Chapter 16 Traveling Waves

Transcription

1 Chapter 16 Traveling Waves GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms as it is used in physics, and use the term in an operational definition: frequency refraction wavelength superposition principle amplitude interference longitudinal wave diffraction transverse wave standing wave phase Fourier's theorem intensity dispersion reflection Wave Forms Sketch a longitudinal wave and a transverse wave. Wave Problems Solve wave problems involving the relationships that exist between the different characteristics of waves. Superposition and Fourier's Theorem Use the superposition principle and Fourier's theorem to explain the wave form of a complex wave. Standing Waves Use the superposition principle to explain the formation of standing waves in different situations. Inverse Square Law Use the inverse square law to calculate the intensity of a wave emanating from a point source. PREREQUISITES Before beginning this chapter you should have achieved the goals of Chapter 5, Energy, Chapter 13, Elastic Properties of Materials, and Chapter 15, Simple Harmonic Motion. 132

2 Chapter 16 Traveling Waves OVERVIEW - The motion of waves is a fascinating topic. Whenever you see a wave of water move along the surface of a quiet lake, you are observing the movement of energy. Most of the cases of energy transported by waves cannot be seen but are detected through our other senses. Examples include both sound and light. This chapter will expand your knowledge of simple harmonic motion (Chapter 15) to include a description of wave motion. As you look over the chapter you should note that in general waves can be classified as Transverse or Longitudinal depending upon their mode of vibrating. SUGGESTED STUDY PROCEDURE - When you begin your study of this chapter, please concentrate your attention on the following Chapter Goals: Definitions, Wave Problems, and Inverse Square Law. For an expanded discussion of each of the terms listed under Definitions, see The Definitions section of this Study Guide chapter. Now, read text sections Be sure to note that the description of wave motion is included in Sections The remaining parts of the Chapter describe many of the properties of waves. Answers to the questions you encounter in your reading can be found in the second section of this Study Guide chapter. Next, read the Chapter Summary and complete Summary Exercises 1-6 and 8. Check your answers carefully. Now, do Algorithmic Problems 1, 3, 5, and 9, and Exercises and Problems 1, 2, 4, 5, 6, and 13. For additional work with the important concepts introduced in this chapter, turn to the Examples section of this Study Guide chapter. Now you should be prepared to attempt the Practice Test included at the end of this Study Guide chapter. Be sure to check your answers after you have considered the problems posed. Refer to the text section or to this Study Guide chapter for additional help if needed Chapter Goals Suggested Summary Algorithmic Exercises Text Readings Exercises Problems & Problems Definitions , 1-3,4,6, Wave Problems , ,3,5 1,2,4,5, 6 Inverse Square Law Wave Forms 16.4,16.8, 4 5, ,16.12 Superposition & 16.8, ,10 Fourier's Theorem Standing Waves

3 DEFINITIONS FREQUENCY - The number of complete oscillations made by a wave in one second is called its frequency. Frequency is measured in complete oscillations, or cycles, per seconds. The unit is named hertz, one hertz means one cycle per second. WAVELENGTH - The distance a wave moves during one time period. As waves travel out from the place a pebble hits still water the distance between successive peaks is one wavelength. AMPLITUDE - The maximum displacement of the oscillating system from its equilibrium position. Large ocean waves are those of the greatest amplitude. LONGITUDINAL WAVES - The individual particle vibrates in a direction parallel to the direction of propagation of the wave. The shaking back and forth in the direction of the stretching of a Slinky toy creates longitudinal waves along the Slinky. TRANSVERSE WAVE - The plane of vibration is perpendicular to the direction of propagation of the wave. The shaking up and down of the end of a Slinky in a plane perpendicular to the length of the toy creates transverse waves that travel along the toy. PHASE - The starting position of a wave with respect to the equilibrium position. Two locations of a traveling wave that differ in phase by 180 ø must be one-half of a wavelength apart. INTENSITY - Energy transported through a unit area in one second. If you make the amplitude of a traveling wave twice as large you make its intensity four times as large. REFLECTION - A phenomenon which occurs at the interface between two media and the wave from the interface is in the same media as the incident wave. An echo is a reflected sound wave that comes back to your ear some time after it first fell on your ear. REFRACTION - A phenomenon which occurs at the interface between two media and the direction of the wave in the second medium is different than in the incident medium. When a water wave passes through a region of abrupt change in depth from shallow water to deep water, its direction of propagation is changed. SUPERPOSITION PRINCIPLE - The resultant effect is equal to the sum of the individual independent effects. A principle that holds true for linear systems. Most complex systems in physics are analyzed into simpler linear systems. Then the resultant is the sum of the individual parts. A free body diagram is an example of the superposition concept applied to a problem of mechanical equilibrium. INTERFERENCE - The superposition of two or more waves with constant phase differences produces interference. Two waves that interfere constructively will have a maximum amplitude greater than either one alone. Two waves that interfere destructively will have a reduced maximum amplitude. DIFFRACTION - Occurs if a wave front encounters an object and there results a superposition of wave fronts. The bending of sound waves around corners is a result of the diffraction of sound waves. 134

