A digital computer is applied to three hydraulic engineering problems with free boundaries. The problems considered are the refraction of periodic waves approaching a coastline, the supercritical flow of water in a spillway bucket, and seepage flow through an anisotropic earth embankment. In each case a solution was sought by use of the numerical methods of finite differences. The application to wave refraction was successful and a fully tested program which solves the refraction equation and the wave intensity equation for arbitrary bottom shapes is presented. A flow chart together with a program listing in FORTRAN IV is given. The application to spillway flows which uses a transformation of variables was also successful though the program has not been fully tested for all configurations. A flow chart and program listing are also given. The application to seepage flows was not completely successful. The program, written in RUSH PL/I, is listed. An appendix is attached in which the cavity shape parameters for the image model of a supercavitating wedge are tabulated, for cavitation numbers in the range 0.05 to 1.0 with wedge angles ranging from 5 to 90 degrees in each case. The results obtained have been compared with other known solutions where possible with good agreement. (Author).