On water waves generated by a bottom obstacle translating at a subcritical speed

This study investigates water waves generated by a bottom obstacle translating at a subcritical speed in constant water depth, using a combination of analytical and numerical approaches. The newly derived analytical solutions reveal two types of waves - the transient free waves that propagate radially outwards, and the trapped wave that stays on top of the translating bottom obstacle. Closed-form asymptotic solutions for both the free surface and the flow velocities are derived in the far field, and near the leading wave or in the shallow water limit. The far-field leading waves are mathematically shown to be insensitive to the exact shape of the obstacle. Numerical long-wave models are employed to examine effects unaccountable by the linear analytical solutions. Nonlinear effects are found to cause only small deviations from the linear solutions. The effects of the obstacle's acceleration and deceleration are also examined numerically. Overall, the idealised linear analytical solutions predict well the characteristics of water waves generated by a bottom obstacle, and therefore can serve as the cornerstone of a theory-based model for quickly predicting the tsunamis generated by a submarine landslide.