2. AP Physics B Giancoli 11 & 12 Waves and Sound

3. Assignments Reading: 11.1-4,7-9,11-13 and 12.1,2,4-7 Problems: Waves: 11.42,43,55,56 Sound: 12.4,5,10,11 Strings/AirColumns12.29,30,35 Interference 12.42,43 Doppler 12.51,52 SHM: 11.4,5,20, 25 Pendulum: 11.31,32

4. Preview What are the two categories of waves with regard to mode of travel? –Mechanical –Electromagnetic Which type of wave requires a medium? –Mechanical An example of a mechanical wave? –Sound

5. Velocity of a Wave The speed of a wave is the distance traveled by a given point on the wave (such as a crest) in a given internal of time. v = d/t d: distance (m) t: time (s) v = f v: speed (m/s)  : wavelength (m) f : frequency (s -1, Hz)

6. Period of a Wave T = 1/f T : Period = (s) F : frequency (s -1, Hz)

7. Problem: Sound travels at approximately 340 m/s, and light travels at 3.0 x 10 8 m/s. How far away is a lightning strike if the sound of the thunder arrives at a location 5.0 seconds after the lightning is seen? Light travels almost instantaneously from strike location to the observer. The sound travels much more slowly: d = v s t = (340 m/s)(5.0 s) = 170m

8. Problem: The frequency of a C key on the piano is 262 Hz. What is the period of this note? What is the wavelength? Assume speed of sound in air to be 340 m/s at 20 o C. T = 1/f = 1/262 s -1 = 0.00382 s V = f = v/f = 340 m/s / 262 /s = 1.30 m

9. Problem A sound wave traveling through water has a frequency of 500 Hz and a wavelength of 3 m. How fast does sound travel through water? v = f = 3m (500 Hz) = 1500 m/s

10. Wave on a Wire v = F T m / L v, velocity, m/s F T, tension on a wire, N m/L mass/unit length, kg/m m/L may be shown as 

11. Problem Ex. 11-11 A wave whose wavelength is 0.30 m is traveling down a 300 m long wire whose total mass is 15 kg. If the tension of the wire is 1000N, what are the speed and frequency of the wave? Using equation on prior slide: v = √[( 1000N) / (15kg)(300m)] = 140m/s f = v / = 140 m/s / 0.30 m = 470 Hz

12. Types of Waves A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves. –Example: Waves on a guitar string A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave moves. These are also called compression waves. –Example: Sound –http://einstein.byu.edu/~masong/HTMstuff/WaveTrans.htmlhttp://einstein.byu.edu/~masong/HTMstuff/WaveTrans.html

13. What are two types of wave shapes? Transverse Longitudinal

14. http://www.school-for-champions.com/science/sound.htm

15. Transverse Wave Type

16. Longitudinal Wave Type

17. Longitudinal vs Transverse

18. Other Waves Types Occurring in Nature Light: electromagnetic Ocean waves: surface Earthquakes: combination Wave demos: http://www.kettering.edu/~drussell/Demos/ waves/wavemotion.htmlhttp://www.kettering.edu/~drussell/Demos/ waves/wavemotion.html http://www.kettering.edu/~drussell/Demos/ doppler/mach1.htmlhttp://www.kettering.edu/~drussell/Demos/ doppler/mach1.html

19. Properties of Waves Reflection occurs when a wave strikes a medium boundary and “bounces back” into the original medium. Those waves completely reflected have the same energy and speed as the original wave.

20. Types of Reflection Fixed-end Reflection- The wave reflects with inverted phase. Open-end Reflection- The wave reflects with The same phase. www.iop.org/activity/education/Teaching_Resources

21. Refraction of Waves Wave is transmitted from one medium to another. Refracted waves may change speed and Wavelength Almost always is accompanied by some reflection. Refracted waves do not change frequency.

