Physics 123 Unit #3 Review I. Definitions and Facts longitudinal wave transverse wave traveling wave standing wave wave front wavelength wave number frequency angular frequency period crest trough node antinode transmission reflection diffraction v = 330 m/s Doppler shift pitch constructive interference destructive interference shock wave beats harmonics overtones amplitude II. Mathematics translation: f(x+x 0 ) is f(x) moved left by x 0. partial derivatives differential equations: be able to show a function is a solution boundary conditions and differential equations linear differential equations: if f 1 and f 2 are solutions, f 1 + f 2 is a solution Overlapping wavefronts can be modeled with Moiré patterns. Fourier transform: gives frequency components of a wave III. Basic Concepts application of Newton s laws to a segment of string gives rise to the wave equation - understand the derivation of the wave equation many (quantities including displacement, pressure, electric and magnetic fields) obey the wave equation any function f(u) with u=kx± t satisfies the wave equation the superposition principle any function can be composed of a sum of sine waves when a wave goes from one medium to another both reflection and transmission occur when a wave goes from a less dense medium to a more dense medium, the reflected wave is inverted - understand qualitatively why standing waves are the superposition of identical waves traveling left and right linear differential equations: if f 1 and f 2 are solutions, f 1 + f 2 is a solution only standing waves which meet boundary conditions can exist Huygens Principle - Each point along a wavefront can be considered as a point source producing a spherical wave (or circular wave in the two-dimensional case). When two overlapping waves have crests on top of crests, constructive interference occurs. When crests are on top of troughs, destructive interference occurs. Sound is a longitudinal wave. Pressure and displacement both obey wave equations. The boundary conditions on standing waves give wavelength. If the velocity is known, frequency can be determined. In organ pipes, the fipple is open pressure node, displacement antinode. A closed end has opposite conditions. When a source and listener are approaching, the frequency is raised. When receding, the frequency is lowered. Be able to use the general Doppler shift equation. - The velocity of sound with respect to the medium is independent of the motion of the source or listener. A shock wave is created when an object exceeds the speed of sound. Know how to do the geometry. When two sources of sound of the same amplitude and nearly the same pitch are heard together, the ear hears the average frequency with pulses at the difference of the frequencies. The overtones of a drum head are not harmonics. The timbre of a sound is produced primarily by transients and overtones.
IV. Equations to Memorize wave equation 2 y x 2 1 v 2 2 y velocity of a wave on a string v T µ traveling wave: y(x)asin (kx ± t ) standing wave: y(x)asin (kx )sin( t ) (for appropriate boundary conditions) Conditions for constructive and destructive interference d 2 d 1 n constructive, d 2 d 1 (n½ ) destructive Sound intensity level: 10 log I I 0.
Physics 123 Unit 3 Reading and Homework Assignments 16.9 Carefully follow the derivation of the wave equation, Eq. 16.26. Show that Eq. 16.2 and Eq. 16.11 are solutions of this equation. 16.7 This section provides a few fundamental definitions. Know it thoroughly. Eq. 16.11 is central, with 16.9, 16.10 to define k and. Also Eq. 16.14 is crucial. Work through Examples 16.3 and 16.4. 16.2 Understand longitudinal and transverse. Note that some waves are combinations of these two types. 16.3 In some ways, the simplest sort of a wave is a single pulse. Note how this pulse moves. Do real pulses on strings behave this way? Look over Ex. 16.1. 