Simple Harmonic Motion continued and Waves. Monday, November 28, 11

Transcription

1 Simple Harmonic Motion continued and Waves

2 Sinusoidal Relationships

3 Mathematical Description of Simple Harmonic Motion

4 Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

5 The Period of a Mass on a Spring Therefore, the period is T = 2π m k

6 Characteristics of SHM

7 Characteristics of SHM

8 X versus t for SHO then simple variations on a theme

9 Energy in Simple Harmonic Motion As a mass on a spring goes through its cycle of oscillation, energy is transformed from potential to kinetic and back to potential.

10 Energy Conservation in Oscillatory Motion In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring: E = K + U = 1 2 mv kx2 Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies: U = 1 max 2 ka2, i.e. when x = A k As v = rω, and in this case r = A, and ω = max m K = 1 max 2 mv2 max = 1 2 ma2 ω 2 max = 1 k 2 ma2 m = 1 2 ka2

11 Energy Conservation in Oscillatory Motion As a function of time, E = K + U = 1 2 mv kx2 v = Aω sin(ωt) v 2 = A 2 ω 2 sin 2 (ωt) x = Acos(ωt) x 2 = A 2 cos 2 (ωt) So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa. ω = k m mω 2 = k E = 1 2 ma2 ω 2 sin 2 (ωt) ka2 cos 2 (ωt) E = 1 2 ka2 sin 2 (ωt) ka2 cos 2 (ωt) E = 1 2 ka2 sin 2 (ωt) + cos 2 (ωt)

12 Energy Conservation in Oscillatory Motion This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

13 Energy in SHM

14 Energy in SHM Energy is conserved during SHM and the forms (potential and kinetic) interconvert as the position of the object in motion changes.

15 The Pendulum A physical pendulum is a solid mass that oscillates around its center of mass. Examples:

16 The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle it makes with the vertical varies with time as a sine or cosine.

17 The Pendulum Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

18 The Pendulum However, for small angles, sin θ and θ in radians are approximately equal.

19 The Pendulum However, for small angles, sin θ and θ in radians are approximately equal.

20 The Pendulum For small angles θ in radians is approximately the sine of θ, θ sinθ The net tangential force acting on m is the tangential component of its weight, and is always direct towards the equilibrium point F = mgsinθ mgθ The arc length displacement of the mass from equilibrium is S = Lθ θ = s F = mgsinθ mgθ = mg L For a spring F=kx We can use this same form for a pendulum if we let s=x, and k=mg/l L s As T = 2π m k = 2π m mg L = 2π L g So the period depends on the length of the pendulum and the acceleration due to gravity

21 Vibrations of molecules Two atoms separated by their internuclear distance r can be pondered as two balls on a spring.

22 Vibrations of molecules Two atoms separated by their internuclear distance r can be pondered as two balls on a spring. The potential energy of such a model is constructed many different ways. The Leonard Jones potential is sketched below. The atoms on a molecule vibrate as shown below.

23 Damped Oscillations In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed: This causes the amplitude to decrease exponentially with time:

24 Damped Oscillations This exponential decrease is shown in the figure:

25 Damped Oscillations The previous image shows a system that is underdamped it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

26 Damping

27 Damped oscillations A person may not wish for the object they study to remain in SHM. Consider shock absorbers and your automobile. Without damping the oscillation, hitting a pothole would set your car into SHM on the springs that support it.

28 Driven Oscillations and Resonance An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.

29 A mass on a spring is driven by a large geared motor apparatus, and exhibits resonance at the appropriate frequency. Below resonance, the driving motion is in phase with the motion of the mass. At resonance the mass is 90 degrees out of phase with the driving motion, and above resonance it is 180 degrees out of phase. Note the amplitude of motion difference when the system is at resonance.

30 Resonance y = Ae t τ The response curve becomes taller and narrower as damping is reduced

31 Driven Oscillations and Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

32 Forced (driven) oscillations and resonance A force applied in synch with a motion already in progress will resonate and add energy to the oscillation A singer can shatter a glass with a pure tone in tune with the natural ring of a thin wine glass.

33 Forced (driven) oscillations and resonance The Tacoma Narrows Bridge suffered spectacular structural failure after absorbing too much resonant energy

34 Summary Period: time required for a motion to go through a complete cycle Frequency: number of oscillations per unit time Angular frequency: Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.

