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Oceanic ridges could act as waveguides transferring tsunami energy to thousands of kilometers away, pumping large energy in to far-field regions as a secondary source. The shallow-water wave velocity $c={(gh)}^{1/2}$ can only predict the arrival time of the early signals accurately but can hardly estimate that of ridge-trapped waves. The present study provides a fast analytic prediction method to estimate the arrival time of the subsequent large trapped waves. The method is based on the energy velocity solution of trapped waves over the uniform parabolic-shaped submerged ridge. Records of two-tide gauge stations, located nearby Sand Island and Nawiliwili for the 2011 Tohoku tsunami are chosen to illustrate the application of this method. The Sand Island record shows typical open-ocean island characteristics that the maximum wave height is followed by rapid amplitude decay. While the Nawiliwili record is strongly affected by topographical trapping effect of the Hawaii Ridge, and several subsequent wave trains carrying large energy arrive within one day duration. Further investigations show that the present analytic method is able to estimate the arrival time for these distinguished subsequent trains.

Due to refraction, the oceanic ridge acts as a waveguide forcing long-period waves to propagate along the topography. Based on the linear shallow water equations, analytical solutions of trapped waves over a hyperbolic-cosine squared ocean ridge are obtained, which are described by combining the associated Legendre functions of the first and second kinds. The spatial distribution pattern for each mode is discussed, and the wave amplitude gets the maximum at the ridge top and decays gradually towards both sides. The higher the mode number, the slower the rate of amplitude decreases, so that more energy is distributed over more of the ridge. The trapped wave number is not only related to the frequency, but also to the varying water depth parameters.