Standing Waves

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Chapter 21. Superposition Chapter 21. Superposition The combination of two or more waves is called a superposition of waves. Applications of superposition range from musical instruments to the colors of an oil film to lasers. Chapter Goal: To understand and use the idea of superposition.

Principal of superposition Chapter 21. Superposition

Topics: • The Principle of Superposition • Standing Waves • Transverse Standing Waves • Standing Sound Waves and Musical Superimposed Superposition Acoustics N • Interference in One Dimension = + + + = • The Mathematics of Interference Dnet D1 D2 ... DN ∑Di i=1 • Interference in Two and Three Dimensions • Beats 3 4 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. When a wave pulse on a string reflects from a hard boundary, how is the reflected pulse related to the incident pulse?

A. Shape unchanged, amplitude unchanged Chapter 21. Reading Quizzes B. Shape inverted, amplitude unchanged C. Shape unchanged, amplitude reduced D. Shape inverted, amplitude reduced E. Amplitude unchanged, speed reduced

When a wave pulse on a string reflects There are some points on a standing from a hard boundary, how is the wave that never move. What are reflected pulse related to the incident these points called? pulse?

A. Shape unchanged, amplitude unchanged A. Harmonics B. Shape inverted, amplitude unchanged B. Normal Modes C. Shape unchanged, amplitude reduced C. Nodes D. Shape inverted, amplitude reduced D. Anti-nodes E. Amplitude unchanged, speed reduced E. Interference

7 8 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. There are some points on a standing Two sound waves of nearly equal wave that never move. What are frequencies are played simultaneously. these points called? What is the name of the acoustic phenomena you hear if you listen to these two waves? A. Harmonics B. Normal Modes C. Nodes A. Beats D. Anti-nodes B. Diffraction E. Interference C. Harmonics D. Chords E. Interference

Two sound waves of nearly equal frequencies are played simultaneously. The various possible standing What is the name of the acoustic waves on a string are called the phenomena you hear if you listen to these two waves? A. antinodes. A. Beats B. resonant nodes. B. Diffraction C. normal modes. C. Harmonics D. incident waves. D. Chords E. Interference

11 12 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The various possible standing The frequency of the third harmonic waves on a string are called the of a string is

A. antinodes. B. resonant nodes. A. one-third the frequency of the fundamental. C. normal modes. B. equal to the frequency of the fundamental. D. incident waves. C. three times the frequency of the fundamental. D. nine times the frequency of the fundamental.

The frequency of the third harmonic of a string is

Chapter 21. Basic Content and Examples A. one-third the frequency of the fundamental. B. equal to the frequency of the fundamental. C. three times the frequency of the fundamental. D. nine times the frequency of the fundamental.

If wave 1 displaces a particle in the medium by D1 and wave 2 simultaneously displaces it by D2, the net displacement of the particle is simply D1 + D2.

Standing Waves

19 20 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Mathematics of Standing Waves The Mathematics of Standing Waves According to the principle of superposition, the net A sinusoidal wave traveling to the right along the x-axis displacement of the medium when both waves are present is with angular frequency ω = 2 πf, wave number k = 2 π/λ and the sum of D and D : amplitude a is R L

We can simplify this by using a trigonometric identity, and An equivalent wave traveling to the left is arrive at

Where the amplitude function A(x) is defined as We previously used the symbol A for the wave amplitude, but here we will use a lowercase a to represent the amplitude of each individual wave and reserve A for the amplitude of the net wave. The amplitude reaches a maximum value of Amax = 2 a at 21 points where sin kx = 1. 22 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Examples: Waves on Strings Examples: Waves on Pipes

23 24 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.1 Node spacing on a string EXAMPLE 21.1 Node spacing on a string

QUESTIONS:

EXAMPLE 21.1 Node spacing on a string EXAMPLE 21.1 Node spacing on a string

27 28 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Standing Waves on a String Standing Waves on a String For a string of fixed length L, the boundary conditions can be satisfied only if the wavelength has one of the values

