Serway & Jewett — Superposition and Standing Waves: Problem 50

In Figures 17.22a and 17.22b, 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 17.22c have an amplitude \( A \) and let \( t = 0 \) be at the extreme left of the figure. So, one period \( T \) of the square wave is described by \[ y(t) = \begin{cases} A & 0 < t < T/2 \\ -A & T/2 < t < T \end{cases} \] Express Equation 17.14 with angular frequencies: \[ y(t) = \sum_{n} (A_n \sin n\omega t + B_n \cos n\omega t) \] Now proceed as follows. (a) Multiply both sides of Equation 17.14 by \( \sin m\omega t \) 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 \( 4A/m\omega \) if \( m \) is odd. (b) Using trigonometric identities, show that all terms on the right-hand side involving \( B_n \) are equal to zero. (c) Using trigonometric identities, show that all terms on the right-hand side involving \( A_n \) are equal to zero except for the one case of \( m = n \). (d) Show that the entire right-hand side of the equation reduces to \( \frac{1}{2} A_m T \). (e) Show that the Fourier series expansion for a square wave is \[ y(t) = \sum_{n} \frac{4A}{n\pi} \sin n\omega t \]