In Figures 18.20a and 18.20b, notice that the amplitude of

Chapter 18, Problem 88

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In Figures 18.20a and 18.20b, notice that the amplitude of the component wave for frequency f is large, that for 3f is smaller, and that for 5f smaller still. How do we know exactly how much amplitude to assign to each frequency component to build a square wave? This problem helps us find the answer to that question. Let the square wave in Figure 18.20c have an amplitude A and let t 5 0 be at the extreme left of the figure. So, one period T of the square wave is described by y1t2 5 A 0 , t , T 2 2A T 2 , t , T Express Equation 18.13 with angular frequencies: y1t 2 5 an 1An sin nvt 1 Bn cos nvt2 Now proceed as follows. (a) Multiply both sides of Equation 18.13 by sin mvt and integrate both sides over one period T. Show that the left-hand side of the resulting equation is equal to 0 if m is even and is equal to 4A/mv if m is odd. (b) Using trigonometric identities, show that all terms on the right-hand side involving Bn are equal to zero. (c) Using trigonometric identities, show that all terms on the right-hand side involving An are equal to zero except for the one case of m 5 n. (d) Show that the entire right-hand side of the equation reduces to 1 2AmT. (e) Show that the Fourier series expansion for a square wave is y1t 2 5 an 4A np sin nvt