22. Sound - a longitudinal wave Sound travels through air about 340 m/s. Sound travels through other media as well, often much faster than 340 m/s. Sound waves are started by vibration of some other material, which starts the air vibrating. www.silcom.com/~aludwig/musicand.htm

23. Hearing Sounds We hear a sound as “high” or “low” pitch depending on the frequency or wavelength. High-pitched sounds have short wavelengths and high frequencies. Low-pitched sounds have long wavelengths and low frequencies. Humans hear from about 20 Hz to about 20,000 Hz. The amplitude of a sound’s vibration is interpreted as its loudness. We measure loudness (also known as sound intensity) on the decibel scale, which is logarithmic. http://www.allegropianoworks.com/assets/rare_co mpress.jpg

24. Doppler Effect The Doppler Effect is the apparent change in pitch of a sound as a result of the relative motion of an observer and the source of a sound. Coming toward you a car horn appears higher pitched because the wavelength has been effectively decreased by the motion of the car relative to you. The opposite occurs when you are behind the car. http://people.finearts.uvic.ca/~aschloss/course_mat/MU207/images/Image2.gif

25. Pure Sound Sounds are longitudinal waves, but they can be shown to look like transverse waves. When air motion is graphed in a pure sound tone versus position, we get what looks like a sine or cosine function. A tuning fork produces a relatively pure tone as does a human whistle.

26. Graphing a Sound Wave

27. Complex Sounds Because of superposition and interference, real world waveforms may not appear to be pure sine or cosine functions. This is because most real world sounds are composed of multiple frequencies. The human voice and most musical instruments are examples.

28. The Oscilloscope With an Oscilloscope we can view waveforms. Pure tones will resemble sine or cosine functions, and complex tones will show other repeating patterns that are formed from multiple sine and cosine functions added together. (Amplitude vs time.)

29. The Fourier Transform The Fourier transform has long been used for characterizing linear systems and for identifying the frequency components making up a continuous waveform. This mathematical technique separates a complex waveform into its component frequencies. The Fourier Transform´s ability to represent time- domain data in the frequency domain and vice- versa has many applications. One of the most frequent applications is analysing the spectral (frequency) energy contained in data that has been sampled at evenly-spaced time intervals. Other applications include fast computation of convolution (linear systems responses, digital filtering, correlation (time-delay estimation, similarity measurements) and time-frequency analysis. convolutiondigital filteringtime-frequency analysis

30. Fourier Transform - showing “time domain” and “frequency domain”.

31. Superposition Principle When two or more waves pass a particular point in a medium simultaneously, the resulting displacement of the medium at that point is the sum of the displacements due to each individual wave. The waves are said to interfere with each other.

32. Superposition of Waves When two or more waves meet, the displacement at any point of the medium is equal to the algebraic sum of the displacements due to the individual waves.

33. Types of Interference If the waves are in phase, when crests and troughs are aligned, the amplitude in increased and this is called constructive interference. If the waves are “out of phase”, when crests and troughs are completely misaligned, the amplitude is decreased and can even be zero. This is called destructive interference.

34. Constructive Interference Crests are Aligned  the waves are “in phase”

35. Destructive Interference Crests are aligned with troughs  Waves are “out of phase”

36. Constructive & Destructive Interference

37. Interference Problem: Draw the waveform from the two components shown below.

38. Standing Waves A standing wave is one which is reflected back and forth between fixes ends of a string or pipe. Reflection may be fixed or open-ended. Superposition of the wave upon itself results in a pattern of constructive and destructive interference and an enhanced wave. Let’s see a simulation http://www.5min.com/Video/The-Rubens-Tube- Frequency-of-Fire-1858291http://www.5min.com/Video/The-Rubens-Tube- Frequency-of-Fire-1858291

39. Fixed-end standing waves - guitar or violin string Fundamental 1 st harmonic = 2L First overtone 2 nd harmonic = L Second Overtone 3 rd harmonic = 2L/3 http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

40. Problem A string of length 12 m that’s fixed at both ends supports a standing wave with a total of 5 nodes. What are the harmonic number and wavelength of this standing wave? L = 4(1/2 )  = 2L/4 4 th harmonic since it matches n = 2L/n for n = 4 wavelength  4 = 2(12m) / 4 = 6 m

41. Open-ended standing waves - flute & clarinet = 4L = (4/3)L = (4/5L = 2L = L = (2/3)L

42. physics.indiana.edu/~p105_f02/standing_waves_...

