Horizontal Refraction in a Three-dimensional Medium of Variable Stratification

In the sea the repeated horizontal deflection of a sound ray due to reflection at a sloping bottom produces a curvature of the horizontal path. The reflection angles in three dimensions are calculated for a sloping bottom or sound velocity interface. Even if the stratification varies in both horizontal directions there is for slow variations a simple relation between the changes in vertical and horizontal ray angle. It is then shown that the horizontal curvature may be predicted by associating a refractive index or phase velocity with each horizontal position. A formula for intensity and some illustrative problems are presented. For example it is shown that on reflection from a coastline the vertical ray angle at closest approach is equal to half the total horizontal angle through which the ray is turned.