A standing wave can exist on the string only if its wavelength is one of the values given by Equation 21.13. Because λf = v for a sinusoidal wave, the oscillation λ frequency corresponding to wavelength m is

Standing Waves on a String EXAMPLE 21.4 Cold spots in a microwave There are three things to note about the normal modes of a oven string. QUESTION: 1. m is the number of antinodes on the standing wave, not the number of nodes. You can tell a string’s mode of oscillation by counting the number of antinodes. λ 2. The fundamental mode , with m = 1, has 1 = 2 L, not λ 1 = L. Only half of a wavelength is contained between the boundaries, a direct consequence of the fact that the spacing between nodes is λ/2.

3. The frequencies of the normal modes form a series: f1, 2f1, 3 f1, …The fundamental frequency f1 can be found as the difference between the frequencies of any two adjacent modes. That is, f = ∆f = f – f . 1 m+1 m 31 32 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.4 Cold spots in a microwave EXAMPLE 21.4 Cold spots in a microwave oven oven

Standing Sound Waves

• A long, narrow column of air, such as the air in a tube or pipe, can support a longitudinal standing sound wave. • A closed end of a column of air must be a displacement node. Thus the boundary conditions—nodes at the ends— are the same as for a standing wave on a string. • It is often useful to think of sound as a pressure wave rather than a displacement wave. The pressure oscillates around its equilibrium value. • The nodes and antinodes of the pressure wave are interchanged with those of the displacement wave.

35 36 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 37 38 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

EXAMPLE 21.6 The length of an organ pipe

QUESTION:

39 40 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.6 The length of an organ pipe EXAMPLE 21.6 The length of an organ pipe

EXAMPLE 21.6 The length of an organ pipe EXAMPLE 21.7 The notes on a clarinet

QUESTION:

43 44 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.7 The notes on a clarinet EXAMPLE 21.7 The notes on a clarinet

Interference in One Dimension The Mathematics of Interference The pattern resulting from the superposition of two waves is As two waves of equal amplitude and frequency travel often called interference. In this section we will look at the together along the x-axis, the net displacement of the interference of two waves traveling in the same direction. medium is

We can use a trigonometric identity to write the net displacement as

Where ∆ø = ø2 – ø1 is the phase difference between the two waves. 47 48 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Mathematics of Interference EXAMPLE 21.10 Designing an antireflection coating The amplitude has a maximum value A = 2 a if cos(∆ø/2) = ±1. This occurs when QUESTION:

Where m is an integer. Similarly, the amplitude is zero if cos(∆ø/2) = 0, which occurs when

EXAMPLE 21.10 Designing an antireflection EXAMPLE 21.10 Designing an antireflection coating coating

51 52 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.10 Designing an antireflection EXAMPLE 21.10 Designing an antireflection coating coating

Interference in Two and Three Dimensions Interference in Two and Three Dimensions

The mathematical description of interference in two or three dimensions is very similar to that of one-dimensional interference. The conditions for constructive and destructive interference are

where ∆r is the path-length difference . 55 56 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Problem-Solving Strategy: Interference of two Problem-Solving Strategy: Interference of two waves waves

Problem-Solving Strategy: Interference of two Problem-Solving Strategy: Interference of two waves waves

59 60 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.11 Two-dimensional EXAMPLE 21.11 Two-dimensional interference between two loudspeakers interference between two loudspeakers QUESTIONS:

EXAMPLE 21.11 Two-dimensional EXAMPLE 21.11 Two-dimensional interference between two loudspeakers interference between two loudspeakers

63 64 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.11 Two-dimensional EXAMPLE 21.11 Two-dimensional interference between two loudspeakers interference between two loudspeakers

