Serway & Jewett — Superposition and Standing Waves: Problem 9

Two waves simultaneously present on a long string have a phase difference \(\phi\) between them so that a standing wave formed from their combination is described by \[y(x, t) = 2A \sin\left(kx + \frac{\phi}{2}\right) \cos\left(\omega t - \frac{\phi}{2}\right)\] (a) Despite the presence of the phase angle \(\phi\), is it still true that the nodes are one-half wavelength apart? Explain. (b) Are the nodes different in any way from the way they would be if \(\phi\) were zero? Explain.