4 STANDING WAVES - These result from the superposition of two traveling waves of equal frequency moving in opposite directions in the medium and 180 ø out of phase. For a given physical system standing waves result at its resonance frequency. The sounds that originate from the vibrating strings of a guitar represent the standing waves of the strings. FOURIER'S THEOREM - Any wave form may be produced by the superposition of sine waves of specific wavelengths and amplitudes. DISPERSION - The speed of a traveling wave may be different for different wavelengths of the wave. This difference results in the spread of the various wavelengths of a complex traveling wave as it passes through various media. ANSWERS TO QUESTIONS FOUND IN THE TEXT SECTION 16.7 Reflection and Refraction Most of our common experiences with reflection and refraction occur with light. Your use of a mirror to check on your appearance is a use of reflection. Have you ever noticed how shallow a very clear lake appears to be? That deception results from refraction. If the energy reaching a surface is to be conserved, then the incident energy must equal the energy leaving the surface. Since both reflected and transmitted waves are leaving the surface, then Incident Energy = Reflected Energy + Transmitted Energy. Divide both sides by the incident energy and multiply by 100 to convert to percentage, then 100% = % Reflected Energy + % Transmitted Energy. SECTION Dispersion If air were a dispersive medium for sound, then the different frequencies, or pitches, of sounds would travel at different speeds. Suppose, for example, that low frequency waves, bass notes, travelled faster than high frequency waves, soprano notes. Then students seated at the back of a large lecture hall would hear different sounding words than the students seated near the front! EXAMPLES WAVE PROBLEMS 1. The equation for a travelling wave y(m) when x is in meters and t is in seconds is given by y = 10 cos(4x + 200t). (a) What is the amplitude of this wave? (b) What is its wavelength? (c) What is its frequency? (d) What is its velocity? (e) Sketch the wave at t = 0. (f) Sketch the wave at t = π/800 sec. (g) In which direction is the wave moving? 135

5 What Data Are Given? The standard travelling wave equation with numerical values is given. What Data Are Implied? The wave is transverse with amplitude in the y - direction and motion along the x - axis. What Physics Principles Are Involved? The basic definitions of the various aspects of travelling waves as presented in Sections 16.4 and 16.5 are needed. What Equations Are to be Used? y = A sin 2πft - (2πx/λ) (16.4) v wave = λf (16.2) Algebraic Solution The problem involves primarily the recognition of the various terms in Equation 16.4 and identifying their values from the numbers given in the problem. Once the wavelength λ and the frequency f have been identified from Equation 16.4, then they can be multiplied together to obtain the wave velocity as shown in Equation Numerical Solution y(m) = 10 cos (4x + 200t) and y = A sin(2πft - (2πx/λ)) (16.4) (a) So the amplitude of the wave A = 10 meters. (b) The wavelength of the wave, 2π/λ = 4; sol = p/2 meters (c) The frequency of the wave, 2πf = 200; so f = 100/π Hz. (d) The wave velocity v = λf = (π/2)(100/π) = 50 m/s. (g) The peaks of the wave have moved to the left in π/800 seconds, so the wave is moving in the negative x - direction. 136

6 INVERSE SQUARE LAW 2. A sound wave of pressure amplitude 3 x 10-2 N/m 2 is eminating from an 80 watt hi-fi speaker. (a) What is the amplitude of the sound wave at a distance of 3 meters from the speaker? (b) What is the intensity of the sound 3 meters from the speaker? (c) What is the amplitude of the sound wave at a distance of 10 m from the speaker? (d) What is the intensity of the sound at a distance of 10 m from the speaker? What Data Are Given? The pressure amplitude is given as 3 x 10-2 N/m 2. The power output of the hi-fi speaker is given as 80 watts. What Data Are Implied? It is implied that the sound wave eminates in a spherically uniform way and that all of the power output of the speaker is carried away by the sound wave. What Physics Principles Are Involved? The definition of amplitude and the inverse square law are needed for this problem. What Equations Are to be Used? Intensity = E / (4πr 2 t) (16.11) Algebraic Solutions Amplitude is independent of distance from the source for ideal, undamped systems. Intensity = (E/t)/(4πr 2 ) = Power/(4πr 2 ) Numerical Solutions (a), (c) Amplitude = 3 x 10-2 N/m 2 at all distances. (b) Intensity 3 = 80 W/(4π(3 m) 2 ) = 80 W/ 36πm 2 = W/m 2 (d) Intensity 10 = 80 W/(4π(10 m) 2 ) = 80 W/ 400π = W/m 2 Thinking About the Answer The answer to (d) should be [(3) 2 /(10) 2 ] times the answer to (b); dividing (d) by (b) we obtain which is equal to 3 2 /

7 PRACTICE TEST 1. The motion of a wave moving along a string is described by the equation y(m) = 10 cos (4.0 x - 200t) where x is in meters and t is in seconds. For this wave find the following: a. Amplitude = b. Wavelength = c. Frequency = d. Velocity = e. In which direction is the wave moving? 2. If the positive pulse shown below is sent down a taut string, describe the pulse as it reflects from a rigid post. 3. A large bell is located in a tall church tower is rung for several minutes. Local physics students are measuring sound levels in the open field nearby. Their measurements show that at 100 meters, the intensity of the ringing bell sound is 10 6 watts/m 2. What would be the intensity of sound received by a person standing only 3 meters from the bell tower? ANSWERS 1. (a) 10 m; (b) 1.6 m; (c) 32/sec; (d) f = 51 m/sec; (e) left to right o phase change; x 10-3 watts/m 2 138