Symmetry relations and decomposition for the mild-slope equation

The reflection and transmission coefficients arising from the scattering of linear water waves by a one-dimensional topography are known to possess certain symmetry properties. In this paper it is shown that the same relations hold in the mild-slope approximation to the full linear theory. These relations are used in the development of a decomposition method where solutions for relatively simple depth profiles may be combined to give solutions for more complicated ones. The use of the decomposition method provides explicit error bounds in cases where they were previously unavailable. Examples are given, including the case where the depth profile consists of a sequence of an arbitrary number of identical, equally spaced humps. An example of how the decomposition may be applied to Kirby's extended mild-slope equation (for which the symmetry relations are also valid) is presented in the case of a ripple bed.