35 Summary The amplitude is the maximum displacement from equilibrium. Position as a function of time: x = Acos 2π T t = Acos( ωt) Velocity as a function of time: ( ) v = Aω sin ωt

36 Summary 2010 Pearson Education, Inc. Slide 14-32

37 Summary Acceleration as a function of time: Period of a mass on a spring: Total energy in simple harmonic motion:

38 Summary Potential energy as a function of time: Kinetic energy as a function of time: A simple pendulum with small amplitude exhibits simple harmonic motion

39 Summary Period of a simple pendulum: Period of a physical pendulum:

40 Summary 2010 Pearson Education, Inc. Slide 14-33

41 Summary Oscillations where there is a nonconservative force are called damped. Underdamped: the amplitude decreases exponentially with time: Critically damped: no oscillations; system relaxes back to equilibrium in minimum time Overdamped: also no oscillations, but slower than critical damping

42 Summary An oscillating system may be driven by an external force This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance Resonance occurs when the driving frequency is equal to the natural frequency of the system

43 Quiz 1. The type of function that describes simple harmonic motion is A. linear B. exponential C. quadratic D. sinusoidal E. inverse

44 Answer 1. The type of function that describes simple harmonic motion is A. linear B. exponential C. quadratic D. sinusoidal E. inverse

45 Quiz 2. A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? A. The time increases. B. The time decreases. C. The time does not change.

46 Answer 2. A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? A. The time increases. B. The time decreases. C. The time does not change.

47 Reading Quiz 3. If you drive an oscillator, it will have the largest amplitude if you drive it at its frequency. A. special B. positive C. resonant D. damped E. pendulum

48 Answer 3. If you drive an oscillator, it will have the largest amplitude if you drive it at its frequency. A. special B. positive C. resonant D. damped E. pendulum

49 Additional Questions Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a > b > c > d B. d > c > b > a C. a = b = c = d

50 Answer Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. A. a > b > c > d B. d > c > b > a C. a = b = c = d Increasing k increases f

51 Additional Questions A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? A. 1 s B. 3.3 s C. 6.7 s D. 13 s

52 Answer A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? A. 1 s B. 3.3 s C. 6.7 s D. 13 s f = 1 T T = 1 f = = 6.67 T 2 = 3.33 Time between highest and lowest point is T/2

53 Answer A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? A. 1 s B. 3.3 s C. 6.7 s D. 13 s f = 1 T T = 1 f = = 6.67 T 2 = 3.33 Time between highest and lowest point is T/2

54 Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed v max? A. T and v max both double. B. T remains the same and v max doubles. C. T and v max both remain the same. D. T doubles and v max remains the same. E. T remains the same and v max increases by a factor of.

55 A13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed v max? A. T and v max both double. B. T remains the same and v max doubles. C. T and v max both remain the same. D. T doubles and v max remains the same. E. T remains the same and v max increases by a factor of.

56 Waves

57 Introduction At right, we see the piles of rubble from a highway that absorbed just a little of the energy from a wave propagating through the earth in California. We are going to look at ripples of disturbance moving through various media.

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59 Water waves spreading out from a source

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61 Types of Waves A wave is a disturbance that propagates from one place to another. The easiest type of wave to visualize is a transverse wave, where the displacement of the medium is perpendicular to the direction of motion of the wave.

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63 Is the velocity of the wave moving along the cord the same as the velocity of a particle of the cord?

64 Is the velocity of the wave moving along the cord the same as the velocity of a particle of the cord? No. The two velocities are different, both in magnitude and direction. The wave on the rope moves to the right along the table top, but each piece of the rope only vibrates horizontally to and fro.

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67 What do you notice about these two waves?

68 Types of Waves In a longitudinal wave, the displacement is along the direction of wave motion.

69 A Longitudinal Wave

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71 Types of Waves Water waves are a combination of transverse and longitudinal waves. Copyright 2010 Pearson Education, Inc.

72 Types of mechanical waves Waves that have compressions and rarefactions parallel to the direction of wave propagation are longitudinal. Waves that have compressions and rarefactions perpendicular to the direction of propagation are transverse.

