Analytical solutions of tsunamis generated by underwater earthquakes

Tsunamis induced by underwater earthquakes are theoretically analyzed by applying the linear potential theory. Special attention is placed on the initial state of tsunami. For instantaneous seabed deformations, analytical wave solutions induced by three fundamental seabed deformations at initial stage are derived rather than integral expressions in past studies. These analytical solutions constitute a fundamental base for analyzing waves generated by arbitrary seabed displacement with the help of Fourier analysis. Tsunamis induced by non-instantaneous seabed deformation are analyzed as well. For the sake of examining the contributions of all wave components involved in the tsunami waveform, the amplitude density is proposed to examine the effects of deformation width, water depth, harmonic mode and rising time on waveforms. Results show that a larger ratio of water depth to deformation width results in a greater difference between initial waveform and seabed deformation, and the effect of the rising time is significant in deeper-water configuration. For cosine and sine seabed liftings, the effects of higher harmonic modes might be ignored.