Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / i / Solution

In coordinates moving with the background fluid, $x^{\prime }=x-Ut$. The stationary hill profile becomesso in the convention $e^{i(k{x}^{\prime }-\omega t)}$ its frequency is $\omega =-kU$; its physical angular frequency has magnitude $kU$.