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74 Graphing wave functions Informative graphic presentation of a wave function is often either y-displacement versus x-position or y-displacement versus x-time.

75 Basic Properties Wavelength λ: distance over which wave repeats Period T: time for one wavelength to pass a given point Frequency f: f = 1 T Speed of a wave: v = distance time = λ T = λ f Copyright 2010 Pearson Education, Inc.

76 If you double the wavelength λ of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged.

77 If you double the wavelength λ of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged. v = distance time = λ T = λ f

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79 λ = v f = = 1.31 m

80 Periodic waves A detailed look at periodic transverse waves will allow us to extract parameters.

81 Periodic waves II A detailed look at periodic longitudinal waves will allow us to extract parameters just as we did with transverse waves.

82 Mathematical description of a wave When the description of the wave needs to be more complete, we can generate a wave function with y(x,t). For a particle at the left end of the string where x=0 y(x = 0,t) = Acosωt = Acos2π ft The wave travels to the right to some point x in time x / v. So the motion at point x is the same as that at x = 0 at an earlier time t - x / v y(x,t) = Acos ω t x v = Acos2π f As T = 1 / f and λ = v / f x y(x,t) = Acos2π f λ t T t x v If we define a wavenumber as k = 2π λ = ω v, as v = λ f, so y(x,t) = Acos(kx ωt)

83 Mathematical description of a wave For a wave traveling in the positive x-direction (to the left) y(x,t) = Acos ω t x v = Acos2π f x λ t T = Acos(kx ωt) So for a wave traveling in the negative x-direction (to the right) y(x,t) = Acos ω t + x v = Acos2π f x λ + t T = Acos(kx + ωt) The quantity (kx ± ωt) is called the phase and is measured in radians

84 Mathematical description of a wave To move along with the wave and keep alongside a point of a given phase, such as a particular crest of a wave on a string traveling in the positive x direction means that kx wt = constant x= ωt k + constant k v = dx dt = ω k where dx is the phase speed v of the wave, dt sometimes called the phase velocity.

85 v = dx dt = ω k 2010 Pearson Education, Inc. Slide 15-40

86 Deriving the wave equation For a wave traveling in the positive x-direction (to the left) y(x,t) = Acos(kx ωt) then differentiate with respect to t holding x constant v y (x,t) = y(x,t) t = ω Asin(kx ωt) and a y (x,t) = 2 y(x,t) t 2 = ω 2 Acos(kx ωt) = ω 2 y(x,t)

87 Deriving the wave equation For a wave traveling in the positive x-direction (to the left) y(x,t) = Acos(kx ωt) then differentiate with respect to t holding x constant v y (x,t) = y(x,t) t = ω Asin(kx ωt) and a y (x,t) = 2 y(x,t) t 2 = ω 2 Acos(kx ωt) = ω 2 y(x,t) The acceleration of a particle equal -ω 2 times its displacement, just like for simple harmonic motion

88 Deriving the wave equation For a wave traveling in the positive x-direction (to the left) y(x,t) = Acos(kx ωt) then differentiate with respect to x holding t constant 2 y(x,t) x 2 = k 2 Acos(kx ωt) = k 2 y(x,t) 2 y(x,t) t 2 2 y(x,t) x 2 2 y(x,t) x 2 = ω 2 y(x,t) k 2 y(x,t) = ω 2 k 2 = 1 v 2 2 y(x,t) t 2 = v2, as ω = vk

89 Deriving the wave equation 2 y(x,t) t 2 2 y(x,t) x 2 = ω 2 k 2 = v2 2 y(x,t) x 2 = 1 v 2 2 y(x,t) t 2 Wave equation One of the most important equations in all of physics

90 Particle velocity and acceleration in a sinusoidal wave

91 Particle velocity and acceleration in a sinusoidal wave From the wave function, we have an expression for the kinematics of a particle at any point on the wave.

92 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the velocity of a particle of the string at x = a? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

93 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the velocity of a particle of the string at x = a? A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

94 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the acceleration of a particle of the string at x = a? A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide

95 A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. 0 y a x At this time, what is the acceleration of a particle of the string at x = a? A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide

96 Waves on a String The speed of a wave is determined by the properties of the material through which it propagates. For a string, the wave speed is determined by: 1. the tension in the string, and 2. the mass of the string. As the tension in the string increases, the speed of waves on the string increases as well.

