1. Chapters 16 – 18
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 linear 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 linear 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 linear 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 linear wave equation
Therefore, solutions of the wave equation should have a form
Considering partial derivatives

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

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

11. The linear wave equation
The linear 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 T

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

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

14. Chapter 16
Problem 44
(a) Show that the function y(x, t) = x2 + v2t2 is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x + vt) + g(x – vt) and determine the functional forms for f and g. (c) Repeat parts (a) and (b) for the function y(x, t) = sin (x) cos (vt).

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 linear wave equation is a sinusoidal wave:
A – 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 18
A transverse sinusoidal wave on a string has a period T = 25.0 ms and travels in the negative x direction with a speed of 30.0 m/s. At t = 0, an element of the string at x = 0 has a transverse position of 2.00 cm and is traveling downward with a speed of 2.00 m/s. (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of an element of the string? (d) Write the wave function for the wave.

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 18
Problem 25
A standing wave pattern is observed in a thin wire with a length of 3.00 m. The wave function is y = (0.002 m) sin (πx) cos (100πt), where x is in meters and t is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) If the original frequency is held constant and the tension in the wire is increased by a factor of 9, how many loops are present in the new pattern?

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

32. 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

33. 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)

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

35. Chapter 17
Problem 12
As a certain sound wave travels through the air, it produces pressure variations (above and below atmospheric pressure) given by ΔP = 1.27 sin (πx – 340πt) in SI units. Find (a) the amplitude of the pressure variations, (b) the frequency, (c) the wavelength in air, and (d) the speed of the sound wave.

36. 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

37. 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

38. 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:

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

40. 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

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

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

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

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

45. 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

46. 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

47. 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)

48. Chapter 17
Problem 26
Two small speakers emit sound waves of different frequencies equally in all directions. Speaker A has an output of 1.00 mW, and speaker B has an output of 1.50 mW. Determine the sound level (in decibels) at point C assuming (a) only speaker A emits sound, (b) only speaker B emits sound, and (c) both speakers emit sound.

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

50. 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

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

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

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

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

55. Chapter 17
Problem 37
A tuning fork vibrating at 512 Hz falls from rest and accelerates at 9.80 m/s2. How far below the point of release is the tuning fork when waves of frequency 485 Hz reach the release point? Take the speed of sound in air to be 340 m/s.

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

57. Supersonic speeds
The Mach cone produces a sonic boom

58. Questions?

59. Answers to the even-numbered problems
Chapter 16
Problem 8
0.800 m/s

60. Answers to the even-numbered problems
Chapter 17
Problem 16
(a) 5.00 × W
(b) 5.00 × 10-5 W

61. Answers to the even-numbered problems
Chapter 17
Problem 40
46.4°

62. Answers to the even-numbered problems
Chapter 17
Problem 54
The gap between bat and insect is closing at 1.69 m/s.