  In this thesis we consider the free surface flow due to a submerged source in
a channel of finite depth. This problem has been considered previously in the
literature, with some disagreement about whether or not a train of waves exist
on the free surface for Froude numbers less than unity. The physical
assumptions behind the accepted model are clearly stated and governing
equations and boundary conditions derived. Complex variable theory is then
employed to obtain a singular nonlinear integral equation, which describes the
flow field completely.
  The integral equation is evaluated numerically using a collocation method,
and implemented with mathematical software {\em Matlab}. The numerical
solutions suggest that indeed waves do exist on the free surface, but are
exponentially small in the limit that the Froude number approaches zero. An
asymptotic expression is also derived to solve the integral equation, valid for
small Froude numbers. This expansion is calculated to second order, which
improves on the leading order solution given previously in the literature. Such
a scheme is unable to predict waves on the free surface, since they are
exponentially small. It is discussed how exponential asymptotics could be used
to derive a more accurate analytic solution that describes waves on the free
surface.
