1. Chapter 11 Waves

2. MFMcGrawCh-11b-Waves - Revised 4-3-102 Chapter 11 Topics Energy Transport by Waves Longitudinal and Transverse Waves Transverse Waves on Strings Periodic Waves Superposition Reflection and Refraction of Waves Interference and Diffraction Standing Waves on a String

3. MFMcGrawCh-11b-Waves - Revised 4-3-103 Waves and Energy Transport A wave is a disturbance that travels, in a medium, outward from its source. Waves carry energy. The energy is transported outward from the source to the destination. The matter in the medium expieriences no net transport.

4. MFMcGrawCh-11b-Waves - Revised 4-3-104 When a stone is dropped into a pond, the water is disturbed from its equilibrium positions as the wave passes; it returns to its equilibrium position after the wave has passed. The water moves up and down as the disturbance moves outward but is not transported by the wave. Waves and Energy Transport

5. MFMcGrawCh-11b-Waves - Revised 4-3-105 A Simple Water Wave The wave travels to the right. The individual water molecules move up and down locally but do not travel with the wave. A surfer CAN travel with the wave, at least for a while. Trough Peak

6. MFMcGrawCh-11b-Waves - Revised 4-3-106 3-D Spherical Wave Energy spreads out, from a point source, uniformly over a 3 dimensional area Intensity = Power/Area Units: Watts/m 2

7. MFMcGrawCh-11b-Waves - Revised 4-3-107 Intensity is a measure of the amount of energy/sec that passes through a square meter of area perpendicular to the wave’s direction of travel. Intensity has units of watts/m 2. This is an inverse square law. The intensity drops as the inverse square of the distance from the source. (Light sources appear dimmer the farther away from them you are.) Waves and Energy Transport

8. MFMcGrawCh-11b-Waves - Revised 4-3-108 Example: At the location of the Earth’s upper atmosphere, the intensity of the Sun’s light is 1400 W/m 2. What is the intensity of the Sun’s light at the orbit of the planet Mercury? Divide one equation by the other (take a ratio): Waves and Energy Transport

9. MFMcGrawCh-11b-Waves - Revised 4-3-109 Longitudinal Wave Amplitude change parallel to propagation direction Transverse Wave Amplitude change perpendicular to propagation direction

10. MFMcGrawCh-11b-Waves - Revised 4-3-1010 The Excitation of a Transverse Wave The speed with which the string is moved vertically is independent of the speed with the wave travels horizontally down the string. Boundary Condition: The end of the string is stationary

11. MFMcGrawCh-11b-Waves - Revised 4-3-1011 Transverse Waves on a String M Attach a wave driver here L Attach a mass to a string to provide tension. The string is then shaken at one end with a frequency f.

12. MFMcGrawCh-11b-Waves - Revised 4-3-1012 A wave traveling on this string will have a speed of where F is the force applied to the string (tension) and  is the mass/unit length of the string (linear mass density). Transverse Waves on a String

13. MFMcGrawCh-11b-Waves - Revised 4-3-1013 Example (text problem 11.8): When the tension in a cord is 75.0 N, the wave speed is 140 m/s. What is the linear mass density of the cord? The speed of a wave on a string is Solving for the linear mass density: Transverse Waves on a String

14. MFMcGrawCh-11b-Waves - Revised 4-3-1014 Speed of Sound in Various Materials Increasing attractive force String Liquid Solid High mass Low Mass

15. MFMcGrawCh-11b-Waves - Revised 4-3-1015 Periodic Waves A periodic wave repeats the same pattern over and over. For periodic waves: v = f v is the wave’s speed f is the wave’s frequency is the wave’s wavelength

16. MFMcGrawCh-11b-Waves - Revised 4-3-1016 A Simple Harmonic Oscillator Its motion, depicted as a function of time, is a wave. A peak-peak = A p-p = 2A peak = 2A p ; A p = A p-p /2

17. MFMcGrawCh-11b-Waves - Revised 4-3-1017 The period T is measured by the amount of time it takes for a point on the wave to go through one complete cycle of oscillations. The frequency is then f = 1/T. Periodic Waves

18. MFMcGrawCh-11b-Waves - Revised 4-3-1018 The maximum displacement from equilibrium is amplitude (A) of a wave. One way to determine the wavelength is by measuring the distance between two consecutive crests. Periodic Waves

19. MFMcGrawCh-11b-Waves - Revised 4-3-1019 Example (text problem 11.13): What is the wavelength of a wave whose speed and period are 75.0 m/s and 5.00 ms, respectively? Solving for the wavelength: Periodic Waves

20. MFMcGrawCh-11b-Waves - Revised 4-3-1020 Wave Properties

21. MFMcGrawCh-11b-Waves - Revised 4-3-1021 The Principle of Superposition For small amplitudes, waves will pass through each other and emerge unchanged. Superposition Principle: When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.

22. MFMcGrawCh-11b-Waves - Revised 4-3-1022 Two traveling wave pulses: left pulse travels right; right pulse travels left.

23. MFMcGrawCh-11b-Waves - Revised 4-3-1023 Reflection and Refraction At an abrupt boundary between two media, a reflection will occur. A portion of the incident wave will be reflected backward from the boundary.

