Wind driven waves on the surfaces of water bodies have great practical importance on all scales (H. Jeffreys in 1926 illustrated his famous paper with photographs of ripples on the Newnham duck pond in Cambridge and huge waves on the Atlantic ocean). Despite 50 years of intense research, there is still disagreement about how idealised mathematical models apply to real air flow over real waves. In this lecture the model of Miles (1957) and Lighthill (1962) for inviscid shear flow over a growing monochromatic wave is explained in terms of critical layer dynamics, but is shown to be invalid for viscous, turbulent flow when the growth rate is asymptotically small. But analytical viscous turbulent shear flow models, also with critical layers, are valid in this limit and agree more closely with experimental wind profile data. But the former type of model (suitably adjusted) is widely used by oceanographers and meteorologists. However for real wind-driven waves both the critical-layer and sheltering mechanisms are significant and affect how waves travel in groups with characteristic asymmetry of the wave-shapes on the windward and leeward sides of the group. (Sajjadi et al. 2013).
The second part of the lecture concerns recent long tsunami-like waves, especially waves where the leading part of the wave is depressed, which was a characteristic feature of the tsunamis that approached the coast-lines of SEAsia in 2004 and Japan in 2011. As such waves travel from the source region, a non-linear Kortweg-de Vries model of R. Grimshaw, K.W. Lam and J.C.R. Hunt (2014) shows how when a depression wave is followed by an elevation (a 'breather) there is a transition at a location which can be estimated when the peak elevation catches up with the peak depression and nearly doubles in height before it then decreases and travels in front of the depression. In situations where the depression reaches the beach, recent modelling and laboratory studies show how the depression deepens, leading to a back flow and drying out of the beach, before there is a transition when the following much amplified elevation (in which the total momentum of the wave is maintained) surges up the beach and moves some kilometres inland, which corresponds with recent and past observations (Klettner et al. 2012).
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