43. http://upload.wikimedia.org/wikiboo ks/en/3/32/Fhsst_waves40.png open ends one end both ends – closed closed

44. Sample Problem 12-30. a) Determine the length of an organ pipe that emits middle C (262Hz). The air temp. is 21 o C. A) v = 331m/s + 0.6 m/s o C(21 o C) = 344m/s A) = 2L v = f  f  L = v/2f = 344m/s/{2(262/s)] L = 0.656m B) What are the wavelength and frequency of the 1 st harmonic? Frequency is 262 Hz Wavelength is twice the length of the pipe, 1.31 m. C) What is the wavelength and frequency in the traveling sound wave produced in the outside air? They are the same because it is air that is resonating in the organ pipe: 262Hz and 1.31 m

45. Superposition of 2 sound waves http://www.ece.utexas.edu/~nodog/me379m/superposition.html http://www.ece.utexas.edu/~nodog/me379m/superposition.html

46. Resonance and Beats Resonance occurs when a vibration from one oscillator occurs at a natural frequency for another oscillator. The first oscillator will cause the second to vibrate. See next slide.

47. Resonance http://www.isd-dc.org/ISD- Wash/GIFS%20Pictures%20&%20Whatnots/tuningforkresonance.jpghttp://www.isd-dc.org/ISD- Wash/GIFS%20Pictures%20&%20Whatnots/tuningforkresonance.jpg

48. Beats The word physicists use to describe the characteristic loud/soft pattern that characterizes two nearly matched frequencies. Musicians call this “being out of tune”.

49. Beats When two sound waves whose frequencies are close but not exactly the same, the resulting sound modulates in amplitude changing from loud to soft to loud. This is called beat frequency and is shown by: f beat = f 1 - f 2

50. Diffraction Bending of a wave around a barrier Diffraction of waves combined with interference of the diffracte waves causes “diffraction patterns”. Here is an example using a “ripple tank”. http://www.falstad.com/ripple/

51. Double-slit or multi-slit diffraction micro.magnet.fsu.edu/.../doubleslit/ Remove frame

53. Single Slit Diffraction n = s sin   n -- dark band number   -- wavelength (m)  s -- slit width (m)   -- angle defined by central band, slit, and dark bank

54. Sample Problem Light of wavelength 360 nm is passed through a diffraction grating that has 10,000 slits per cm. If the screen is 2.0 m from the grating, how far from the central bright band is the first order bright band?

55. Sample Problem Light of wavelength 560 nm is passed through two slits. It is found that, on a screen 1.0 m from the slits, a bright spot is formed at x = 0, and another is formed at x = 0.03m. What is the spacing between the slits?

56. Sample Problem Light is passed through a single slit of width 2.1 x 10 -6 m. How far from the central bright band do the 1 st and 2 nd order dark bands appear if the screen is 3.0 m away from the slit?

57. Mathematical Description of a Traveling Wave Y = A sin (  t +  x ) Y dependent of x and t; y(x,t) or “y of x & t” If the - sign is used, wave is traveling in +x direction  A is amplitude of the wave  (omega) is angular frequency (  = 2  f)  angular wave # (  = 2  k, k = 1/ )

58. Other forms Important features of the wave: amplitude, frequency f (through  ), period T (which is 1/f = 2  /  wavelength ( = 2  /  ) and wave speed v (which is f =  /k) y = Asin2  [ft + (1/  x] or y = Asin(2  / )(vt + x)

