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1 E WAVES E-RT0: WAVES WAVELENGTH The drawings represent snapshots taken of waves traveling to the right along strings. The grids shown in the background are identical. The waves all have speed, but their amplitudes vary. A B C D Rank the wavelength of the waves. Answer: C = D > A > B The wavelength is the horizontal distance between point of the wave. C and D are 6 horizontal units while A is 4 horizontal units and B is 2 horizontal units. 3
2 E-RT02: WAVES FREQUENCY The drawings represent snapshots taken of waves traveling to the right along strings. The grids shown in the background are identical. The waves all have speed, but their amplitudes and wavelengths vary. A B C D Rank the frequency of the waves. Answer: B > A > C = D The number of waves per second (the frequency) times the length of the waves (the wavelength) equals the speed of the waves along the string. of these waves have wave speed, so the frequency must be inversely proportional to the wavelength. The frequency is independent of amplitude. The wavelength can be measured it is the horizontal distance taken by one complete cycle of the wave, or the horizontal distance between point on two successive cycles of the wave, so from the drawings the ranking sequence for the wavelengths is C = D > A > B. Thus the frequency ranking will be the inverse or B > A > C = D. 4
3 E-RT03: WAVES WITH SAME FREQUENCY WAVE SPEED The drawings represent snapshots taken of waves traveling to the right along strings. The grids shown in the background are identical. The waves all have frequency, but their amplitudes and wavelengths vary. A B C D Rank the speed of the waves on the string. Answer: C = D > A > B The number of waves per second (the frequency) times the length of the waves (the wavelength) equals the speed of the waves along the string. of these waves have frequency, so the wave speed is proportional to the wavelength. The wave speed is independent of amplitude. The wavelength can be measured it is the horizontal distance taken by one complete cycle of the wave. 5
4 E-RT04: WAVE PULSES LEADING EDGE TIME TO TRAVEL The drawings represent snapshots taken of waves traveling along a rope. The grids shown in the background are identical. These pulses, which vary in amplitude, are all sent down identical ropes under equal tension. The ropes are all length, and there is no distortion of the pulses as they travel down the ropes. Rank the time it takes the leading edge of the pulse to travel 3 m. Answer:. The time to move 3 m will be d by the wave speed, which since they are all traveling in medium (equal length, identical ropes under tension), is for all cases. 6
5 E-RT05: PAIRS OF TRANSVERSE WAVES SUPERPOSITION Rectangular transverse wave pulses are traveling toward each other along a string. The grids shown in the background are identical, and the pulses vary in height and length. The pulses will meet and interact soon after they are in the positions shown. Rank the maximum amplitude of the string at the instant that the positions of the centers of the two pulses coincide. Answer: A > B > C > D. The peak height is the sum of the two amplitudes taking account of their signs. For cases A and B, the pulses are on side of the string, and they will interfere constructively. For cases C and D, the pulses are on opposite sides of the string, and they will interfere destructively. For C there will be a non- amplitude, but for D the amplitude will be. 7
6 E-RT06: WAVE SOURCES SEPARATED BY ONE WAVELENGTH INTENSITY Two identical point sources are generating waves with frequency and amplitude. The two sources are in phase with each other, so the two sources generate wave crests at instant. The wavelength of the waves is equal to the distance between the two sources. λ D A B C Point sources Rank the maximum amplitude of the wave at the labeled points. Answer: A = C = D > B, or A > D > C > B, depending on the level of detail included in the explanation. The intensity at each point depends on the relative phases of the two waves when they reach the point. The sources are in phase, so if they travel distance from each source to some point, they will still be in phase when they reach that point (as at point D), and the maximum amplitude will be twice the amplitude of a single wave at that point. The distance from the left source to point A is one-half wavelength, and the distance from the right source to point A is.5 wavelengths, so the difference in path length from the sources to point A is one wavelength. The crests and troughs coincide, and the maximum amplitude will be twice the amplitude of a single wave at that point. At point C the difference in distance for the waves from the two sources is again one wavelength (2.5 wavelengths from the left source an.5 wavelengths from the right source). As at points A and D, the crests and troughs coincide, and the maximum amplitude will be twice the amplitude of a single wave at that point. For point B the path distance from the left source is one-quarter wavelength and the distance from the right source is three-quarters of a wavelength, so the path difference is one-half wavelength. At point B, then, a crest from one source arrives at time as a trough from the other source, and there is destructive interference. The wave amplitude at point B is close to. If we only take into account whether the waves constructively or destructively interfere, then, our ranking for the maximum wave amplitudes is A = C = D > B. However, as waves get farther from their sources the maximum amplitude of the waves diminishes. For this reason, the maximum amplitude at point A will in reality be greater than the maximum amplitude at point D, which will in turn be greater than the maximum amplitude at point C. Taking the distance into account, then, the ranking will be A > D > C > B. 8
