the ship hull including the propeller influences. In this panel method, a quadratic dipole distribution is distributed over each panel on the body surface, and an internal Dirichlet boundary condition is imposed at the control point of each panel. The pressure fluctuations on the hull surface can then be calculated from solutions of the dipole strength by applying Bernoulli's equation.

In order to calculate the propeller generated pressure fluctuations, we need first to develop a numerical method to represent the ship hulls. In the present work, the free surface effect is approximated by a positive or negative hull image. Therefore, it is equivalent to solving the flow around submerged bodies. Hess and Smith [4] first developed a velocity based panel method to calculate the flow around a submerged body. In their method, the source singularities are distributed over the body surface, and the Neumann boundary condition is imposed for solutions. In the present work, the flow around a ship hull is analyzed by a higher order, potential based panel method. A dipole sheet is distributed on the surface of hull, and an internal Dirichlet boundary condition is imposed.

Assuming the flow around, the body is incompressible, inviscid, and irrotational, namely, potential flow, then the governing equation of this flow is the Laplace's equation:

^{2}=0 (1)

where is the velocity potential.

If we let to be the external flow velocity potential, and *ˉ* to be the internal flow velocity potential (Figure 1), then from Lamb [13], and Kerwin, etc.[10], we have the following equation:

where *P* is a field point in the flow field, *Q* is a point on the body surface, and *R* is the distance between *P* and *Q*. In equation (2), we can interpret the term *[**(Q)–**ˉ(Q)]* as a dipole strength, and the term as a source strength. Most of panel methods used in the fluid dynamics area are based on equations derived from equation (2) with a proper boundary condition imposed [6].

When solving the problems of non-lifting bodies, velocity based panel methods are commonly used. That is, a source sheet is distributed on the body surface, and the Neumann boundary condition is imposed.

(3)

If we define the total potential as the sum of the inflow potential, _{∞}, and the perturbation potential, *,* then, we can rewrite equation (3) as follows:

(4)

where is the total velocity.

However, for the convenience of calculating the pressure fluctuations on the hull surface, a potential based panel method which a dipole sheet distributed on the body surface is preferred [2]. As we know, the normal velocity is continuous through a dipole sheet, therefore,

(5)