16.5 This is really a consequence of the wave equation, so you ve already seen it. Eq. 16.4 is very useful for homework problems. Be sure you understand Ex. 16.2. It s easy. 16.4 The principle of superposition is very important to many areas in physics. We ll treat it more mathematically in class, but be sure you understand it. When two waves add to make a larger wave, we call it constructive interference. When they tend to cancel, we call it destructive interference. The term constructive interference seems like an oxymoron, but we use it anyway. 16.6 Note that there are two different ways waves can reflect. We ll need to use this idea later with light also. Note that if a wave passes a boundary between two different ropes, some of the wave reflects and some transmits. Qualitatively understand all these results. 16.8 We won t hit this section so heavily, but be aware that waves carry energy and momentum. Work through Ex. 16.5. 18.1 These sections from chapter 18 are included to introduce standing waves. We create standing waves mathematically from the superposition of traveling waves. This section just gives the math of adding sine waves together. Do not go over Interference of Sound Waves for now! 18.2 The steps leading to Eq. 18.3 are often used in physics. You may need to remember a few trig identities here. The main ones are: sin(a±b)sin a cosb±cosa sinb cos(a±b)cosa cosbsin a sinb Work through Ex. 18.2. 18.3 Normal modes are important to many applications in physics. Recognize that a string may vibrate in any linear combination of normal modes, not in just one normal mode or another. You don t really need to use Eq. 18.7 and 18.8 as long as you understand Fig. 18.6. Work Ex. 18.3. 17.1 We won t use Eq. 17.1, but we will derive it in class. This section is mostly for your information. We ll often
use v = 343 m / sec for the velocity of sound. 17.2 Note that both displacement and pressure satisfy the wave equation. In a plane wave the displacement and phase are 90 out of phase, but this isn t always the case. We ll usually speak of waves in terms of pressure. Although sound is a longitudinal wave, it s usually easier to visualize a sound wave if we draw it as a transverse wave. 17.3 Intensity is proportional to the square of the amplitude. More precisely, if we measure the energy passing through a frame perpendicular to the wave, intensity is this energy divided by (time area of frame). Eq. 17.5 and 17.6 and not crucial. The definition of db is important. Note that I 0 is somewhat arbitrary. Two good rules of thumb: If intensity doubles, sound level increases by about 3dB. If intensity increases by a factor of 10, sound level increases by 10dB. 17.4 This is mostly qualitative. Eq. 17.9 is useful, but we ll mostly stick to plane waves, Eq. 17.10, which look like waves on a string mathematically. 35.6 Huygens Principle is primarily used in optics, but it is a useful model for understanding how sound waves propagate around corners. The model is quite limited in its application, however. Don t read the part about reflection and refraction yet. 18.1 We looked at part of this earlier. This time, look carefully at the section on interference of sound waves. Work example 18.1 carefully. 18.8 The idea of a Fourier transform is that we can make any regularly repeating wave shape out of a combination of sine waves of a fundamental frequency and multiples of the fundamental. Each sine wave has a different frequency or pitch. Our ears are sensitive to the different pitches in a wave they re biological Fourier analyzers. Don t worry about the details of how to calculate Fourier transforms, we ll just use them as tools for now. 17.5 Understand this section carefully. The qualitative aspects are essential and usually fairly easy. Don t worry about reproducing the algebra leading to the important results, though you should be able to follow it. Eq. 17.13 is an approximation for objects moving considerably less than the speed of speed of sound. Use logic to decide which sign is right. Eq. 17.17 is more general. Note the sign convention is just under the box. Carefully study Fig. 17.11 to understand shock waves. Try working through Example 17.6. 18.5 Pipes work the same way as strings, except that the boundary conditions are different. The pressure at an open hole is fixed at atmospheric pressure, so it is a pressure node. Note the book graphs displacement rather than pressure.