7 E-QRT07: WAVE SOURCES SEPARATED BY ONE WAVELENGTH I INTERFERENCE Two identical point sources are generating waves with frequency and amplitude. The two sources are in phase with each other, so the two sources generate wave crests at instant. The wavelength of the waves is equal to the distance between the two sources. λ A B C D E Point sources List all the labeled points where the waves from the two sources constructively interfere. If there are no such points, indicate that by stating none of them. Answer: A, B, D, and E Constructive interference occurs at points where the crests from the two sources arrive at time, as do the troughs from the two sources. At these points the maximum amplitude of the wave is the sum of the amplitudes of the waves from the two sources, and is bigger than the individual waves at that point. If the path length to a point from the farthest source is an integer multiple of a wavelength larger than the path length to that point from the closest source, then the waves constructively interfere. This occurs at point A (where the distance from the left source is one wavelength, the distance from the right source is two wavelengths, and the difference in distances is one wavelength) and at point B (where the distance from the left source is one-half wavelength, the distance from the right source is one and one-half wavelengths, and the difference in distances is one wavelength) and at point D (where the distance from the right source is 0.75 wavelengths, the distance from the left source is.75 wavelengths, and the difference in distances is one wavelength) and at point E (where the distance from the right source is.5 wavelengths, the distance from the left source is 2.5 wavelengths, and the difference in distances is one wavelength). List all the labeled points where the waves from the two sources destructively interfere. If there are no such points, indicate that by stating none of them. Answer: C Destructive interference occurs at points where the crest from one source arrives at time as the trough from the other source. At these points the maximum amplitude of the wave is the difference the amplitudes of the waves from the two sources, and is smaller than the individual waves at that point. (The amplitude is close to at all times if the distances from the sources is about.) If the path length to a point from the farthest source is one-half wavelength larger than the path length to that point from the closest source, (or.5 wavelengths, or 2.5 wavelengths, etc) then the waves destructively interfere. This occurs at point C (where the distance from the left source is 0.25 wavelengths, the distance from the right source is 0.75 wavelengths, and the difference in distances is one-half wavelength). 9
8 E-QRT08: WAVE SOURCES SEPARATED BY ONE AND ONE-HALF WAVELENGTHS INTERFERENCE Two identical point sources are generating waves with frequency and amplitude. The two sources are in phase with each other, so the two sources generate wave crests at instant. The distance between the two sources is equal to one and one-half times the wavelength, or.5. C F A B D E Point sources List all the labeled points where the waves from the two sources constructively interfere. If there are no such points, indicate that by stating none of them. Answer: None of them. Constructive interference occurs at points where the crests from the two sources arrive at time, as do the troughs from the two sources. At these points the maximum amplitude of the wave is the sum of the amplitudes of the waves from the two sources, and is bigger than the individual waves at that point. If the path length to a point from the farthest source is an integer multiple of a wavelength larger than the path length to that point from the closest source, then the waves constructively interfere. For points A and B, the left source is one and one-half wavelengths closer than the right source, not an integer multiple. For points D and E, the nearer source is onehalf wavelengths away and the farther source is a full wavelength away, leaving a difference of one-half wavelength, also not an integer multiple. For point C, the nearer source is four half-wavelengths away, and since the distance between sources is three half-wavelengths, the distance from C to the more distant source is five half-wavelengths (point C and the sources form a right triangle), and the difference in path lengths is onehalf wavelength. Point F also has a path difference of one-half wavelength for reason as point C. There are no points where the difference in path length is an integer multiple of one wavelength, and no points where constructive interference occurs. List all the labeled points where the waves from the two sources destructively interfere. If there are no such points, indicate that by stating none of them. Answer: of