Generation of two-dimensional unsteady motion in a viscous incompressible fluid of finite depth

The problem of generation of two-dimensional unsteady motion in a viscous incompressible fluid of finite depth is investigated here. The motion is generated due to initial disturbances in the form of prescribed surface pressure or displacement at the free surface. The Fourier transform with respect to space and the Laplace transform with respect to time are used to obtain the form of the free surface in terms of multiple integrals. Finally, an asymptotic form of the free surface is obtained using the method of steepest descent.