97 Waves on a String The total mass of the string depends on how long it is; what makes a difference in the speed is the mass per unit length. We expect that a larger mass per unit length results in a slower wave speed. Definition of Mass per unit length, µ is µ= m L

98 Speed of a transverse wave One of the key properties of any wave is the wave speed. Light waves in air have a speed much greater than that of sound (3x10 8 m/s compared to 344 m/s). That is why we see the lightning before hearing the thunder. Let us take a look at the speed of a wave in a string.

99 Waves on a String Speed of a Wave on a string, v is v = F µ As we can see, the speed increases when the force increases, and decreases when the mass increases.

100 The speed of a transverse wave Consider a pulse on a perfectly flexible string. In the equilibrium position the tension is F, and the linear mass density (mass per unit length is) µ. The wave is traveling with constant speed, v. According to the impulse-momentul theorem F y t = mv y So the total momentum must increase proportinal to the time. As the point P is moving with constant speed, the total mass in motion is also proportional to the time. So the change in momentum is associated with the changing mass in motion, not an increasing velocity v.

101 The speed of a transverse wave At time t, the left end of the string has moved up a distance v y t, and the boundary point P has adavanced a distance vt. The total force at the left end of the string has components F horizontaly and F y vertically. In the displaced position the tension is F 2 + F y 2. The impulse of the transverse force F y at time t is F y t. The right-angled triangle at point P with sides vt and v y t is simillar to the right-angled triangle at Point P with sides F and F y. Therefore, F y F = v t y vt F = F v y y v and the Transverse Impulse=F y t = F v y t. The mass of the moving part of the string is the product of the v mass per unit length, µ, and the length, vt, i.e. µvt. So Transverse Momentum=(µvt)v y

102 The speed of a transverse wave Transverse Impulse=F v y v t Transverse Momentum=µvtv y So by the Impulse-momentum theorem F v y v t = (µvt)v y Solving for the speed of a transverse wave on a string, v, gives v = F µ So the wave speed increases with the tension, F, and decreases with the mass per unit length, µ

103 The speed of mechanical waves It turns out that for many types of mechanical waves, the expression for wave speed has the same general form: v = v = Restoring force returning the system to equilibrium Inertia resisting the return to equilibrium elastic force factor inertia factor For a longitudinal wave traveling down a solid rod of elastic modulus E v = E ρ For a longitudinal wave traveling in a fluid of bulk modulus B v = B ρ

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105 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide

106 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide

107 Sound Waves The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it. Copyright 2010 Pearson Education, Inc.

108 Sound Waves Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz. Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic. Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves. Copyright 2010 Pearson Education, Inc.

109 A transverse wave so the net horizontal force is zero

110 The speed of a transverse wave v = F µ F=weight of samples= =196 N µ= m rope L = v = = kg/m = 88.5 m/s λ= v f = = 44.3 m The length of the rope is 80.0 m, so the number of wave cycles in the rope is 80.0 m/s = 1.81 cycles 44.3 m/cycle

111 Energy in Wave Motion Every wave motion has energy associated with it. This is borne out by the energy we receive from sunlight on the destructive effects of earthquakes and ocean surf.

112 Energy in Wave Motion The ratio F y / F is equal to the negative of the slope of the string at a, y/ x. F y (x,t) = F y(x,t) x When a point a moves in the y-direction, the force F y does work on this point and therefore transfers energy into the part of the string to the right of a. The power P (rate of doing work) at point a is the transverse force at a times the transverse velocity: P(x,t) = F y (x,t)v y (x,t) = F y(x,t) x y(x,t) t y(x,t) = Acos(kx ωt) y(x,t) x = kasin(kx ωt), and y(x,t) t P(x,t) = Fkω A 2 sin 2 (kx ωt), since ω = vk and v 2 = F / µ, P(x,t) = = ω Asin(kx ωt) µfω 2 A 2 sin 2 (kx ωt) The sin 2 function is never negative, so the instantaneous power in a sinusoidal wave is either positive (so that energy flows in the positive x-direction) or zero (no energy transfer). Energy is never transferred in the direction opposite to the wave propogation.

