Harbor oscillations on a piecewise bottom of two different slopes

Abstract

Analytical and numerical investigations are carried out for harbor oscillations on a piecewise bottom of two different slopes. With the shallow water approximation, the longitudinal and transverse oscillations are analyzed by coupling the solutions on each segment of the bottom with the coupling conditions at the point of inflection in ignoring the nonlinear interactions between them. The longitudinal ones are the combinations of the Bessel functions of the first and second kinds, and the transverse ones are expressed with Kummer’s equations. As the transverse oscillations trapped within the harbor are typically standing edge waves, their dispersion relationship is also derived for the two-slope-piecewise bottom. With the analytical predictions of the oscillation behaviors and based on Boussinesq equations, a range of numerical tests are made for the oscillations induced by seafloor motions on this type of piecewise bottom. Different combinations of the two slopes of the bottom are firstly tested and the numerical eigenfrequencies and wave profiles are compared with analytical ones. The transverse oscillations are revealed to be sensitive to the positions of the seafloor movements, and the importance of the position of the seafloor motions is manifested in their relative positions against the nodal lines and anti-nodal lines of the transverse modes. Although the longitudinal oscillations generated by seafloor motions of harbor width may not reach a steady state without forcing terms at the entrance, some patterns of several low-mode ones occur and are also sensitive to the position of the moveable seafloor. Wavelet spectra are used to analyze their evolutions and comparisons are made with theoretical predictions for the three principle longitudinal modes.