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Steady and unsteady linearised flow past a submerged source are studied inthe small-surface-tension limit, in the absence of gravitational effects. Thefree-surface capillary waves generated are exponentially small in the surfacetension, and are determined using the theory of exponential asymptotics. In thesteady 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 theposition of the free surface.  In the unsteady problem, transient effects cause the solution to display morecomplicated asymptotic behaviour, such as higher-order Stokes lines. The theoryof exponential asymptotics is applied to show how the capillary waves evolveover time, and eventually tend to the steady solution.

Three-dimensional capillary waves due to a submerged source with small surface tension