Analytical solutions for oscillations in a harbor with a hyperbolic-cosine squared bottom

• Analytical solutions are derived for wave oscillations in a harbor with a hyperbolic-cosine squared bottom.
• Analytical solutions are presented for longitudinal and transverse oscillations.
• Effects of topographic parameters on longitudinal response are investigated.
• Transverse eigenvalues are related to the geometric and topographic parameters.

Based on the linear shallow water approximation, longitudinal oscillations in a rectangular harbor with a hyperbolic-cosine squared bottom induced by incident perpendicular waves are analytically investigated, which could be described by combining the associated Legendre functions of the first and second kinds. The effects of topographic parameters on the resonant spectrum and response are examined in detail. When the width of the harbor is of the same order magnitude as wavelengths, transverse oscillations may exist due to the wave refraction. Analytic solutions for transverse oscillations within a harbor of hyperbolic-cosine squared bottom are derived. These oscillations are typically standing edge waves. The transverse eigenfrequency is found to be related not only to the width of the harbor, but also to the varying water depth parameters.