59. Sample Problem: The vertical position y of any point on a rope that supports a transverse wave traveling horizontally is given by the equation y = 0.1 sin (6  t + 8  x) Find: amplitude: 0.1 angular frequency:  = 6  s -1 frequency: f =  / (2  ) = (6  s -1 )/ (2  ) = 3 Hz angular wave number: k =  / (2  ) = 8  m -1 ) / ((2  ) = 4 m -1 wavelength: = 1/ k = 1/ (4 m -1 ) = 0.25 m period: T = 1/f = 1 / 3 Hz = 0.33s wave speed: v = f = 0.25m (3 Hz) = 0.75 m/s

60. Assignment P15/MC1-4

61. Sound Level Intensity: Rate at which sound waves transmit energy is measured in energy per unit area: watt/m 2 or watt/cm 2 Intensity level or loudness level,  B = 10log I/I o where I o = 1x10 -12 w/m 2 or 1x10 -16 w/cm 2

62. More math…  = 10 log I I o 10 -16 w/cm 2 Intensity level = 10 log Intensity / threshhold of hearing We all don’t hear the same, so this is a comparative measurement in decibels

63. Flow chart for  problem If I = 4.7 x 10^-10 w/cm^2: 10xlog(4.7 2 nd EE -10 / 1 2 nd EE -16) = 66.7 dB If I = 2.9 x 10^-3 w/cm^2: 10xlog(2.9 2 nd EE -3 / 1 2 nd EE -16) = 135 dB

64. Problem Now we are going backwards from intensity level (dB) to intensity (w/cm 2 ) If the intensity level is 83 dB, convert that to intensity in w/cm 2. B = 10 log I / Io get to a working eqtn: B /10 = log I / Io Log -1 (B/10) = Log -1 (log I/Io) Log -1 (B/10) = I/Io

65. Let’s say that the intensity level of a sound is 25.3 dB. What is the intensity of the sound in w/cm 2 ? B = 10 log I / Io 25.3 dB = 10 log (I/10 -16 w/cm 2 ) 2.53 = log I – log 10 -16 2.50 + log 10 -16 = log I 2.50 – 16 = log I

66. 25.3 dB = 10 log (I/10 -16 w/cm2) 2.53 = log I – log 10 -16 2.50 + log 10 -16 = log I  what power do you raise 10 to, to get 10 -16 ? 2.50 – 16 = log I Adding on the left  -13.5 = log I Raise 10 to the -13.5 power by this sequence: 2nd 10x (-13.5)  3.16 x 10 -14 w/cm 2 = I

67. Doppler Effect Doppler Effect is the apparent change in frequency as a result of relative motion between the source of a sound and an observer. f’ frequency heard by observer f frequency of source v velocity of sound in air v d velocity of detector v s velocity of source

68. Sample Problem A source of 4 kHz sound waves travel at 1/9 the speed of sound toward a detector that’s moving at 1/9 the speed of sound, toward the source. –a. what is the frequency of the waves as they’re received by the detector? –b. how does the wavelength of the detected waves compare to the wavelength of the emitted waves?

69. = v + 1/9 v x f = 5/4 f = 5/4 (4 kHz) = 5 kHz v - 1/9 v + sign on top as detector moves toward source -sign on bottom as source moves toward det. -Frequency is shifted up by a factor of 5/4, the will shift down by the same factor. d =  s / 5/4 = 4/5  s

70. Sample A person yells, emitting a constant frequency of 200 Hz, as he runs at 5m/s toward a stationary brick wall. When the reflected waves reach the person, how many beats per second will he hear? (Use 343 m/s for the speed of sound.)

71. Determine what f will be reflected and heard by the runner. person is source and wall is detector fwall = v f v - vrunner Reflected sound wave (no change in f): wall is the source and runner is the detector. frunner = v + v runner f v Combine these two formulae: frunner = v + vrunner f v - vrunner f = (343+5)m/s (200Hz) = 206 Hz (343-5)m/s Beat frequency is fbeat = 206Hz - 200 Hz = 6 Hz. p. 343 Princeton

72. Doppler Effect for Light wps.prenhall.com/.../ch25_SWA/images/8.gif c = 3 x 10 8 m/s u = speed of the source and the detector.

