Randall D. Knight — Traveling Waves: Problem 85

A water wave is a shallow-water wave if the water depth \(d\) is less than \(\approx \lambda/10\). It is shown in hydrodynamics that the speed of a shallow-water wave is \(v = \sqrt{gd}\), so waves slow down as they move into shallower water. Ocean waves, with wavelengths of typically \(100\text{ m}\), are shallow-water waves when the water depth is less than \(\approx 10\text{ m}\). Consider a beach where the depth increases linearly with distance from the shore until reaching a depth of \(5.0\text{ m}\) at a distance of \(100\text{ m}\). How long does it take a wave to move the last \(100\text{ m}\) to the shore? Assume that the waves are so small that they don’t break before reaching the shore.