them. Destructive interference occurs at points where the crest from one source arrives at time as the trough from the other source. At these points the maximum amplitude of the wave is the difference the amplitudes of the waves from the two sources, and is smaller than the individual waves at that point. (The amplitude is close to at all times if the distances from the sources is about.) If the path length to a point from the farthest source is one-half wavelength larger than the path length to that point from the closest source, (or.5 wavelengths, or 2.5 wavelengths, etc) then the waves destructively interfere. This occurs at points A and B (where the difference in path lengths is one and one-half wavelengths), at points D and E (where the difference in path lengths is one-half wavelengths), and at points C and F (where the difference in path lengths is also one-half wavelengths). λ 0
9 E-QRT09: WAVE SOURCES SEPARATED BY TWO WAVELENGTHS INTERFERENCE Two identical point sources are generating waves with frequency and amplitude. The two sources are in phase with each other, so the two sources generate wave crests at instant. The distance between the two sources is equal to two wavelengths, or 2. C F A B D E Point sources λ List all the labeled points where the waves from the two sources constructively interfere. If there are no such points, indicate that by stating none of them. Answer: of them Constructive interference occurs at points where the crests from the two sources arrive at time, as do the troughs from the two sources. At these points the maximum amplitude of the wave is the sum of the amplitudes of the waves from the two sources, and is bigger than the individual waves at that point. If the path length to a point from the farthest source is an integer multiple of a wavelength larger than the path length to that point from the closest source, then the waves constructively interfere. For points A and B, the left source is two wavelengths closer than the right source, an integer multiple. For point D, both sources are one wavelength away and there is no difference in path distance from the sources, so the crests arrive at time. For point E, the right source is one-half wavelengths away and the left source is one and one-half wavelengths away, leaving a path difference of one wavelength. For point C, the nearer source is three half-wavelengths away, and since the distance between sources is four half-wavelengths, the distance from C to the more distant source is five halfwavelengths (point C and the sources form a right triangle), and the difference in path lengths is one wavelength. Point F also has a path difference of one wavelength for reason as point C. Thus all points have a difference in path length of an integer multiple of one wavelength, and there is constructive interference at all points. List all the labeled points where the waves from the two sources destructively interfere. If there are no such points, indicate that by stating none of them. Answer: None of them. Destructive interference occurs at points where the crest from one source arrives at time as the trough from the other source. At these points the maximum amplitude of the wave is the difference the amplitudes of the waves from the two sources, and is smaller than the individual waves at that point. (The amplitude is close to at all times if the distances from the sources is about.) If the path length to a point from the farthest source is one-half wavelength larger than the path length to that point from the closest source, (or.5 wavelengths, or 2.5 wavelengths, etc) then the waves destructively interfere. As described above, all points listed here have integer-wavelength differences in path lengths, and so there are no points where there is destructive interference.
10 E-QRT0: WAVE SOURCES SEPARATED BY ONE WAVELENGTH II INTERFERENCE Two identical point sources are generating waves with frequency and amplitude. The two sources are out of phase with each other, so at the instant that one source is creating a crest, the other source is creating a trough. The wavelength of the waves is equal to the distance between the two sources. λ D A B C Point sources List all the labeled points where the waves from the two sources constructively interfere. If there are no such points, indicate that by stating none of them. Answer: B Constructive interference occurs at points where the crests from the two sources arrive at time, as do the troughs from the two sources. At these points the maximum amplitude of the wave is the sum of the amplitudes of the waves from the two sources, and is bigger than the individual waves at that point. Since in this case the waves are out of phase, and one wave is creating a crest when the other is creating a trough, if the path lengths from the sources to a point are (as they are at point D) then a crest will arrive from one source at instant that a trough arrives from the other source, and the waves will cancel. Only if the path length to a point from the farthest source is one-half wavelength larger than the path length to that point from the closest source, (or.5 wavelengths, or 2.5 wavelengths, etc) will the waves constructively interfere, because then the path distance from the further source will mean that crest meets crest and trough meets trough at that point. For point B the