# Divergent series, exponential asymptotics and surface waves

For a wide class of water wave problems, the characteristic velocity field is measured by a dimensionless parameter called the Froude number, $F$. In the singular limit $F\rightarrow 0$, a standard perturbation approach gives rise to a divergent series in powers of $F^2$. Typically, the water waves that we are trying to analyse cannot be captured by such a divergent series, as their amplitude is exponentially small, and so they appear {\em beyond all orders} of the perturbation expansion. Thus this sort of problem belongs to the realm of exponential asymptotics. Here the goal is to generate a so-called ``hyperasymptotic’’ approximation, which is tricksy but interesting. No real expertise is required for this talk, but some idea about formal asymptotics and complex variables may help. Inconvenient notions about converging series may be left in your office.