Homework 3-1 Sections 16.7, 16.9 Questions: Question 16.6 Essential Problems: Problems 16.35, 16.46 Application Problems: Problems 16.29, 16.31 Computational Problems: None. Context-rich Problems: Problems 16.48 Homework 3-2 Sections 16.1-16.3,16.5 Questions: Question 16.10 Essential Problems: Problem 16.12, 16.14 Application Problems: Problems 16.1, 16.17 Computational Problems: A. Plot different waves of the form y = A sin (kx t + ) and see what each of the parameters do to the wave. How does the wave change in time? How does one point on the wave as a function of time? Context-rich Problems: Problems 16.3, 16.12 Homework 3-3 Sections 16.4, 16.6, 16.8 Questions: Questions 16.5 Essential Problems: Problems 16.7, 16.38 Application Problems: Problems 16.39 A. Two waves are traveling in the same direction along a stretched string. Each has an amplitude A and they are out of phase by an angle. Find the amplitude of the resultant wave. Computational Problems: None Context-rich Problems: Problem 16.44 Homework 3-4 Section 18.1-18.3 Questions: Questions 18.4 Essential Problems: Problems 18.9, 18.15, 18.18 Application Problems: Problems 18.17 A. Two harmonic waves are described by y 1 (6.0m) sin 15 m x 0.0050 s t y 2 (6.0m) sin 15 m x 0.0050 s t (a) What is the amplitude of the resultant wave when = /6 rad? (b) For what values of will the amplitude of the resultant wave be maxima? B. A stretched string is 160 cm long and has a linear density of 0.015 g /cm. What tension in the string will result in a second harmonic of 460 Hz? Computational Problems:
C. Add two identical waves traveling in opposite directions. Plot the waveform at several times to demonstrate that the result is a standing wave. Context-rich Problems: Problem 18.23 Homework 3-5 Sections 17.1-17.4 Questions: Questions 17.14 Essential Problems: Problems 17.45, 17.14 Application Problems: Problems 17.13, 17.22, 17.25 A. The speed of sound in air is given by the expression v P / where is the adiabatic constant. If the density of air is 1.29 kg / m 3, calculate v. Computational Problems: None. Context-rich Problems: Question 17-13, Problems 17.5, 17.15 Homework 3-6 Sections 35.6, 18.1, 18.8 Questions: Question 18.1 Essential Problems: Problem 18.5 Application Problems: Problems 18.7 (assume the listener is at the same height as the lower speaker). A. In figure 18.2 let r 1 =1.20 m and r 2 =0.80 m. (a) Calculate the three lowest speaker frequencies that will result in intensity maxima at the receiver? (b) What is the highest frequency with the audible range (20 20000 Hz) that will result in a minimum at the speaker? Use v = 340 m/s. Computational Problems: B. Speakers are located at x = +5m and 5 m. If the sound intensity level is 70 db at the point (0, 10m), find the intensity along the line y=10m. For convenience, choose the wavelength to be 1.00 m. Context-rich Problems: Problems 17.24, 18.5 Homework 3-7 Sections 17.5 Questions: Questions 17.7 Essential Problems: Problems 17.41 Application Problems: Problems 17.35, 17.39 Computational Problems: A. A speaker emits a sound at 440 Hz. Plot the frequency a stationary receiver hears if the source is moving at velocities up v sound to + v sound. repeat for a stationary source with the receiver moving. Context-rich Problems: Problem 17.38. Homework 3-8 Section 18.5 Questions: A. Give examples of sounds that are harmonic and sounds that are not harmonic. Essential Problems: Problem 18.31 Application Problems: Problems 18.33, 18.41 Computational Problems: None. Context-rich Problems: Problem 18.42
Physics 123 Section 2 Sample Exam #3 You are allowed to use your textbook, calculator, unit handouts, and anything you have personally written. There are no time limits. Be sure to show your work, as we can only give credit for what is on your paper. Be sure your copy of the test has fourteen problems. For the multiple choice problems, mark the one best answer. sin( ± )sin cos ±cos cos( ± )cos cos sin sin sin sin( ), if «1 cos( )1, if «1 2 y 2 y x 2 1 v 2 Possibly useful information: Terminology 1. (5 points) Define antinode. f ' = f v ± vo ( vv ) S 2. (5 points) How are angular frequency and period related to each other? 3. (5 points) Define amplitude. 4. (5 points) Define destructive interference.
