Linear wave theory has a special place in applied mathematics. For example, the powerful concepts of linear wave theory, such as dispersion, group velocity or wave action conservation, are fundamental for describing the behaviour of solutions to many commonly occurring partial differential equations (PDEs). Also, whilst it is certainly not true that every linear wave problem has an explicit general solution, it is true that every linear problem can be approached by using linear thinking, i.e., by building up more complex solutions out of superpositions of simpler solutions. In some cases, this procedure can be carried to its logical conclusion and the complete general solution to a problem can be formulated as a sum over special solutions. For example, this works for PDEs with constant coefficients in a periodic domain, for which the general solution can be written as a sum of plane waves described mathematically by a Fourier series.

But even in cases where there is no explicit general solution, the possibility to develop special solutions using asymptotic methods and the ability to combine several simple solutions to form a more complex solution always deepens our understanding of the underlying problem, and such an improved understanding could then be used to aid a numerical simulation for situations of particular interest, for example. Thus time spent studying linear wave theory is time well spent.

We are particularly interested in the behaviour of small-scale waves propagating on an inhomogeneous basic state, because this is the natural setting for unresolved waves interacting with a resolved mean flow in a numerical model.