16 - Wave Motion

168b the wave function for the resulting complex wave

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Unformatted text preview: etric.) When the waves begin to overlap (Fig. 16.8b), the wave function for the resulting complex wave is given by y 1 y 2 . y2 (a) y1 y2 (b) y1 (c) y 1+ y 2 y2 (d) y1 y2 (e) y1 (f) Figure 16.9 (a – e) Two wave pulses traveling in opposite directions and having displacements that are inverted relative to each other. When the two overlap in (c), their displacements partially cancel each other. (f) Photograph of superposition of two symmetric pulses traveling in opposite directions, where one pulse is inverted relative to the other. 16.5 The Speed of Waves on Strings When the crests of the pulses coincide (Fig. 16.8c), the resulting wave given by y 1 y 2 is symmetric. The two pulses finally separate and continue moving in their original directions (Fig. 16.8d). Note that the pulse shapes remain unchanged, as if the two pulses had never met! The combination of separate waves in the same region of space to produce a resultant wave is called interference. For the two pulses shown in Figure 16.8, the displacement of the medium is in the positive y direction for both pulses, and the resultant wave (created when the pulses overlap) exhibits a displacement greater than that of either individual pulse. Because the displacements caused by the two pulses are in the same direction, we refer to their superposition as constructive interference. Now consider two pulses traveling in opposite directions on a taut string where one pulse is inverted relative to the other, as illustrated in Figure 16.9. In this case, when the pulses begin to overlap, the resultant wave is given by y 1 y 2 , but the values of the function y 2 are negative. Again, the two pulses pass through each other; however, because the displacements caused by the two pulses are in opposite directions, we refer to their superposition as destructive interference. Quick Quiz 16.2 Two pulses are traveling toward each other at 10 cm/s on a long string, as shown in Figure 16.10. Sketch the shape of the string at t 0.6 s. 1 cm Figure 16.10 16.5 The pulses on this string are traveling at 10 cm/s. THE SPEED OF WAVES ON STRINGS In this section, we focus on determining the speed of a transverse pulse traveling on a taut string. Let us first conceptually argue the parameters that determine the speed. If a string under tension is pulled sideways and then released, the tension is responsible for accelerating a particular segment of the string back toward its equilibrium position. According to Newton’s second law, the acceleration of the segment increases with increasing tension. If the segment returns to equilibrium more rapidly due to this increased acceleration, we would intuitively argue that the wave speed is greater. Thus, we expect the wave speed to increase with increasing tension. Likewise, we can argue that the wave speed decreases if the mass per unit length of the string increases. This is because it is more difficult to accelerate a massive segment of the string than a light segment. If the tension in the string is T (not to be confused with the same symbol used for the period) and its mass per 499 500 The strings of this piano vary in both tension and mass per unit length. These differences in tension and density, in combination with the different lengths of the strings, allow the instrument to produce a wide range of sounds. unit length is (Greek letter mu), then, as we shall show, the wave speed is Speed of a wave on a stretched string ∆s ar = v v2 R R O (a) v ∆s θ θ T R Fr T θ O (b) Figure 16.11 (a) To obtain the speed v of a wave on a stretched string, it is convenient to describe the motion of a small segment of the string in a moving frame of reference. (b) In the moving frame of reference, the small segment of length s moves to the left with speed v. The net force on the segment is in the radial direction because the horizontal components of the tension force cancel. √ T (16.4) First, let us verify that this expression is dimensionally correct. The dimensions of T are ML/T 2, and the dimensions of are M/L. Therefore, the dimensions of T/ are L2/T 2; hence, the dimensions of √T/ are L/T — indeed, the dimensions of speed. No other combination of T and is dimensionally correct if we assume that they are the only variables relevant to the situation. Now let us use a mechanical analysis to derive Equation 16.4. On our string under tension, consider a pulse moving to the right with a uniform speed v measured relative to a stationary frame of reference. Instead of staying in this reference frame, it is more convenient to choose as our reference frame one that moves along with the pulse with the same speed as the pulse, so that the pulse is at rest within the frame. This change of reference frame is permitted because Newton’s laws are valid in either a stationary frame or one that moves with constant velocity. In our new reference frame, a given segment of the string initially to the right of the pulse moves to the left, rises up and follows the shape of...
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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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