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Standing Waves 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. 1 2 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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 5 6 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 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 9 10 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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. 13 14 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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. 15 16 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Principle of Superposition 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. 17 18 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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: 25 26 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 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 29 30 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 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 33 34 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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 41 42 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 21.7 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 45 46 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 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.
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