path distance from the left source is one-quarter wavelength and the distance from the right source is three-quarters of a wavelength, so the path difference is one-half wavelength. For this reason a crest from the left source arrives at B at time as a crest from the right source, and there is constructive interference. List all the labeled points where the waves from the two sources destructively interfere. If there are no such points, indicate that by stating none of them. Answer: A, C and D. Destructive interference occurs at points where the crests from the two sources arrive at time, as do the troughs from the two sources. At these points the maximum amplitude of the wave is the difference of the amplitudes of the waves from the two sources, and is smller than the individual waves at that point. Since in this case the waves are out of phase, and one wave is creating a crest when the other is creating a trough, if the path lengths from the sources to a point are (as they are at point D) then a crest will arrive from one source at instant that a trough arrives from the other source, and the waves will cancel. The waves will also cancel if the difference in distance is an integer multiple of one wavelength. The distance from the left source to point A is one-half wavelength, and the distance from the right source to point A is.5 wavelengths, so the difference in path length from the sources to point A is one wavelength, and destructive interference occurs. For point C, right source is.5 wavelengths away and the left source is 2.5 wavelengths away, giving a path difference of one wavelengths and again destructive interference occurs. 2
11 E-RT: STANDING WAVES FREQUENCY A string is stretched so that it is under tension and is tied at both ends so that the endpoints don t move. A mechanical oscillator then vibrates the string so that a standing wave is created. The dark line in each diagram represents a snapshot of a string at an instant in time when the amplitude of the standing wave is a maximum. The lighter lines represent the string at other times during a complete cycle. of the strings are identical except for their lengths, and all strings have tension. The number of nodes and antinodes in each standing wave is different. The lengths of the strings (L) and the amplitudes at the antinodes (A) are given in each figure. A B A = 2 cm L = 25 cm A = 2 cm L = 28 cm C A = 8 cm L = 27 cm D A = 6 cm L = 28 cm Rank the frequencies of the waves. Answer: B > A > D > C. Since all of these waves are on the strings under tension, and the mass per unit length of the strings are identical, they all have wave speed. The wave speed is equal to the product of the frequency and wavelength, so the shorter the wavelength the higher the frequency. The wave frequency is independent of the amplitude of the standing wave. 3
12 E-RT2: STANDING WAVES WAVELENGTH A string is stretched so that it is under tension and is tied at both ends so that the endpoints don t move. A mechanical oscillator then vibrates the string so that a standing wave is created. The dark line in each diagram represents a snapshot of a string at an instant in time when the amplitude of the standing wave is a maximum. The lighter lines represent the string at other times during a complete cycle. of the strings are identical except for their lengths, and all strings have tension. The number of nodes and antinodes in each standing wave is different. The lengths of the strings (L) and the amplitudes at the antinodes (A) are given in each figure. A B A = 2 cm L = 25 cm A = 2 cm L = 28 cm C A = 24 cm L = 20 cm D A = 36 cm L = 30 cm Rank the wavelengths of the waves. Answer: D > A = C > B. The wavelength is the distance of one complete cycle, which in the figures is the length of a complete cycle of the black line. In case A this is two-fifths of the length of the string, or 0 cm; in case B two-sevenths of the length of the string, or 8 cm; in case C two-fourths of the length of the string, or 0 cm; and in case D two-thirds of the length of the string, or 20 cm. The wavelength is independent of the amplitude of the standing wave. 4
13 E-RT3: STANDING WAVES SYSTEMS WAVE SPEED A string is stretched so that it is under tension and is tied at both ends so that the endpoints don t move. A mechanical oscillator then vibrates the string so that a standing wave is created. The dark line in each diagram represents a snapshot of a string at an instant in time when the amplitude of the standing wave is a maximum. The lighter lines represent the string at other times during a complete cycle. of the strings have length but may not have mass. The number of nodes and antinodes in the standing wave is in Cases A and D. The tensions in the strings (T) and the standing wave frequencies (f) are given in each figure. A T = 2 N f = 500 Hz B T = 5 N f = 300 Hz C T = 6 N f = 300 Hz D T = 2 N f = 400 Hz Rank the speeds of the waves in the strings. Answer: C > A = B > D. The wave speed is equal to the product of the frequency and wavelength. The frequencies are given and the wavelength is the distance of one complete cycle, which in the figures is the length of a complete cycle of the black line. In cases A and D this is two-tenths of the length of the string; in case B two-sixths of the length of the string; and in case C this is two-fifths of the length of the string. The amplitudes of the waves is not given, but the speed of the waves is independent of the amplitude of the standing wave. If we label the length of the string L, then for case A, the wave speed will be (500Hz)(0.2L); for case B (300Hz)(0.33L); for case C (300Hz)(0.4L); and for case D (400Hz)(0.2L). 5
