Three-dimensional exponential asymptotics and Stokes surfaces for flows past a submerged point source

When studying two-dimensional fluid–body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free surface at every order. The waves are, in fact, exponentially small and it has been well-established that such waves ‘switch on’ seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes-line concept extends to higher-dimensional Stokes surfaces. This article is concerned with the archetypal problem of uniform flow over a point source, reminiscent of, but separate to, the famous Kelvin wave problem. In prior research, the intersection of the Stokes surface with the free surface was found, in implicit form, for this case of a point source. However, on account of the algebraic manipulations required, it is not clear how this approach can be extended to more challenging settings. Here we develop a numerical-based procedure that allows the Stokes surface to be computed. The intersections of the Stokes surfaces with both the free surface and the deeper fluid are discussed for the case of the point source. Crucially, this procedure provides an important tool for generalising exponential asymptotics to the case of nonlinear (non-point-source) wave-generating bodies.