1. Chapters 16, 17
Waves

2. Types of waves
Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.)
Electromagnetic – governed by electricity and magnetism equations, may exist without any medium
Matter – governed by quantum mechanical equations

3. Types of waves
Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:
Transverse – if the direction of displacement is perpendicular to the direction of propagation
Longitudinal – if the direction of displacement is parallel to the direction of propagation

4. Types of waves
Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:
Transverse – if the direction of displacement is perpendicular to the direction of propagation
Longitudinal – if the direction of displacement is parallel to the direction of propagation

5. The wave equation
Let us consider transverse waves propagating without change in shape and with a constant wave velocity v
We will describe waves via vertical displacement y(x,t)
For an observer moving with the wave
the wave shape doesn’t depend on time y(x’) = f(x’)

6. The wave equation For an observer at rest:
the wave shape depends on time y(x,t)
the reference frame linked to the wave is moving with the velocity of the wave v

7. The wave equation We considered a wave propagating with velocity v
For a medium with isotropic (symmetric) properties, the wave equation should have a symmetric solution for a wave propagating with velocity –v

8. The wave equation
Therefore, solutions of the wave equation should have a form
Considering partial derivatives

9. The wave equation
Therefore, solutions of the wave equation should have a form
Considering partial derivatives

10. The wave equation
Therefore, solutions of the wave equation should have a form
Considering partial derivatives

11. The wave equation
The wave equation (not the only one having solutions of the form y(x,t) = f(x ± vt)):
It works for longitudinal waves as well
v is a constant and is determined by the properties of the medium. E.g., for a stretched string with linear density μ = m/l under tension τ

12. Superposition of waves
Let us consider two different solutions of the wave equation
Superposition principle – a sum of two solutions to the wave equation is a solution to the wave equation +

13. Superposition of waves
Overlapping solutions of the wave equation algebraically add to produce a resultant (net) wave
Overlapping solutions of the wave equation do not in any way alter the travel of each other

14. Chapter 16
Problem 27

15. Reflection of waves at boundaries
Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries
Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted
Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

16. Sinusoidal waves
One of the most characteristic solutions of the wave equation is a sinusoidal wave:
ym - amplitude, φ - phase constant

17. Wavelength
“Freezing” the solution at t = 0 we obtain a sinusoidal function of x:
Wavelength λ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

18. Wave number
On the other hand:
Angular wave number: k = 2π / λ

19. Angular frequency Considering motion of the point at x = 0
we observe a simple harmonic motion (oscillation) :
For simple harmonic motion (Chapter 15):
Angular frequency ω

20. Frequency, period
Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion:
Therefore, for the wave velocity

21. Chapter 16
Problem 7

22. Interference of waves
Interference – a phenomenon of combining waves, which follows from the superposition principle
Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation
The resultant wave:

23. Interference of waves If φ = 0 (Fully constructive)
If φ = π (Fully destructive)
If φ = 2π/3 (Intermediate)

24. Interference of waves
Considering two sinusoidal waves of the same amplitude, wavelength, but running in opposite directions
The resultant wave:

25. Interference of waves
If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave
Nodes
Antinodes

26. Chapter 16
Problem 54

27. Standing waves and resonance
For a medium with fixed boundaries (hard reflection) standing waves can be generated because of the reflection from both boundaries: resonance
Depending on the number of antinodes, different resonances can occur

28. Standing waves and resonance
Resonance wavelengths
Resonance frequencies

29. Harmonic series
Harmonic series – collection of all possible modes - resonant oscillations (n – harmonic number)
First harmonic (fundamental mode):

30. More about standing waves
Longitudinal standing waves can also be produced
Standing waves can be produced in 2 and 3 dimensions as well

31. Phasors
For superposition of waves it is convenient to use phasors – vectors that have magnitude equal to the amplitude of the wave and rotating around the origin
Two phase-shifted waves with the same frequency can be represented by phasors separated by a fixed angle

32. Phasors
To obtain a resultant wave (add waves) one has to add phasors as vectors
Using phasors one can add waves of different amplitudes

33. Rate of energy transmission
As the wave travels it transports energy, even though the particles of the medium don’t propagate with the wave
The average power of energy transmission for the sinusoidal solution of the wave equation
Exact expression depends on the medium or the system through which the wave is propagating

34. Sound waves
Sound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies detectable by human ears (between ~ 20 Hz and ~ 20 KHz)
Ultrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies higher than detectable by human ears (> 20 KHz)
Infrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies lower than detectable by human ears (< 20 Hz)