113 Energy in Wave Motion The maximum value of the instantaneous power occurs when the sin 2 function has the value unity. P(x,t) = µfω 2 A 2 sin 2 (kx ωt) P max = µfω 2 A 2 The average of the sin 2 function is 1 2 so P av = 1 2 µfω 2 A 2

114 Energy in Wave Motion The maximum value of the instantaneous power occurs when the sin 2 function has the value unity. P(x,t) = P max = µfω 2 A 2 µfω 2 A 2 sin 2 (kx ωt) The average of the sin 2 function is 1 2 so P av = 1 2 µfω 2 A 2 Note: The average rate of energy transfer is proportional to the square of the amplitude and the square of the frequency. Energy is never transferred in the direction opposite to wave propagation.

115 Waves on a clothesline

116 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0

117 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0 b) y(x,t) = Acos2π x λ t T = cos2π ( ) y(x,t) = cos 1.05x 12.6t x 6.00 t 0.500

118 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0 b) y(x,t) = Acos2π x λ t T = cos2π ( ) y(x,t) = cos 1.05x 12.6t x 6.00 t c) Substitute into the wave equation x = 0 and x = m. ( ) ( ) y(x = 0,t) = cos 0 x 12.6t y(x = 3,t) = cos 3.00 x 12.6t

119 Waves on a clothesline

120 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0

121 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0 b) y(x,t) = Acos2π x λ t T = cos2π ( ) y(x,t) = cos 1.05x 12.6t x 6.00 t 0.500

122 Waves on a clothesline a) A = m f = 2.00 Hz ω = 2π f = 2π 2.00 = 4π rad/s λ= v f = = 6.00 m k= 2π λ = 2π = 1.05 rad/m, 6.00 or k = ω v = 4π = 1.05 rad/m 12.0 b) y(x,t) = Acos2π x λ t T = cos2π ( ) y(x,t) = cos 1.05x 12.6t x 6.00 t c) Substitute into the wave equation x = 0 and x = m. ( ) ( ) y(x = 0,t) = cos 0 x 12.6t y(x = 3,t) = cos 3.00 x 12.6t

123 Power in a wave

124 Power in a wave We already derived that P max = µfω 2 A 2, and P av = 1 2 µfω 2 A 2 So P max = µfω 2 A 2 = (4π ) 2 (0.075) 2 = 2.66W P av = = 1.33W The new amplitude is 1 of the original value, the average power is 10 proportional to the square of the amplitude so the new average power is P av = = W = 13.3mW

125 Sound Intensity The intensity of a sound is the amount of energy that passes through a given area in a given time. Copyright 2010 Pearson Education, Inc.

126 Sound Intensity Expressed in terms of power, Copyright 2010 Pearson Education, Inc.

127 Sound Intensity Sound intensity from a point source will decrease as the square of the distance. Copyright 2010 Pearson Education, Inc.

128 Sound Intensity Bats can use this decrease in sound intensity to locate small objects in the dark. Copyright 2010 Pearson Education, Inc.

129 Sound Intensity The quantity β is called a bel; a more common unit is the decibel, db, which is a tenth of a bel. The intensity of a sound doubles with each increase in intensity level of 10 db. Copyright 2010 Pearson Education, Inc.

130 Wave intensity Go beyond the wave on a string and visualize, say a sound wave spreading from a speaker. That wave has intensity dropping as 1/r 2.

131 Wave intensity

132 Wave intensity

133 The inverse square law

134 The inverse square law

135 The inverse square law

136 Wave interference, boundaries, and superposition

137 Wave interference, boundaries, and superposition

138 Wave interference, boundaries, and superposition

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141 Waves on a String When a wave reaches the end of a string, it will be reflected. If the end is fixed, the reflected wave will be inverted: Copyright 2010 Pearson Education, Inc.

142 Waves on a String If the end of the string is free to move transversely, the wave will be reflected without inversion. Copyright 2010 Pearson Education, Inc.

143 Superposition and Interference Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition. Copyright 2010 Pearson Education, Inc.

144 Superposition and Interference If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right). Copyright 2010 Pearson Education, Inc.

145 Overlapping pulses As wave pulses travel, reflect, travel back, and repeat the whole cycle again, waves in phase will add and waves out of phase will cancel.