14 E-QRT4: WAVE PULSE ON HIZONTAL SPRING WITH FIXED END REFLECTION SHAPE A long spring is firmly connected to a stationary metal rod at one end. A student holding the other end moves her hand rapidly up and down to create a pulse with the shape shown in the figure. The pulse moves along the taut spring toward the rod. In the space below, draw what the pulse on the spring looks like after it has completely reflected from the wall and is moving to the left. Rod Rod Answer: When the pulse reaches the fixed end of the spring, the rod exerts downward forces on the spring. These forces serve to invert the pulse, which reflects to the left with the deflection beneath the spring rather than above it. The steeper leading edge of the pulse remains the leading edge as the pulse moves to the left. 6
15 E-QRT5: WAVE PULSE ON HIZONTAL SPRING WITH FREE END REFLECTION SHAPE A long spring is connected to a loop that passes around a stationary metal rod at one end. The loop is free to move vertically without friction along the rod. A student holding the other end moves her hand rapidly up and down to create a pulse with the shape shown in the figure. The pulse moves along the taut spring toward the rod. Rod In the space below, draw what the pulse on the spring looks like after it has completely reflected from the wall and is moving to the left. Rod Rod Answer: When the pulse reaches the free end of the spring, the rod exerts no vertical forces on the spring. The pulse reverses direction without changing shape, and stays on side of the spring. The steeper leading edge of the pulse remains the leading edge as the pulse moves to the left. 7
16 E-CT6: WAVE PULSES TRAVELING TOWARD EACH OTHER SPEED Two pulses travel toward each other along a long stretched spring as shown. Pulse A is wider than pulse B, but not as high. A B Is the speed of pulse A (i) larger than, (ii) smaller than, or (iii) equal to the speed of pulse B? If there is not enough information to tell, state that explicitly. Answer: The speed of pulse A is as the speed of pulse B. The pulse speed depends only on the tension in the spring and the mass per unit length of the spring, which is the same for both pulses. 8
17 E-WWT7: TWO WAVE PULSES INTERACTING IMPACT A student states: If two wave pulses traveling in opposite directions along string meet, they reflect from one another and go back the way they came from. What, if anything, is wrong with this statement? If something is wrong, identify it and explain how to correct it. If this statement is correct, explain why. Answer: The student is incorrect. The string obeys the principal of superposition at all times: The shape of the string is the sum of the individual pulse shapes as they move along the string. When the pulses are in location on the string, the shape of the string is d by the sum of the pulse shapes as each shape continues along the string at speed and in direction. After the trailing edge of each pulse has passed the other pulse, the shape of the pulses is as the shape before there was any interaction, and the two pulses appear to have passed through one another. 9
18 E-QRT8: WAVE PULSES TRAVELING TOWARD EACH OTHER SHAPE AND DIRECTION Two pulses travel toward each other along a long, stretched spring. The speed of each pulse is 00 cm/s. Each square on the grid is cm long, as can be seen on the ruler markings below the grid. The left pulse is about 6 cm wide, and the right pulse is about 2 cm wide and twice as tall as the left pulse. The figure below represents a snapshot of the spring taken at time t = 0. (a) In the space below, sketch the shape of the spring at time t = 0. seconds cm Answer: After 0. seconds, each pulse will have moved 0 cm (0.s)(00cm/s). Their wave shapes do not change. (b) In the space below, sketch the shape of the spring at time t = 0.2 seconds. Answer: After 0.2 seconds, each pulse will have moved 20 cm (0.2s)(00cm/s). The centers of the two pulses are now both at 24 cm on the scale. The amplitude at each location is the sum of the amplitudes of the two pulses. (c) In the space below, sketch the shape of the spring at time t = 0.3 seconds. Answer: After 0.3 seconds, each pulse will have moved 30 cm (0.3s)(00cm/s). Their wave shapes do not change. 20
19 E-RT9: WAVE SOURCE INTENSITY AT VARIOUS LOCATIONS Two point sources can generate waves with frequency and amplitude. The sources can be turned on and off individually. D A B C Point sources (a) Rank the intensity of the wave at the labeled points if the left point source is turned on and the right point source is turned off. Explain. Answer: B > A > D > C. The intensity at each point depends on the distance from the point source, diminishing as the distance increases. (b) Rank the intensity of the wave at the labeled points if the right point source is turned on and the left point source is turned off. Explain. Answer: B > D > A = C. The intensity at each point depends on the distance from the point source, diminishing as the distance increases. 2
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