Conceptual Applications 5. (5 points) Two waves have the same frequency, amplitude, and direction of travel; however, they have a phase difference of 2. When they are added, the result is A. A wave with twice the frequency of the original wave. B. A wave with twice the amplitude of the original wave. C. A wave with twice the wavelength of the original wave. D. Nothing. E. None of the above 6. (5 points) Oscillatory motion always results from systems where there is 1) there is a force which tries to restore the system to its equilibrium state and (2) an inertia which tends to carry the system past equilibrium. (Depending on the system, these quantities may not represent an actual force and inertia, hence the quotation marks.) On a stretched string: A. The force is gravity and the inertia is the interatomic force in the string. B. The force is tension and the inertia is the string s mass. C. The force is tension and the inertia is gravity. D. The force is gravity and the inertia is tension. E. None of the above 7. (5 points) Which statement about stringed musical instruments is NOT true? A. Lower pitched notes often have heavier strings than higher pitched notes. B. Lower pitched notes often have longer strings than higher pitched notes. C. Pitch can be changed by adjusting the tension of the string. D. Waves on strings are always sine functions of a single frequency. E. All of the above are true.
Equations and Tools 8. (5 points) A violin produces a wave which is a complex combination of different harmonics. What does a Fourier transform of the wave do? For each of the following equations: A) Tell what each symbol means B) Write a short problem which can be solved with this equation The problem must be a word problem, not just a=5, b=4, what is c? 9. (5 points) y(x)asin (kx )sin( t ) 10. (5 points) 10 log I I 0.
Problems We can usually evaluate how well you understand the physics of a problem by the way you attempt to solve the problem. Occasionally, however, you may be unable to work a problem or a part of a problem even if you do understand the physics. If you cannot work a problem, please do the following so that you may receive partial credit: 1. Discuss the basic physics concepts involved. Include enough detail that the grader will know if you understand the physics. 2. Write down the equation(s) you need to solve the problem. Explain the meaning of each symbol in the equations. 3. Explain as much of the problem as you understand. 4. Explain why you feel you have difficulties with the problem. 11. (25 points) A wave is given by the function y( x, t) = f ( x) + Asin( kx ω t), where f(x) is a function of x ONLY. If y(x,t) obeys the usual wave equation, what can you conclude about f(x)? (Use the wave equation. You may use the relationship v= / k. Be sure your proposed solution has NO t dependence.)
12. (25 points) A string of linear mass density µ is stretched between two poles. A wave is then produced on the string. The amplitude of the wave is large enough that we cannot assume the angle is small. (a) Using a Taylor series expansion, f(x+ x)f(x)+f (x) x, show that: sin( )sin( )cos( ) cos( )cos( )sin( ) (b) Draw a free body diagram for the segment of the string shown above. Be sure to include all forces. (c) Assuming that the string can not move horizontally, write down the equation of motion for the horizontal direction. Show that this reduces to dt d Ttan (d) Show that the mass of the string segment can be given by: dm µ cos dx (e) Write the equation of motion for the vertical direction. Show that this reduces to: T x µ2 y (f) Show that if is small, this reduces to the ususal wave equation, 2 y x 2 µ T 2 y tan dy dx Hint: 13. (25 points) Two trains are approaching each other on the same track. They are each traveling at the same speed. Each train sounds its warning whistle, a 400 Hz tone. The velocity of sound in the air is 330 m/s. (a) If each engineer hears the whistle of the other train at 480 Hz, what is the speed of each train? (10 points)
