Randall D. Knight — Superposition: Problem Challenge_Problem_80

A water wave is called a deep-water wave if the water’s depth is more than one-quarter of the wavelength. Unlike the waves we’ve considered in this chapter, the speed of a deep-water wave depends on its wavelength: \[ v = \sqrt{\frac{g\lambda}{2\pi}} \] Longer wavelengths travel faster. Let’s apply this to standing waves. Consider a diving pool that is 5.0 m deep and 10.0 m wide. Standing water waves can set up across the width of the pool. Because water sloshes up and down at the sides of the pool, the boundary conditions require antinodes at \( x = 0 \) and \( x = L \). Thus a standing water wave resembles a standing sound wave in an open-open tube. (a) What are the wavelengths of the first three standing-wave modes for water in the pool? Do they satisfy the condition for being deep-water waves? (b) What are the wave speeds for each of these waves? (c) Derive a general expression for the frequencies \( f_m \) of the possible standing waves. Your expression should be in terms of \( m, g \), and \( L \). (d) What are the oscillation periods of the first three standing wave modes?