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

The mountain-wave cutoff follows because the hills force intrinsic frequency $Uk$, whereas internal gravity waves require $|\omega |\le N$. Thus $k_c=N/U$ separates long hills, which radiate vertically propagating waves and carry energy upward, from short hills, whose response is evanescent and carries no mean vertical energy flux.