  Steady and unsteady linearised flow past a submerged source are studied in
the small-surface-tension limit, in the absence of gravitational effects. The
free-surface capillary waves generated are exponentially small in the surface
tension, and are determined using the theory of exponential asymptotics. In the
steady problem, capillary waves are found to extend upstream from the source,
switching on across curves on the free surface known as Stokes lines.
Asymptotic predictions and compared with computational solutions for the
position of the free surface.
  In the unsteady problem, transient effects cause the solution to display more
complicated asymptotic behaviour, such as higher-order Stokes lines. The theory
of exponential asymptotics is applied to show how the capillary waves evolve
over time, and eventually tend to the steady solution.
