Tom Bridges gave a talk in the Industrial and Applied Mathematics Seminar series at the University of Nottingham on 20th February and a talk in the Applied Mathematics Series at the University of Reading on 26th February.  The title of the talk in both cases was “Reappraisal of two myths in the theory of nonlinear waves“.  The first myth is that the shallow water equations (SWEs) are a model for shallow water hydrodynamics.  It is shown that in fact the shallow water equations can not control the growth of horizontal vorticity and so the equations will in general break down.  Although the SWEs are a perfectly valid mathematical model, they may be invalid when applied to realistic shallow water oceanographic flows.  The second myth is how the KdV equation arises as a model PDE.  The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of PDEs has the appropriate form, and the nonlinearity is quadratic.  A new mechanism is presented which shows that the KdV equation always takes a universal form where the coefficients in the KdV equation are completely determined from the modulation properties of the background state — even when the background state is the trivial solution!   One outcome is that “shallow water” is neither necessary nor sufficient for the emergence of the KdV equation.