An analytical investigation for oscillations in a harbor of a parabolic bottom

Abstract

Based on the linear shallow water approximation, longitudinal and transverse oscillations in a rectangular harbor with a parabolic bottom are analyzed. The longitudinal ones are combinations of the Legendre functions of the first and second kinds and the transverse ones are expressed with modified Bessel equations. Analytic results for longitudinal oscillations show that the augmentation of rapidity of variation of the water depth shifts the resonant wave frequencies to larger values and slightly changes the positions of the nodes for the resonant modes. For the transverse oscillations trapped within the harbor which are typically standing edge waves, the dispersion relationship is derived and the spatial structures of the first four modes are presented. The solutions illustrate that all the trapped modes are affected by the varying water depth parameters, especially for the higher modes whose profiles extend farther and the distribution of the energy of transverse oscillations is influenced by the rapidity of variation of the bottom within the harbor.

Numerical investigation of oscillations within a harbor of parabolic bottom induced by small-scale landslides

“…The frequencies of the three first components, extracted from the wavelet ridges, are compared with theoretical predictions as shown in Table 2. From Table 2, it can be seen that these resonant frequencies are close to those predicted by Shao et al 1 with slightly larger differences than the comparisons of transverse modes. The reasons of these larger differences are similar to the ones given by Wang et al 17 in their study of longitudinal oscillations induced by landslides on a constant slope which may be described as follows.…”

“…Oscillations and trapped modes in harbors and bays may cause many problems such as preventing of cargo operations, breaking of mooring ropes, damaging of infrastructures or moored vessels and can be triggered by the match of the eigenvalues of the free oscillations of a harbor and the external forces coming from wave groups, atmospheric pressure disturbances, seafloor motions, landslides or shear flows, and so on. With linear shallow water approximation, the companion article of Shao et al 1 presents formulary descriptions of longitudinal and transverse oscillations inside a rectangular harbor with a parabolic bottom open to the sea. Some characteristics of the oscillations on this type of bottom are revealed analytically.…”

“…The transverse oscillation modes due to refraction of the waves on the constant slope and the dispersion relationship are also discussed. Following them, Shao et al 1 derived the formulations for a harbor of parabolic bottom.…”

Standing edge waves reinvestigated: A comparison between two different bathymetries

“…Although for a sloping beach the water depth at the shoreline may be zero (h 1 = 0), as conventionally found in the research of edge waves, here in this article, the term h 1 is reserved to make the profiles of the beach on each slope equivalent in the following formulations. With equation (15), the water depth is expressed in a generalized form h slopej = s j x + h j , j = 1, 2, . .…”

“…with h only a function of x given by equation (15) as the water depth varies only in the offshore direction. While c = v=k, equation (49) can be transformed into…”