16 - Wave Motion

This expression therefore represents the equation of

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Unformatted text preview: dless of how x and t change individually, we must require that x vt x 0 in order to stay with the crest. This expression therefore represents the equation of motion of the crest. At t 0, the crest is at x x 0 ; at a y y vt v v P A P O (a) Pulse at t = 0 x O x (b) Pulse at time t Figure 16.6 A one-dimensional wave pulse traveling to the right with a speed v. (a) At t 0, the shape of the pulse is given by y f (x). (b) At some later time t, the shape remains unchanged and the vertical displacement of any point P of the medium is given by y f (x vt ). 496 CHAPTER 16 Wave Motion time dt later, the crest is at x x 0 v dt. Therefore, in a time dt, the crest has moved a distance dx (x 0 v dt ) x 0 v dt. Hence, the wave speed is dx dt v EXAMPLE 16.1 A Pulse Moving to the Right A wave pulse moving to the right along the x axis is represented by the wave function y(x, t ) 2 3.0t )2 (x Solution First, note that this function is of the form y f (x vt ). By inspection, we see that the wave speed is v 3.0 cm/s. Furthermore, the wave amplitude (the maximum value of y ) is given by A 2.0 cm. (We find the maximum value of the function representing y by letting x 3.0t 0.) The wave function expressions are 2 x2 at t 1 0 1 (x 2 6.0)2 1 at t 1.0 s at t 2.0 s Likewise, at x 1.0 cm, y(1.0, 0) 1.0 cm, and at x 2.0 cm, y(2.0, 0) 0.40 cm. Continuing this procedure for other values of x yields the wave function shown in Figure 16.7a. In a similar manner, we obtain the graphs of y (x, 1.0) and y (x, 2.0), shown in Figure 16.7b and c, respectively. These snapshots show that the wave pulse moves to the right without changing its shape and that it has a constant speed of 3.0 cm/s. y(cm) 2.0 2.0 3.0 cm/s 1.5 3.0 cm/s 1.5 t=0 t = 1.0 s 1.0 y(x, 0) y(x, 1.0) 0.5 0 2 3.0)2 We now use these expressions to plot the wave function versus x at these times. For example, let us evaluate y(x, 0) at x 0.50 cm: 2 y(0.50, 0) 1.6 cm (0.50)2 1 y(cm) 1.0 (x y(x, 1.0) y(x, 2.0) 1 where x and y are measured in centimeters and t is measured in seconds. Plot the wave function at t 0, t 1.0 s, and t 2.0 s. y(x, 0) (16.3) 0.5 1 2 3 4 5 x(cm) 6 0 1 2 3 (a) 4 5 6 x(cm) 7 (b) y(cm) 3.0 cm/s 2.0 t = 2.0 s 1.5 1.0 y(x, 2.0) 0.5 Figure 16.7 at (a) t Graphs of the function y(x, t ) 0, (b) t 1.0 s, and (c) t 2.0 s. 2/[(x 3.0t )2 1] 0 1 2 3 4 (c) 5 6 7 8 x(cm) 497 16.4 Superposition and Interference 16.4 SUPERPOSITION AND INTERFERENCE Many interesting wave phenomena in nature cannot be described by a single moving pulse. Instead, one must analyze complex waves in terms of a combination of many traveling waves. To analyze such wave combinations, one can make use of the superposition principle: If two or more traveling waves are moving through a medium, the resultant wave function at any point is the algebraic sum of the wave functions of the individual waves. Waves that obey this principle are called linear waves and are generally characterized by small amplitudes. Waves that violate the superposition principle are called nonlinear waves and are often characterized by large amplitudes. In this book, we deal only with linear waves. One consequence of the superposition principle is that two traveling waves can pass through each other without being destroyed or even altered. For instance, when two pebbles are thrown into a pond and hit the surface at different places, the expanding circular surface waves do not destroy each other but rather pass through each other. The complex pattern that is observed can be viewed as two independent sets of expanding circles. Likewise, when sound waves from two sources move through air, they pass through each other. The resulting sound that one hears at a given point is the resultant of the two disturbances. Figure 16.8 is a pictorial representation of superposition. The wave function for the pulse moving to the right is y 1 , and the wave function for the pulse moving (a) y1 y2 (b) y 1+ y 2 (c) y 1+ y 2 (d) y2 y1 (e) Figure 16.8 (a – d) Two wave pulses traveling on a stretched string in opposite directions pass through each other. When the pulses overlap, as shown in (b) and (c), the net displacement of the string equals the sum of the displacements produced by each pulse. Because each pulse displaces the string in the positive direction, we refer to the superposition of the two pulses as constructive interference. (e) Photograph of superposition of two equal, symmetric pulses traveling in opposite directions on a stretched spring. Linear waves obey the superposition principle 498 CHAPTER 16 Wave Motion Interference of water waves produced in a ripple tank. The sources of the waves are two objects that oscillate perpendicular to the surface of the tank. to the left is y 2 . The pulses have the same speed but different shapes. Each pulse is assumed to be symmetric, and the displacement of the medium is in the positive y direction for both pulses. (Note, however, that the superposition principle applies even when the two pulses are not symm...
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This note was uploaded on 03/24/2010 for the course PHYSICS 2202 taught by Professor Mihalisin during the Spring '09 term at Temple.

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