. What is the speed of purple light?
Answer: The same speed as every other colour.

21. Graphic Representation 1: Displacement-Time graph
This looks at the movement of one point of the wave over a period of time
Time s -1 -2
0.1
0.2
0.3
0.4
displacement cm 1

22. Displacement/Time graph
This looks at the movement of one point of the wave over a period of time
Time s -1 -2
0.1
0.2
0.3
0.4
displacement cm
IMPORTANT NOTE: Although the representation here looks transverse it may be a longitudinal wave (eg sound) 1
PERIOD

23. Graphical Representation 2: Displacement - Distance graph
This is a “snapshot” of the wave at a particular moment
displacement cm
IMPORTANT NOTE: This wave could also be either transverse or longitudnal 1
WAVELENGTH
Distance cm
0.4
0.8
1.2
1.6 -1 -2

24. Wave intensity
This is defined as the amount of energy per unit time flowing through a unit area. It is normally measured in W.m-2 For example, imagine a window with an area of 1m2. If one joule of light energy flows through that window every second we say the light intensity is 1 W.m-2.

25. Intensity at a distance from a light source
I = P/4πd2 d is the distance from the light source (in m) and P is the power of the light source (in Watts)

26. Intensity and amplitude
The intensity of a wave is proportional to the square of its amplitude I α a2 (or I = ka2) This means if you double the amplitude of a wave, its intensity quadruples! I = ka2 If amplitude = 2a, new intensity = k(2a)2 new intensity = 4ka2

27. Electromagnetic spectrum
λ ≈ nm
λ ≈ m
λ ≈ m
λ ≈ m
λ ≈ m
λ ≈ m
λ ≈ m

28. What do they all have in common?
They can travel in a vacuum
They travel at 3 x 108m.s-1 in a vacuum (the speed of light)
They are transverse
They are electromagnetic waves (electric and magnetic fields at right angles to each oscillating perpendicularly to the direction of energy transfer)

29. Refraction
Occurs when a wave changes speed (normally when entering another medium). It may refract (change direction/bend).

30. Examples Sound travels faster in warmer air:
Light slows down as it goes from air to glass/water:

31. Snell’s law
There is a wonderful relationship between the speed of the wave in the two media and the angles of incidence and refraction. There is a nice geometrical proof as well, if interested (which you should be…) i
Derivation: r

32. Snell’s law =
speed in substance 1 sinθ1 speed in substance 2 sinθ2

33. Snell’s law
In the case of light only, we usually define a quantity called the index of refraction for a given medium as
n = c = sinθ1/sinθ2 cm
where c is the speed of
light in a vacuum and cm
is the speed of light in the
medium c
vacuum cm

34. Snell’s law
Thus for two different media
sinθ1/sinθ2 = c1/c2 = n2/n1

35. Refraction – a few notes
The wavelength changes, the speed changes, but the frequency stays the same

36. Refraction – a few notes
When the wave enters at 90°, no change of direction takes place.

37. Diffraction
Waves spread as they pass an obstacle or through an opening. This can be described using Huygen’s construction.

38. Diffraction
Diffraction is strongest when the opening or obstacle is similar in size to the wavelength of the wave.
That’s why we can hear people from around a wall but not see them!

39. Diffraction of radio waves
                                                                
                                                                
                                                                
                                                                
Diffraction of radio waves
Aerials in hilly areas can often receive radio waves but not shorter
wavelength TV or Microwaves (mobile phone).

40. The Principle of Superposition
“When two or more waves meet, the resultant displacement is the sum of the individual displacements.” Constructive and Destructive Interference: When two waves of the same frequency superimpose, we can get constructive interference or destructive interference. + = = +

41. Phase difference
is the time difference or phase angle by which one wave/oscillation leads or lags another.
180° or π radians

42. Superposition
In general, the displacements of two (or more) waves can be added to produce a resultant wave. (Note, displacements can be negative)

43. Interference patterns
In areas of constructive interference, bright bands occur. Destructive interference = no intensity.

44. Path difference
Whether there is constructive or destructive interference observed at a particular point depends on the path difference of the two waves

45. Constructive interference if path difference is a whole number of wavelengths

46. Constructive interference if path difference is a whole number of wavelengths
antinode

47. Destructive interference occurs if the path difference is a half number of wavelengths
node

48. Simple harmonic motion (SHM)
periodic motion in which the restoring force is proportional and in the opposite direction to the displacement

49. Simple harmonic motion (SHM)
periodic motion in which the restoring force is in the opposite durection and proportional to the displacement
F = -kx

50. Graph of motion A graph of the motion will have this form displacement
Time
displacement

51. Graph of motion A graph of the motion will have this form Amplitude x0
Period T
Time
displacement

52. Graph of motion Notice the similarity with a sine curve 2π radians
angle
π/2 π
3π/2 2π

53. x = x0sinθ Graph of motion Notice the similarity with a sine curve
Amplitude x0
x = x0sinθ
2π radians
angle
π/2 π
3π/2 2π

54. Graph of motion
Amplitude x0
Period T
Time
displacement

55. x = x0sinωt Graph of motion
Amplitude x0
Period T
x = x0sinωt
Time
displacement
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

56. When x = 0 at t = 0
Amplitude x0
Period T
x = x0sinωt
Time
displacement
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

57. When x = x0 at t = 0
Amplitude x0
Period T
x = x0cosωt
displacement
Time
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

58. x = x0sinωt v = v0cosωt When x = 0 at t = 0
Amplitude x0
Period T
Time
displacement
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

59. x = x0cosωt v = -v0sinωt When x = x0 at t = 0
Amplitude x0
Period T
displacement
Time
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

60. To summarise! where ω = 2π/T = 2πf = (angular frequency in rad.s-1)
When x = 0 at t = 0
x = x0sinωt and v = v0cosωt
When x = x0 at t = 0
x = x0cosωt and v = -v0sinωt
It can also be shown that v = ±ω√(x02 – x2) and a = -ω2x
where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

61. Maximum velocity? When x = 0
At this point the acceleration is zero (no resultant force at the equilibrium position).

62. Maximum acceleration? When x = +/– x0 Here the velocity is zero
amax = -ω2x0

63. Oscillating spring
We know that F = -kx and that for SHM, a = -ω2x (so F = -mω2x) So -kx = -mω2x k = mω2 ω = √(k/m) Remembering that ω = 2π/T T = 2π√(m/k)

64. S.H.M.
Where is the kinetic energy maxiumum? Where is the potential energy maximum?

66. It can be shown that….
Ek = ½mω2(xo2 – x2) ET = ½mω2xo2 Ep = ½mω2x2 where ω = 2πf = 2π/T

67. Resources
- Walter Lewin’s famous Pendulum derivation and demo
Great Intro:

68. Focault Pendulum
Sixty Symbols description:
Wolfram Alpha model:
Sydney Uni:

69. Damping
In most real oscillating systems energy is lost through friction. The amplitude of oscillations gradually decreases until they reach zero. This is called damping

70. Underdamped
The system makes several oscillations before coming to rest

71. Overdamped
The system takes a long time to reach equilibrium

72. Critical damping
Equilibrium is reached in the quickest time

73. Natural frequency
All objects have a natural frequency that they prefer to vibrate at.

74. Forced vibrations
If a force is applied at a different frequency to the natural frequency we get forced vibrations

75. Resonance
If the frequency of the external force is equal to the natural frequency we get resonance
YouTube - Ground Resonance - Side View
YouTube - breaking a wine glass using resonance