24. MFMcGrawCh-11b-Waves - Revised 4-3-1024 When you have a wave that travels from a “low density” medium to a “high density” medium, the reflected wave pulse will be inverted. (180 o phase shift.) The frequency of the reflected wave remains the same. The Reflected Wave & Phase Change

25. MFMcGrawCh-11b-Waves - Revised 4-3-1025 When a wave is incident on the boundary between two different media, a portion of the wave is reflected, and a portion will be transmitted into the second medium. Reflected ray is 180 o out of phase. Light Ray Example

26. MFMcGrawCh-11b-Waves - Revised 4-3-1026 The frequency of the transmitted wave remains the same. However, both the wave’s speed and wavelength are changed such that: The transmitted wave will also suffer a change in propagation direction (refraction). The Frequency is Constant

27. MFMcGrawCh-11b-Waves - Revised 4-3-1027 Example (text problem 11.36) Light of wavelength 0.500  m in air enters the water in a swimming pool. The speed of light in water is 0.750 times the speed in air. What is the wavelength of the light in water? Since the frequency is unchanged in both media:

28. MFMcGrawCh-11b-Waves - Revised 4-3-1028 Interference Two waves are considered coherent if they have the same frequency and maintain a fixed phase relationship. Two waves are considered incoherent if the phase relationship between them varies randomly.

29. MFMcGrawCh-11b-Waves - Revised 4-3-1029 When waves are in phase, their superposition gives constructive interference. When waves are one-half a cycle out of phase, their superposition gives destructive interference. This is referred to as: “exactly out of phase” or “180 o out of phase.”

30. MFMcGrawCh-11b-Waves - Revised 4-3-1030 Constructive Interference

31. MFMcGrawCh-11b-Waves - Revised 4-3-1031 Destructive Interference

32. MFMcGrawCh-11b-Waves - Revised 4-3-1032 Interference

33. MFMcGrawCh-11b-Waves - Revised 4-3-1033 Constructive Interference. Means that the waves ADD together and their amplitudes are in the same direction Destructive Interference. Means that the waves ADD together and their amplitudes are in the opposite directions. Interference = Bad choice of words The two waves do not interfere with each other. They do not interact with each other. No energy or momentum is exchanged.

34. MFMcGrawCh-11b-Waves - Revised 4-3-1034 When two waves travel different distances to reach the same point, the phase difference is determined by: Note: This is a ratio comparison. λ is not equal to 2π Phase Difference

35. MFMcGrawCh-11b-Waves - Revised 4-3-1035 Diffraction is the spreading of a wave around an obstacle in its path and it is common to all types of waves. The size of the obstacle must be similar to the wavelength of the wave for the diffraction to be observed. Larger by 10x is too big and smaller by (1/10)x is too small. Diffraction

36. MFMcGrawCh-11b-Waves - Revised 4-3-1036 Standing Waves Pluck a stretched string such that y(x,t) = A sin(  t + kx) (This is a wave that moves to the left.) When the wave strikes the wall, there will be a reflected wave that travels back along the string.

37. MFMcGrawCh-11b-Waves - Revised 4-3-1037 The reflected wave will be 180° out of phase with the wave incident on the wall. Its form is y(x,t) =  A sin (  t  kx). (This is a wave that moves to the right.) Apply the superposition principle to the two waves on the string: The time variation and the spatial variation have been separated.

38. MFMcGrawCh-11b-Waves - Revised 4-3-1038 The previous expression is the mathematical form of a standing wave. N N N N A A A A node (N) is a point of zero oscillation. An antinode (A) is a point of maximum displacement. All points between nodes oscillate up and down.

39. MFMcGrawCh-11b-Waves - Revised 4-3-1039 The nodes occur where y(x,t) = 0. The nodes are found from the locations where sin kx = 0, which happens when kx = 0, , 2 ,…. That is when kx = n  where n = 0,1,2,… The antinodes occur when sin kx =  1; that is where

40. MFMcGrawCh-11b-Waves - Revised 4-3-1040 If the string has a length L, and both ends are fixed, then y(x = 0, t) = 0 and y(x = L, t) = 0. (Boundary conditions) The wavelength of a standing wave: where n = 1, 2, 3,…

41. MFMcGrawCh-11b-Waves - Revised 4-3-1041 These are the permitted wavelengths of standing waves on a string; no others are allowed. The speed of the wave is: The allowed frequencies are then: n =1, 2, 3,…

42. MFMcGrawCh-11b-Waves - Revised 4-3-1042 The n = 1 frequency is called the fundamental frequency. All allowed frequencies (called harmonics) are integer multiples of f 1.

43. MFMcGrawCh-11b-Waves - Revised 4-3-1043 Example (text problem 11.51): A Guitar’s E-string has a length 65 cm and is stretched to a tension of 82 N. It vibrates with a fundamental frequency of 329.63 Hz. Determine the mass per unit length of the string. For a wave on a string: Solving for the linear mass density:

44. MFMcGrawCh-11b-Waves - Revised 4-3-1044

45. MFMcGrawCh-11b-Waves - Revised 4-3-1045 Two Methods of Classification Just to make life interesting 1st Harmonic 2nd Harmonic 3rd Harmonic Fundamental 1st Overtone 2nd Overtone The labels on the right are more general. If the frequencies are integer multiples of one another then they are referred to as harmonics

46. MFMcGrawCh-11b-Waves - Revised 4-3-1046 Summary Energy Transport by Waves Longitudinal and Transverse Waves Transverse Waves on Strings Periodic Waves Superposition Reflection and Refraction of Waves Interference and Diffraction Standing Waves on a String

47. MFMcGrawCh-11b-Waves - Revised 4-3-1047 Extra

48. MFMcGrawCh-11b-Waves - Revised 4-3-1048 Wave Properties The sound in an acoustic instrument comes from the vibrating strings moving the air and coupling into the resonant cavity of the instrument whose walls vibrate and in turn cause vibrations in the surrounding air pressure that we interpret as sound. It acts as a sound amplifier