(b) If the trains are traveling at different speeds, which engineer will hear the higher pitch? Justify your answer. (Hint: It may be easier if you think of extreme cases.) (4 points) (c) An observer standing on the ground is behind one train. What frequencies does he hear for each whistle? (6 points)
14. (25 points) (a) Moiré patterns can be used to described the interference of two point sources emitting identical waves in phase with each other. Below is a typical pattern. Show regions of constructive and destructive interference. 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1 (b) Describe what is different about the figure below compared to the figure of part (a). 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1
Answers Terminology 1. Points along a standing wave where the amplitude reaches its largest value. 2. Period is the number of seconds per cycle of oscillation, whereas the angular frequency is the number of radians per second of oscillation or 2 times the number of cycles per second. ( = 2 / T). 3. Amplitude is the maximum displacement for a mechanical wave (or by analogy, the maximum value of pressure, electric field, etc. for other waves). 4. When the crests of one wave are superimposed with the troughs of another wave, the result is no wave at all. (In general destructive interference is when the superposition of two waves is smaller than the amplitude of the larger wave.) Conceptual Applications 5. B 6. B 7. D Equations and Tools 8. A Fourier transform gives the amplitude of each of the harmonics (fundamental oscillations) from which the wave can be composed. A 3.00 liter volume of gas has a pressure of 1.2 10 5 Pa and a temperature of 150C. How many molecules are present? 9. y(x) the vertical displacement of a standing wave on a string. A the amplitude (maximum displacement) of the wave. k the wavenumber the angular frequency Where are the nodes of all possible waves on a 2.0 m long string? 10. the sound intensity level in db I the intensity of a sound I 0 the reference intensity (10 12 W/m 2.) 20 Harleys are cruising past your house. How much larger is the sound intensity level than if only one Harley were there? Problems 11. Therefore, f(x) must be a linear function. 2 y 2 y x 1 2 v 2 x t d 2 f dx 2k 2 y(x, t) 1 2 y(x, t) v 2 d 2 f dx 2k 2 y(x, t)k 2 y( x,t) d 2 f dx 2 0 f (x)axb 12. (a) (b) sin ( )sin ( ) dsin ( ) d cos( )cos( ) dcos( ) d sin( )cos( ) cos( )sin( )
( c) (d) (T T)cos( )Tcos 0 (T T)(cos sin )Tcos 0 Tcos Tsin Tcos Tcos 0 Tcos Tsin Ttan T dt d dmµ dµ dx cos (e) (TdT)sin( d )Tsin g dmdm a y (TdT)(sin cos d )Tsin dm a y ignoringtheweight Tsin Tcos d dtsin Tsin µ dx cos Tcos 2 d sin cos dtµdx 2 y Tcos 2 d sin cos Ttan d µdx 2 y Td (cos 2 sin 2 )µdx 2 y Td µdx 2 y T d dx µ 2 y 2 y using the result of part (c) (f) 13. (a) if is small, then tan y x so T x y µ2 becomes T 2 y x y 2 µ2 2 y x µ 2 y 2 T f f vv o vv s f vv t vv t (vv t )f (vv t )f v(f f )v t (f f) v t v f f f f v t 330m /s 480400 480400 30m/ s
(b) When the source is moving near the speed of sound and the observer is at rest, the pitch approaches infinity. As the observer moves near the speed of sound with source at rest, the pitch doubles. In general, then, the more slowly moving train hears the higher pitch. ( c) For the train going away from the observer: f v f vv t f 330 400Hz 33030 367Hz For the train approaching the observer: 14. (a) f v f vv t f 330 400Hz 33030 440Hz 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1 (b)the two sources are producing sound out of phase (as easily seen by comparing the inner circles). The regions of constructive and destructive interference are therefore reversed.
Physics 123 Unit Summary Unit #3 ID Number: Unit Score: (sum all boxes below): 1. Homework problems completed on time 10 = Homework problems completed on time 8 = Sum of previous lines 41 problems =, average score per problem Homework score: Average score per problem 2.5 = ( Maximum = 25 ) 2 Hours Item Hours Item Hours Total Hours (expected = 23, maximum = 40): Standard Work Score: Total Standard Hours 1.75 = ( Maximum = 70 ) 4. Reading Checks Reading score: 10 # correct = ( Maximum = 10 ) 9 4. Quizzes Quiz score: 10 # correct = ( Maximum = 10 ) 9 5. Walk-in Labs Lab score: = 7.5 #completed = ( Maximum = 15 )