146 Overlapping pulses

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148 Standing Waves A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air. Copyright 2010 Pearson Education, Inc.

149 Standing Waves The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics. Copyright 2010 Pearson Education, Inc.

150 Standing Waves There must be an integral number of halfwavelengths on the string; this means that only certain frequencies are possible. Points on the string which never move are called nodes; those which have the maximum movement are called antinodes. Copyright 2010 Pearson Education, Inc.

151 Standing Waves In order for different strings to have different fundamental frequencies, they must differ in length and/or linear density. A guitar has strings that are all the same length, but the density varies. Copyright 2010 Pearson Education, Inc.

152 Standing Waves In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape. Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension. Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage. Copyright 2010 Pearson Education, Inc.

153 Standing Waves Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes. As indicated in the drawing, one end is a node (N), and the other is an antinode (A). Copyright 2010 Pearson Education, Inc.

154 Standing Waves In this case, the fundamental wavelength is four times the length of the pipe, and only oddnumbered harmonics appear. Copyright 2010 Pearson Education, Inc.

155 Standing Waves If the tube is open at both ends, both ends are antinodes, and the sequence of harmonics is the same as that on a string. Copyright 2010 Pearson Education, Inc.

156 Beats Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency. Copyright 2010 Pearson Education, Inc.

157 Standing waves on a string Fixed at both ends, the resonator was have waveforms that match. In this case, the standing waveform must have nodes at both ends. Differences arise only from increased energy in the waveform.

158 The formation of a standing wave The process seems complicated at first, but it is nothing more than waveforms adding constructively when they re in phase and destructively when they re not.

159 Complex standing waves As the shape and composition of the resonator change, the standing wave changes also. Regard Figure 15.27, a multidimensional standing wave. Figure provides many such multidimensional shapes.

160 Waves on Strings and in Air v string = T s µ where T s is the string tension and µ is the linear density v sound = γ RT M where T is the string tension γ = C p Cv is the adiabatic index R is the molar gas constant M is the molar mass

161 Snapshot Graphs

162 Sound waves may be graphed several ways

163 Sound waves may be graphed several ways

164 At a compression in a sound wave, A. particles are displaced by the maximum distance in the same direction as the wave is moving. B. particles are displaced by the maximum distance in the direction opposite to the direction the wave is moving. C. particles are displaced by the maximum distance in the direction perpendicular to the direction the wave is moving. D. the particle displacement is zero.

165 At a compression in a sound wave, A. particles are displaced by the maximum distance in the same direction as the wave is moving. B. particles are displaced by the maximum distance in the direction opposite to the direction the wave is moving. C. particles are displaced by the maximum distance in the direction perpendicular to the direction the wave is moving. D. the particle displacement is zero.

166 Different instruments give the same pitch but different flavor The same frequency, say middle c at 256 Hz, played on a piano, on a trumpet, on a clarinet, on a tuba they will all be the same pitch but they will all sound different to the listener.

167 Speed of sound in liquids and solids v sound = γ RT M where T is the string tension γ = C p Cv is the adiabatic index R is the molar gas constant M is the molar mass Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

168 Speed of sound in liquids and solids The speed of sound will increase with the density of the material. v sound = γ RT M where T is the string tension γ = C p Cv is the adiabatic index R is the molar gas constant M is the molar mass Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

169 Sound intensity Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

170 Sound intensity The in amplitude term in our wave equation can be related to the sound intensity, but perception of the listener often complicates the physics (location, weather, voice, or sound) in question. Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

171 Sound Intensity The intensity of a sound is the amount of energy that passes through a given area in a given time. I = P A = E At = P 4πr 2 Copyright 2010 Pearson Education, Inc.

