Active Figure 18.4 The superposition of two identical waves y1 and y2 (blue and green) to yield a resultant wave (red). (a) When y1 and y2 are in phase, the result is constructive interference. (b) When y1 and y2 are  rad out of phase, the result is destructive interference. (c) When the phase angle has a value other than 0 or  rad, the resultant wave y falls somewhere between the extremes shown in (a) and (b). At the Active Figures link at http://www.pse6.com, you can change the phase relationship between the waves and observe the wave representing the superposition. Fig. 18.4, p.547

Active Figure 18.4 The superposition of two identical waves y1 and y2 (blue and green) to yield a resultant wave (red). (a) When y1 and y2 are in phase, the result is constructive interference. At the Active Figures link at http://www.pse6.com, you can change the phase relationship between the waves and observe the wave representing the superposition. Fig. 18.4a, p.547

Active Figure 18.4 The superposition of two identical waves y1 and y2 (blue and green) to yield a resultant wave (red). (b) When y1 and y2 are  rad out of phase, the result is destructive interference. At the Active Figures link at http://www.pse6.com, you can change the phase relationship between the waves and observe the wave representing the superposition. Fig. 18.4b, p.547

Active Figure 18.4 The superposition of two identical waves y1 and y2 (blue and green) to yield a resultant wave (red). (c) When the phase angle has a value other than 0 or  rad, the resultant wave y falls somewhere between the extremes shown in (a) and (b). At the Active Figures link at http://www.pse6.com, you can change the phase relationship between the waves and observe the wave representing the superposition. Fig. 18.4c, p.547

Active Figure 18.22 Beats are formed by the combination of two waves of slightly different frequencies. (a) The individual waves. (b) The combined wave has an amplitude (broken line) that oscillates in time. At the Active Figures link at http://www.pse6.com, you can choose the two frequencies and see the corresponding beats. Fig. 18.22, p.565

Figure 18.5  An acoustical system for demonstrating interference of sound waves. A sound wave from the speaker (S) propagates into the tube and splits into two parts at point P. The two waves, which combine at the opposite side, are detected at the receiver (R). The upper path length r2 can be varied by sliding the upper section. Fig. 18.5, p.548

Figure 18.6 Two speakers emit sound waves to a listener at P. Fig. 18.6, p.549

Figure 18. 7 Two speakers emit sound waves toward each other Figure 18.7 Two speakers emit sound waves toward each other. Between the speakers, identical waves traveling in opposite directions combine to form standing waves. Fig. 18.7, p.550

Active Figure 18. 10 (a) A string of length L fixed at both ends Active Figure 18.10  (a) A string of length L fixed at both ends. The normal modes of vibration form a harmonic series: (b) the fundamental, or first harmonic; (c) the second harmonic; (d) the third harmonic. At the Active Figures link at http://www.pse6.com, you can choose the mode number and see the corresponding standing wave. Fig. 18.10, p.553

Figure 18.18  Motion of elements of air in standing longitudinal waves in a pipe, along with schematic representations of the waves. In the schematic representations, the structure at the left end has the purpose of exciting the air column into a normal mode. The hole in the upper edge of the column assures that the left end acts as an open end.The graphs represent the displacement amplitudes, not the pressure amplitudes. (a) In a pipe open at both ends, the harmonic series created consists of all integer multiples of the fundamental frequency: f1, 2f1, 3f1, . . . .  (b) In a pipe closed at one end and open at the other, the harmonic series created consists of only odd-integer multiples of the fundamental frequency: f1, 3f1, 5f1, . . . .  Fig. 18.18, p.560

Figure 18.18   (a) In a pipe open at both ends, the harmonic series created consists of all integer multiples of the fundamental frequency: f1, 2f1, 3f1, . . . .  Fig. 18.18a, p.560

Figure 18.18   (b) In a pipe closed at one end and open at the other, the harmonic series created consists of only odd-integer multiples of the fundamental frequency: f1, 3f1, 5f1, . . . .  Fig. 18.18b, p.560