35. Speed of sound ρ – density of a medium, B – bulk modulus of a medium
Traveling sound waves

36. Chapter 17
Problem 12

37. Intensity of sound
Intensity of sound – average rate of sound energy transmission per unit area
For a sinusoidal traveling wave:
Decibel scale
β – sound level; I0 = W/m2 – lower limit of human hearing

38. Chapter 17
Problem 18

39. Sources of musical sound
Music produced by musical instruments is a combination of sound waves with frequencies corresponding to a superposition of harmonics (resonances) of those musical instruments
In a musical instrument, energy of resonant oscillations is transferred to a resonator of a fixed or adjustable geometry

40. Open pipe resonance
In an open pipe soft reflection of the waves at the ends of the pipe (less effective than form the closed ends) produces standing waves
Fundamental mode (first harmonic): n = 1
Higher harmonics:

41. Organ pipes

42. Organ pipes Organ pipes are open on one end and closed on the other
For such pipes the resonance condition is modified:

43. Musical instruments
The size of the musical instrument reflects the range of frequencies over which the instrument is designed to function
Smaller size implies higher frequencies, larger size implies lower frequencies

44. Musical instruments
Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
Guitar resonances (exaggerated) at low frequencies:

45. Musical instruments
Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
Guitar resonances at medium frequencies:

46. Musical instruments
Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
Guitar resonances at high frequencies:

47. Beats
Beats – interference of two waves with close frequencies +

48. Sound from a point source
Point source – source with size negligible compared to the wavelength
Point sources produce spherical waves
Wavefronts – surfaces over which oscillations have the same value
Rays – lines perpendicular to wavefronts indicating direction of travel of wavefronts

49. Interference of sound waves
Far from the point source wavefronts can be approximated as planes – planar waves
Phase difference and path length difference are related:
Fully constructive interference
Fully destructive interference

50. Variation of intensity with distance
A single point emits sound isotropically – with equal intensity in all directions (mechanical energy of the sound wave is conserved)
All the energy emitted by the source must pass through the surface of imaginary sphere of radius r
Sound intensity
(inverse square law)

51. Chapter 17
Problem 29

52. Doppler effect
Doppler effect – change in the frequency due to relative motion of a source and an observer (detector)
Andreas Christian
Johann Doppler
( )

53. Doppler effect For a moving detector (ear) and a stationary source
In the source (stationary) reference frame:
Speed of detector is –vD
Speed of sound waves is v
In the detector (moving) reference frame:
Speed of detector is 0
Speed of sound waves is v + vD

54. Doppler effect For a moving detector (ear) and a stationary source
If the detector is moving away from the source:
For both cases:

55. Doppler effect For a stationary detector (ear) and a moving source
In the detector (stationary) reference frame:
In the moving (source) frame:

56. Doppler effect For a stationary detector and a moving source
If the source is moving away from the detector:
For both cases:

57. Doppler effect For a moving detector and a moving source
Doppler radar:

58. Chapter 17
Problem 52

59. Supersonic speeds For a source moving faster than the speed of sound
the wavefronts form the Mach cone
Mach number
Ernst Mach
( )

60. Supersonic speeds
The Mach cone produces a sonic boom

61. Answers to the even-numbered problems
Chapter 16:
Problem 2
(a) 3.49 m−1;
(b) 31.5 m/s

62. Answers to the even-numbered problems
Chapter 16:
Problem 24
198 Hz

63. Answers to the even-numbered problems
Chapter 16:
Problem 26
1.75 m/s

64. Answers to the even-numbered problems
Chapter 16:
Problem 30
(a) 82.8º;
(b) 1.45 rad;
(c) 0.23 wavelength

65. Answers to the even-numbered problems
Chapter 16:
Problem 46
260 Hz

66. Answers to the even-numbered problems
Chapter 17:
Problem 6
44 m

67. Answers to the even-numbered problems
Chapter 17:
Problem 8
1.50 Pa;
(b) 158 Hz;
(c) 2.22 m;
(d) 350 m/s

68. Answers to the even-numbered problems
Chapter 17:
Problem 14
4.12 rad

69. Answers to the even-numbered problems
Chapter 17:
Problem 36
57.2 cm;
(b) 42.9 cm

70. Answers to the even-numbered problems
Chapter 17:
Problem 50
zero