172 Energy and Intensity 2010 Pearson Education, Inc.

173 Sound Intensity I = P A = E At = P 4πr 2 Copyright 2010 Pearson Education, Inc.

174 The Decibel Scale Sound intensity level is measured in decibels Pearson Education, Inc.

175 Wave intensity

176 Wave intensity For a wave traveling in the positive x-direction (to the left) y(x,t) = Acos ω t x v = Acos2π f x λ t T = Acos(kx ωt) then differentiate with respect to t holding x constant v y (x,t) = y(x,t) t = ω Asin(kx ωt) The pressure variation of the wave for a bulk modulus, B, is p(x,t) = BkAsin(kx ωt) So the maximum pressure is p max = BkA The Intensity is the power per unit area, the product of the pressure and velocity I = P A = p(x,t)v y (x,t) = [ BkAsin(kx ωt) ][ ω Asin(kx ωt) ] I = BωkA 2 sin 2 (kx ωt)

177 Q16.2 Increasing the pressure amplitude of a sound wave by a factor of 4 (while leaving the frequency unchanged) A. causes the intensity to increase by a factor of 16. B. causes the intensity to increase by a factor of 4. C. causes the intensity to increase by a factor of 2. D. has no effect on the wave intensity. E. The answer depends on the frequency of the sound wave.

178 A16.2 Increasing the pressure amplitude of a sound wave by a factor of 4 (while leaving the frequency unchanged) A. causes the intensity to increase by a factor of 16. B. causes the intensity to increase by a factor of 4. C. causes the intensity to increase by a factor of 2. D. has no effect on the wave intensity. E. The answer depends on the frequency of the sound wave. The Intensity is the power per unit area I = P A = BωkA2 sin 2 (kx ωt)

179 Q16.3 Increasing the frequency of a sound wave by a factor of 4 (while leaving the pressure amplitude unchanged) A. causes the intensity to increase by a factor of 16. B. causes the intensity to increase by a factor of 4. C. causes the intensity to increase by a factor of 2. D. has no effect on the wave intensity. E. The answer depends on the frequency of the sound wave.

180 A16.3 Increasing the frequency of a sound wave by a factor of 4 (while leaving the pressure amplitude unchanged) A. causes the intensity to increase by a factor of 16. B. causes the intensity to increase by a factor of 4. C. causes the intensity to increase by a factor of 2. D. has no effect on the wave intensity. E. The answer depends on the frequency of the sound wave. The Intensity is the power per unit area I = P A = BωkA2 sin 2 (kx ωt)

181

182 Waves on a String When a wave reaches the end of a string, it will be reflected. If the end is fixed, the reflected wave will be inverted: Copyright 2010 Pearson Education, Inc.

183 Waves on a String If the end of the string is free to move transversely, the wave will be reflected without inversion. Copyright 2010 Pearson Education, Inc.

184

185

186 Wave interference, boundaries, and superposition

187 Wave interference, boundaries, and superposition

188 Wave interference, boundaries, and superposition

189 Superposition and Interference Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition. Copyright 2010 Pearson Education, Inc.

190 Superposition and Interference If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right). Copyright 2010 Pearson Education, Inc.

191 Overlapping pulses As wave pulses travel, reflect, travel back, and repeat the whole cycle again, waves in phase will add and waves out of phase will cancel.

192 Overlapping pulses

193 Summary

194 Checking Understanding The graph below shows a snapshot graph of a wave on a string that is moving to the right. A point on the string is noted. Which of the choices is the history graph for the subsequent motion of this point?

195 Answer The graph below shows a snapshot graph of a wave on a string that is moving to the right. A point on the string is noted. Which of the choices is the history graph for the subsequent motion of this point? (b)

196 Answer The graph below shows a snapshot graph of a wave on a string that is moving to the right. A point on the string is noted. Which of the choices is the history graph for the subsequent motion of this point? (b)

197 Standing Waves A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air. Copyright 2010 Pearson Education, Inc.

198 Standing Waves There must be an integral number of halfwavelengths on the string; this means that only certain frequencies are possible. Points on the string which never move are called nodes; those which have the maximum movement are called antinodes. Copyright 2010 Pearson Education, Inc.

199 Standing Waves The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics. Standing waves on a string Fundamental frequency and wavelength f 1 = v 2L,λ 1 = 2L Frequency and wavelength of the nth harmonic, with n=1,2,3, f n = n v 2L = nf 1,λ n = λ 1 n = 2L n Copyright 2010 Pearson Education, Inc.

200 Standing Waves In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape. Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension. Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage. Copyright 2010 Pearson Education, Inc.

201 Fundamental frequency and wavelength f 1 = v 2L,λ 1 = 2L Frequency and wavelength of the nth harmonic, with n=1,2,3, f n = n v 2L = nf 1,λ n = λ 1 n = 2L n Copyright 2010 Pearson Education, Inc.