Figure 18.23 Sound wave patterns produced by (a) a tuning fork, (b) a flute, and (c) a clarinet, each at approximately the same frequency. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956, p. 128.) Fig. 18.23, p.566

Figure 18. 24 Harmonics of the wave patterns shown in Figure 18. 23 Figure 18.24  Harmonics of the wave patterns shown in Figure 18.23. Note the variations in intensity of the various harmonics. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956.) Fig. 18.24a, p.567

Figure 18. 24 Harmonics of the wave patterns shown in Figure 18. 23 Figure 18.24  Harmonics of the wave patterns shown in Figure 18.23. Note the variations in intensity of the various harmonics. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956.) Fig. 18.24b, p.567

Figure 18. 24 Harmonics of the wave patterns shown in Figure 18. 23 Figure 18.24  Harmonics of the wave patterns shown in Figure 18.23. Note the variations in intensity of the various harmonics. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956.) Fig. 18.24c, p.567

Active Figure 18. 10 (a) A string of length L fixed at both ends Active Figure 18.10  (a) A string of length L fixed at both ends. The normal modes of vibration form a harmonic series: (b) the fundamental, or first harmonic; (c) the second harmonic; (d) the third harmonic. At the Active Figures link at http://www.pse6.com, you can choose the mode number and see the corresponding standing wave. Fig. 18.10, p.553

Figure 41.7 Potential-energy diagram of a well of finite height U and length L. The total energy E of the system is less than U. Fig 41-7, p.1334

Active Figure 41. 8 (a) Wave functions y Active Figure 41.8 (a) Wave functions y. The wave functions and probability densities are plotted vertically from separate axes that are offset vertically for clarity. The positions of these axes on the potential-energy function suggest the relative energies of the states, but the positions are not shown to scale Fig 41-8a, p.1335

Figure 18.21 Representation of some of the normal modes possible in a circular membrane fixed at its perimeter. The pair of numbers above each pattern corresponds to the number of radial nodes and the number of circular nodes. Below each pattern is a factor by which the frequency of the mode is larger than that of the 01 mode. The frequencies of oscillation do not form a harmonic series because these factors are not integers. In each diagram, elements of the membrane on either side of a nodal line move in opposite directions, as indicated by the colors. (Adapted from T. D. Rossing, The Science of Sound, 2nd ed, Reading, Massachusetts, Addison-Wesley Publishing Co., 1990) Fig. 18.21, p.563

Fig 41-CO A quantum corral shows two aspects of current technological advances in physics. The first aspect involves control over individual atoms. This corral is formed by positioning iron atoms in a stadium-shaped ring on a copper surface. The second aspect is the ability to image the individual atoms with a scanning tunneling microscope. The corral can be used to study the quantized states of electrons trapped in a small region. (Courtesy of IBM Research, Almaden Research Center. Unauthorized use prohibited.) Fig 41-CO, p.1321

Figure 37.3 An interference pattern involving water waves is produced by two vibrating sources at the water’s surface. The pattern is analogous to that observed in Young’s double-slit experiment. Note the regions of constructive (A) and destructive (B) interference. Fig 37-3, p.1179

Figure 37.3 An interference pattern involving water waves is produced by two vibrating sources at the water’s surface. The pattern is analogous to that observed in Young’s double-slit experiment. Note the regions of constructive (A) and destructive (B) interference.