73. www.sv.vt.edu/.../class95/physics/doppler.gif

74. Periodic Motion … Motion that repeats itself over a fixed and reproducible period of time An example is a planet moving about its sun. This is called the period (T) or year of the planet Mechanical devices on earth that have periodic motion are useful timers and are called oscillators.

75. Simple Harmonic Motion Attach a weight to a spring, stretch the spring past its equilibrium point and release. The weight bobs up and down with a reproducible period, T Plot position vs time to get a graph that resembles a sine or cosine function. The graph is “sinusoidal”, so the motion is referred to as simple harmonic motion. Springs and pendulums undergo simple harmonic motion and are referred to as simple harmonic oscillators.

76. Graph Analysis

78. Oscillator Definitions Amplitude –Maximum displacement from equilibrium –Related to energy Period –Length of time required for one oscillation Frequency –How fast the oscillator is oscillating –f = 1/T in Hz or s -1

79. Problem 11-5 An elastic cord vibrates with a frequency of 3.0 Hz when a mass of 0.60 kg is hung from it. What is its frequency if only 0.38 kg hangs from it?

80. Springs Springs are a common type of simple harmonic oscillator. Our springs are “ideal springs”, which means: They are massless They are both compressible and extensible. They will follow Hooke’s Law. F = -kx

81. Hooke’s Law Review The force constant of a spring can be determined by attaching a weight and seeing how far it stretches.

82. Period of a Spring T = period (s) m = mass (kg) k = force constant (N/m)

83. Sample Problem Calculate the period of a 350 g mass attached to an ideal spr5ing with a force constant of 35 N/m.

84. Sample Problem A 300 g mass attached to a spring undergoes simple harmonic motion with a frequency of 25 Hz. What is the force constant of the spring?

85. Sample Problem An 80 g mass attached to a spring hung vertically causes it to stretch 30 cm from its unstretched position. If the mass is set into oscillation on the end of the spring, what is its period?

86. Combinations of Springs In parallel, springs work together In series, springs work independently

87. What do you think? Does this combination of springs act as parallel or series? Answer: parallel !

88. Problem If you want to double the force constant of a spring, you A. double its length by connecting it to another one just like it. B. cut it in half. C. add twice as much mass. D. take half of the mass off. Answer: B

89. Conservation of Energy Springs and pendulums obey conservation of energy The equilibrium position has high kinetic energy and low potential energy. The positions of maximum displacement have high potential energy and low kinetic energy. Total energy of the oscillating system is constant.

90. Problem 11-20 A block of mass m is supported by 2 identical parallel vertical springs, each with spring stiffness constant k. What will be the frequency of vibration? x General form of a restoring force producing SHM with spring constant of 2k

91. Sample Problem

92. Spring Problem

93. Another spring problem

94. Pendulums Pendulums can be thought of as simple harmonic oscillators. The displacement needs to be small for it to work properly.

95. Conservation of Energy Pendulums also obey conservation of energy. The equilibrium position has high kinetic energy and low potential energy. The positions of maximum displacement have high potential energy and low kinetic energy. Total energy of the oscillating system is constant.

96. Pendulum Forces 2005 Pearson Prentice Hall Fig 11-12

97. Period of a Pendulum T -- period (s) l -- length of string (m) g -- gravitational acceleration (m/s 2 )

98. Problem: Predict the period of a pendulum consisting of a 500 g mass attached to a 2.5 m long string.

99. Problem: Suppose you notice that a 5 kg weight fixed to a string swings back and forth 5 times in 20 seconds. How long is the string?

100. Last problem! The period of a pendulum is observed to be T. Suppose you want to make the period 2T. What do you do to the pendulum?