202 Complex standing waves As the shape and composition of the resonator change, the standing wave changes also.

203 Standing Waves In order for different strings to have different fundamental frequencies, they must differ in length and/or linear density. A guitar has strings that are all the same length, but the density varies. Copyright 2010 Pearson Education, Inc.

204 Standing waves on a string Fixed at both ends, the standing waveform must have nodes at both ends. Differences arise only from increased energy in the waveform.

205 The formation of a standing wave The process seems complicated at first, but it is nothing more than waveforms adding constructively when they re in phase and destructively when they re not.

206 Copyright 2010 Pearson Education, Inc.

207 Standing Waves Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes. As indicated in the drawing, one end is a node (N), and the other is an antinode (A). Copyright 2010 Pearson Education, Inc.

208 Cross-sectional views reveal harmonic waves - open pipe Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

209 Cross-sectional views reveal harmonic waves - closed pipe Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

210 The air in an organ pipe is replaced by helium (which has a lower molar mass than air) at the same temperature. How does this affect the normal-mode wavelengths of the pipe? A. The normal-mode wavelengths are unaffected. B. The normal-mode wavelengths increase. C. The normal-mode wavelengths decrease. D. The answer depends on whether the pipe is open or closed.

211 A16.4 The air in an organ pipe is replaced by helium (which has a lower molar mass than air) at the same temperature. How does this affect the normal-mode wavelengths of the pipe? A. The normal-mode wavelengths are unaffected. B. The normal-mode wavelengths increase. C. The normal-mode wavelengths decrease. D. The answer depends on whether the pipe is open or closed.

212 Q16.6 When you blow air into an open organ pipe, it produces a sound with a fundamental frequency of 440 Hz. If you close one end of this pipe, the new fundamental frequency of the sound that emerges from the pipe is A. 110 Hz. B. 220 Hz. C. 440 Hz. D. 880 Hz. E Hz.

213 A16.6 When you blow air into an open organ pipe, it produces a sound with a fundamental frequency of 440 Hz. If you close one end of this pipe, the new fundamental frequency of the sound that emerges from the pipe is A. 110 Hz. B. 220 Hz. C. 440 Hz. D. 880 Hz. E Hz.

214 Beats Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency. Copyright 2010 Pearson Education, Inc.

215 Slightly mismatched frequencies cause audible beats Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

216

217

218 The Doppler Effect

219

220 moving listener, stationary source f L = v + v L λ = v + v L v f s = v + v L v f s = 1+ v L v f s

221 in front of moving source λ in front = v v s f s behind the moving source λ behind = v + v s f s frequency of listener behind the source f L = v + v L = v + v L λ behind v + v s f s = v + v L v + v s f s

222 f = v + v L L v + v s f s

223 The Doppler Effect 2010 Pearson Education, Inc. Slide 15-33

224 Q16.9 On a day when there is no wind, you are at rest and a source of sound waves is moving toward you. Compared to what you would hear if the source were not moving, the sound that you hear has A. a higher frequency and a shorter wavelength. B. the same frequency and a shorter wavelength. C. a higher frequency and the same wavelength. D. the same frequency and the same wavelength.

225 A16.9 On a day when there is no wind, you are at rest and a source of sound waves is moving toward you. Compared to what you would hear if the source were not moving, the sound that you hear has A. a higher frequency and a shorter wavelength. B. the same frequency and a shorter wavelength. C. a higher frequency and the same wavelength. D. the same frequency and the same wavelength.

226 Doppler Effect: Wavelengths

227 Doppler Effect: Wavelengths

228 Doppler Effect: Wavelengths

229 Doppler Effect: Wavelengths

230 Doppler Effect: Frequencies

231 Doppler Effect: Frequencies

232 Doppler Effect: A moving listener

233 Doppler Effect: A moving listener

234 Doppler Effect: Moving source, moving listener

235 Doppler Effect: Moving source, moving listener

236 Doppler Effect: A double doppler shift

237 Doppler Effect: A double doppler shift

238 Doppler Effect: A double doppler shift

239 Very fast aircraft can outrun the sound they generate A sonic boom can be heard when an aircraft s speed overcomes the sound it generates. Before Chuck Yeager s flight, designers were not sure the plane would survive.