16 - Wave Motion

16 Wave Motion

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Unformatted text preview: pshot at some later time t. 504 CHAPTER 16 Wave Motion so represents a wave traveling to the right. If the wave were traveling to the left, the quantity x vt would be replaced by x vt, as we learned when we developed Equations 16.1 and 16.2. By definition, the wave travels a distance of one wavelength in one period T. Therefore, the wave speed, wavelength, and period are related by the expression v (16.7) T Substituting this expression for v into Equation 16.6, we find that x A sin 2 y t T (16.8) This form of the wave function clearly shows the periodic nature of y. At any given , time t (a snapshot of the wave), y has the same value at the positions x, x x 2 , and so on. Furthermore, at any given position x, the value of y is the same at times t, t T, t 2T, and so on. We can express the wave function in a convenient form by defining two other quantities, the angular wave number k and the angular frequency : k Angular wave number 2 (16.9) 2 T Angular frequency (16.10) Using these definitions, we see that Equation 16.8 can be written in the more compact form Wave function for a sinusoidal wave y A sin(kx (16.11) t) The frequency of a sinusoidal wave is related to the period by the expression f Frequency 1 T (16.12) The most common unit for frequency, as we learned in Chapter 13, is second 1, or hertz (Hz). The corresponding unit for T is seconds. Using Equations 16.9, 16.10, and 16.12, we can express the wave speed v originally given in Equation 16.7 in the alternative forms v Speed of a sinusoidal wave v General expression for a sinusoidal wave (16.13) k (16.14) f The wave function given by Equation 16.11 assumes that the vertical displacement y is zero at x 0 and t 0. This need not be the case. If it is not, we generally express the wave function in the form y A sin(kx t ) (16.15) 505 16.7 Sinusoidal Waves where is the phase constant, just as we learned in our study of periodic motion in Chapter 13. This constant can be determined from the initial conditions. EXAMPLE 16.3 A Traveling Sinusoidal Wave A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t 0 and x 0 is also 15.0 cm, as shown in Figure 16.18. (a) Find the angular wave number k, period T, angular frequency , and speed v of the wave. 2f 2 (8.00 s 1) T 1 f 1 8.00 s v f 1 50.3 rad/s 0.125 s (40.0 cm)(8.00 s 1) 320 cm/s Solution (a) Using Equations 16.9, 16.10, 16.12, and 16.14, we find the following: k 2 2 rad 40.0 cm 0.157 rad/cm (b) Determine the phase constant expression for the wave function. Solution at x Because A 15.0 cm and because y 15.0 cm 0 and t 0, substitution into Equation 16.15 gives 15.0 y(cm) or (15.0) sin sin We may take the principal value Hence, the wave function is of the form 40.0 cm y 15.0 cm , and write a general A sin kx x(cm) t 2 1 /2 rad (or 90°). A cos(kx t) By inspection, we can see that the wave function must have this form, noting that the cosine function has the same shape as the sine function displaced by 90°. Substituting the values for A, k, and into this expression, we obtain Figure 16.18 A sinusoidal wave of wavelength 40.0 cm and amplitude A 15.0 cm. The wave function can be written in the form y A cos(kx t ). y (15.0 cm) cos(0.157x Sinusoidal Waves on Strings In Figure 16.2, we demonstrated how to create a pulse by jerking a taut string up and down once. To create a train of such pulses, normally referred to as a wave train, or just plain wave, we can replace the hand with an oscillating blade. If the wave consists of a train of identical cycles, whatever their shape, the relationships f 1/T and v f among speed, frequency, period, and wavelength hold true. We can make more definite statements about the wave function if the source of the waves vibrates in simple harmonic motion. Figure 16.19 represents snapshots of the wave created in this way at intervals of T/4. Note that because the end of the blade oscillates in simple harmonic motion, each particle of the string, such as that at P, also oscillates vertically with simple harmonic motion. This must be the case because each particle follows the simple harmonic motion of the blade. Therefore, every segment of the string can be treated as a simple harmonic oscillator vibrating with a frequency equal to the frequency of oscillation of the blade.3 Note that although each segment oscillates in the y direction, the wave travels in the x direction with a speed v. Of course, this is the definition of a transverse wave. 3 In this arrangement, we are assuming that a string segment always oscillates in a vertical line. The tension in the string would vary if a segment were allowed to move sideways. Such motion would make the analysis very complex. 50.3t ) 506 CHAPTER 16 Wave Motion λ y P A P (a) Vibrating blade (b) P P (c) (d) Figure 16.19 One method for producing a train of sinusoidal wave pulses on a string. The left end of the string is connected to a blade that is set into oscillation. Every segment of the string, such as the point...
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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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