Surface Waves Normally Incident on a Submerged Horizontal Cylinder

Abstract

When small amplitude surface waves on infinitely deep water are normally incident on a submerged circular cylinder whose axis is horizontal, there is zero reflection at all frequencies (Ursell (1950)) and an exponentially small phase shift at high frequency (Davis (1974)). Here the latter result is generalized to finite depth and cylinders of arbitrary cross section, after which it is shown that zero reflection at all frequencies only occurs when the cross section of the fluid region can be mapped by a bilinear transformation into a concentric circular annulus, i.e., only for Ursell’s geometry.