Irregular frequencies and iterative methods in the solution of steady surface-wave problems in hydrodynamics

Abstract

Two linearized problems in free-surface hydrodynamics are discussed. The first concerns flow due to a submerged line vortex in a running stream, and the second investigates three-dimensional flow about a moving pressure distribution at the surface of the fluid. Closed-form solutions to both linearized problems are well known, and therefore are not of interest; however, it is shown that the solution of either problem by a boundary-integral technique utilizing “simple” (Rankine) sources as fundamental singular solutions leads to Fredholm integral equations of the second kind, for which the irregular frequencies do not occur discretely, but as a continuum. Consequently, Neumann-type iteration schemes for the solution of these equations necessarily diverge for any Froude number. Ramifications of this result in the attempted numerical solution of the corresponding non-linear problems are discussed, and the convergence difficulties encountered by Hess [1] are analyzed.