Figure 18.12  Playing an F note on a guitar. (Charles D. Winters) Fig. 18.12, p.556

Figure 18.13 When the sphere hangs in air, the string vibrates in its second harmonic. When the sphere is immersed in water, the string vibrates in its fifth harmonic. Fig. 18.13, p.557

Figure 18.14  Graph of the amplitude (response) versus driving frequency for an oscillating system. The amplitude is a maximum at the resonance frequency f0. Fig. 18.14, p.558

Figure 18.16  Standing waves are set up in a string when one end is connected to a vibrating blade. When the blade vibrates at one of the natural frequencies of the string, large-amplitude standing waves are created. Fig. 18.16, p.558

Figure 18. 17 (a) Standing-wave pattern in a vibrating wine glass Figure 18.17 (a) Standing-wave pattern in a vibrating wine glass. The glass shatters if the amplitude of vibration becomes too great. (photo courtesy Professor Thomas D. Rossing, Northern Illinois University) (b) A wine glass shattered by the amplified sound of a human voice. Fig. 18.17, p.559

Figure 18.19  (a) Apparatus for demonstrating the resonance of sound waves in a tube closed at one end. The length L of the air column is varied by moving the tube vertically while it is partially submerged in water. (b) The first three normal modes of the system shown in part (a). Fig. 18.19, p.562

Figure 18.19  (a) Apparatus for demonstrating the resonance of sound waves in a tube closed at one end. The length L of the air column is varied by moving the tube vertically while it is partially submerged in water. Fig. 18.19a, p.562

Figure 18.19  (b) The first three normal modes of the system shown in part (a). Fig. 18.19b, p.562

Figure 18. 23 Sound wave patterns produced by (a) a tuning fork Figure 18.23 Sound wave patterns produced by (a) a tuning fork. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956, p. 128.) Fig. 18.23a, p.566

Figure 18. 23 Sound wave patterns produced by (b) a flute Figure 18.23 Sound wave patterns produced by (b) a flute. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956, p. 128.) Fig. 18.23b, p.566

Figure 18. 23 Sound wave patterns produced by (c) a clarinet Figure 18.23 Sound wave patterns produced by (c) a clarinet. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956, p. 128.) Fig. 18.23c, p.566

Figure 18. 24 Harmonics of the wave patterns shown in Figure 18. 23 Figure 18.24  Harmonics of the wave patterns shown in Figure 18.23. Note the variations in intensity of the various harmonics. (Adapted from C. A. Culver, Musical Acoustics, 4th ed., New York, McGraw-Hill Book Company, 1956.) Fig. 18.24, p.567

Active Figure 18.25  Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f. (a) Waves of frequency f and 3f are added. (b) One more odd harmonic of frequency 5f is added. (c) The synthesis curve approaches closer to the square wave when odd frequencies up to 9f are added. At the Active Figures link at http://www.pse6.com, you can add in harmonics with frequencies higher than 9f to try to synthesize a square wave. Fig. 18.25, p.568

Active Figure 18.25  Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f. (a) Waves of frequency f and 3f are added. Active Figures link at http://www.pse6.com, you can add in harmonics with frequencies higher than 9f to try to synthesize a square wave. Fig. 18.25a, p.568

Active Figure 18.25  Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f. (b) One more odd harmonic of frequency 5f is added. At the Active Figures link at http://www.pse6.com, you can add in harmonics with frequencies higher than 9f to try to synthesize a square wave. Fig. 18.25b, p.568

Active Figure 18.25  Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f. (c) The synthesis curve approaches closer to the square wave when odd frequencies up to 9f are added. At the Active Figures link at http://www.pse6.com, you can add in harmonics with frequencies higher than 9f to try to synthesize a square wave. Fig. 18.25c, p.568

Fig. P18.2, p.570

Fig. P18.21, p.572

Figure 18.21 Representation of some of the normal modes possible in a circular membrane fixed at its perimeter. The pair of numbers above each pattern corresponds to the number of radial nodes and the number of circular nodes. Below each pattern is a factor by which the frequency of the mode is larger than that of the 01 mode. The frequencies of oscillation do not form a harmonic series because these factors are not integers. In each diagram, elements of the membrane on either side of a nodal line move in opposite directions, as indicated by the colors. (Adapted from T. D. Rossing, The Science of Sound, 2nd ed, Reading, Massachusetts, Addison-Wesley Publishing Co., 1990) Fig. 18.21, p.563