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

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.