where S_F is the mean free-surface level, S_B is the hull surface below the mean free-surface level, Z is the free-surface height, and is the unit normal directed into the fluid. Eqs. (3) and (4) are the kinematic and dynamic free-surface boundary conditions, which are to be satisfied on the mean free-surface level. To ensure a unique solution, a radiation condition that no waves propagate upstream of the hull must also be satisfied.

After normalizing lengths by the length L of the ship and velocities by the speed U, the equations take the form

(10)

where F is the Froude number given by . The nondimensional kinematic and dynamic boundary conditions, eqs. (8) and (9), can be combined into the single boundary condition

which is to be applied on the mean free-surface level.

Green's second identity can be used to obtain an integral equation for the solution of eqs. (7)–(10). If the integral over a hemispherical surface beneath the mean free-surface level vanishes as that surface is expanded to infinity, then the integral equation

(12)

holds for . All integrals other than the first or third one are regular. If , then the first integral is a principal-value integral; otherwise and the third integral is a principal-value integral. Eq. (11), the combined free-surface boundary condition, was used to obtain the last term in its present form.

It is intended to solve the boundary value problem with a low-order Rankine singularity panel code in which the source and dipole distributions over each panel are uniform. Such codes often use finite differences to approximate the derivative of the potential in the last term of the integral equation. However, as is stated in the recent paper by Lechter (6), there is really no satisfactory finite difference scheme for Rankine singularity methods. The situation is worsened by the appearance of the second derivative rather than a first derivative. In this case, however, the need for finite differencing can be greatly reduced by applying Stokes ' theorem to the last term of eq. (12). Then a collocation point shifting scheme together with analytic differentiation can be used over most of the free surface. This scheme is similar to the scheme used by Jensen (7). As is pointed out in (6), there are difficulties associated with Jensen's method, especially at low speeds; even so, at the moderate to high speed flows with which this paper deals, the method is an improvement over one in which is approximated by forward finite differences. The first application of Stokes' theorem follows a simple use of the chain rule for differentiation. The result is

(13)

in which the line integral is counterclockwise around the outer edges of the computational region and clockwise around the hull. The x-derivative in front of the surface integral was obtained from the symmetry of r with respect to and and from the fact that the integration variable is independent of x. The range of integration for the surface integrals is taken to be the finite portion of the mean free-surface level which is to be paneled later. If the computational domain has sides parallel to the x-z plane, then the line integral has nonvanishing contributions only from around the hull, along the upstream boundary, and along the downstream boundary. The technique can be used again to remove the remaining derivative from inside the surface integral. The result is