LONG OCEAN WAVES ON LINEARLY VARYING BOTTOM TOPOGRAPHIE

The effects of bottom topography on long free surface gravity ocean waves is studied. Both wave trapping and transmission-reflection problems are considered. The topographies are described by straight parallel bottom contours which in cross section are composed of segments of constant and linear depth variations. Over linearly varying depth segments, the solution to the vertically integrated governing equations is given in terms of two Kummer (or Whittaker) functions. Coriolis effects are included but primary interest is on class I (high frequency) free surface gravity waves. The wave trapping problems considered are waves trapped over a submerged ridge composed of constant slope segments bounded on either side by constant depth segments and wave trapping over a continental slope and shelf composed of a constant depth ocean segment followed by two linearly varying depth segments of different slopes. The shelf terminates with zero depth at the coastline. In each case, solutions are found for time harmonic waves periodic in the along contour direction and of exponential decay behavior normal to the contours over the constant depth segments. Dispersion equations are found relating the wave frequency to the along ridge wave number for trapped waves. A comparison of the dispersion equations for the piecewise continuous ridge and a piecewise discontinuous constant depth ridge is made with emphasis on the cutoff frequencies and phase and group velocities. The transmission-reflection problems consider a progressive plane wave field obliquely incident from a constant depth ocean onto the submerged ridge, a submerged trench analogous to the submerged rigde, and a continental slope and shelf. The ridge and trench problems examine the portion of the incident wave energy reflected by or transmitted past the topography while the continental slope and shelf problem examines the amplification of the incident wave at the coast and resonances over the shelf.