Beats Beats • With beats, the sound intensity rises and falls twice during one cycle of the modulation envelope. • Each “loud-soft-loud” is one beat, so the beat frequency

fbeat , which is the number of beats per second, is twice the modulation frequency fmod . • The beat frequency is

where, to keep fbeat from being negative, we will always let f1 be the larger of the two frequencies. The beat is simply the difference between the two individual

67 frequencies. 68 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 21.13 Listening to beats EXAMPLE 21.13 Listening to beats

QUESTIONS:

EXAMPLE 21.13 Listening to beats

Chapter 21. Summary Slides

71 72 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. General Principles Important Concepts

Important Concepts Applications

75 76 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Applications • Beat occurs due to the superposition of two waves of different f and same A Loud Soft Loud Soft

= Asin(2 π t) y1 f1

= Asin(2 π t) y2 f2

Two pulses on a string approach each other Chapter 21. Clicker Questions at speeds of 1 m/s. What is the shape of the string at t = 6 s?

79 80 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Two pulses on a string approach each other A standing wave on a string vibrates as shown at the at speeds of 1 m/s. What is the shape of the top. Suppose the tension is quadrupled while the string at t = 6 s? frequency and the length of the string are held constant. Which standing wave pattern is produced?

An open-open tube of air supports standing waves at frequencies of 300 Hz and 400 Hz, and at no frequencies A standing wave on a string vibrates as shown at the between these two. The second harmonic top. Suppose the tension is quadrupled while the of this tube has frequency frequency and the length of the string are held constant. Which standing wave pattern is produced? A. 800 Hz. B. 200 Hz. C. 600 Hz. D. 400 Hz. E. 100 Hz. 83 84 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Two loudspeakers An open-open tube of air supports emit waves with λλλ = standing waves at frequencies of 300 Hz 2.0 m. Speaker 2 is 1.0 m in front of speaker and 400 Hz, and at no frequencies 1. What, if anything, between these two. The second harmonic must be done to cause of this tube has frequency constructive interference between A. 800 Hz. the two waves? B. 200 Hz. A. Move speaker 1 forward (to the right) 0.5 m. B. Move speaker 1 backward (to the left) 1.0 m. C. 600 Hz. C. Move speaker 1 forward (to the right) 1.0 m. D. 400 Hz. D. Move speaker 1 backward (to the left) 0.5 m. E. Nothing. The situation shown already causes E. 100 Hz. constructive interference. 85 86 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Two loudspeakers emit waves with λλλ = The interference at 2.0 m. Speaker 2 is 1.0 m in front of speaker point C in the 1. What, if anything, figure at the right is must be done to cause constructive A. maximum constructive. interference between B. destructive, but not the two waves? perfect. A. Move speaker 1 forward (to the right) 0.5 m. C. constructive, but less than B. Move speaker 1 backward (to the left) 1.0 m. maximum. C. Move speaker 1 forward (to the right) 1.0 m. D. perfect destructive. D. Move speaker 1 backward (to the left) 0.5 m. E. there is no interference at E. Nothing. The situation shown already causes point C. constructive interference. 87 88 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The interference at point C in the

figure at the right is These two loudspeakers are in phase. They emit equal-amplitude A. maximum constructive. sound waves with a wavelength of B. destructive, but not 1.0 m. At the point indicated, is perfect. the interference maximum C. constructive, but less than constructive, perfect destructive maximum. or something in between? D. perfect destructive. E. there is no interference at A. perfect destructive point C. B. maximum constructive C. something in between 89 90 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The These two loudspeakers are in frequency of the other is phase. They emit equal-amplitude sound waves with a wavelength of 1.0 m. At the point indicated, is A. 604 Hz. the interference maximum B. 607 Hz. constructive, perfect destructive or something in between? C. 613 Hz. D. 616 Hz. A. perfect destructive B. maximum constructive E. Either b or c. C. something in between 91 92 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The frequency of the other is

A. 604 Hz. B. 607 Hz. C. 613 Hz. D. 616 Hz. E. Either b or c.