Initial Stage of the Finite-Amplitude Cauchy–Poisson Problem

The nonlinear Cauchy–Poisson problem for an incompressible inviscid fluid to start flowing under gravity is investigated analytically. The general nonlinear initial/boundary-value problem is formulated, including both an initial surface deflection and an initial velocity generated by a pressure impulse on the surface. Two subproblems are: (1) a finite-amplitude surface deflection released from rest; (2) the fluid is forced into motion by a pressure impulse on the initially horizontal surface. Solutions for these two subproblems are given to the leading order. One exact solution is given for the fully nonlinear initial-value problem, where a surface pressure impulse is applied on a surface with finite initial deflection. The concept of the highest non-breaking wave is illustrated by dipole acceleration fields at a state of gravitational release from rest. This is done for two phenomena: run-up of a non-breaking solitary wave on a sloping beach, and free nonlinear sloshing in an open container.