The analysis of interaction between waves and ship by singularity distribution method involves calculation for Green function [1,2]. After computer was used to compute hydrodynamics, finding a fast way to calculate Green function became a science research work [3,4]. Newman [5] published theories and numeric methods for computing the velocity potential, and its derivatives, for linearized wave motions due to a unit source with harmonic time dependence beneath a free surface. Shen [6] gave an approximated algorithm to estimate Green function and its derivatives by using truncated series expansion of the Green function were to avoid the conventional timeconsuming numerical integration. Zhu [7] showed a subdomain approximate method to evaluate frequency domain free surface Green function with sufficient accuracy. Yang [8] compared the classical Green function and a simpler Green function associated with the linearized free-surface boundary condition for diffraction radiation by a ship advancing through regular waves. Shen [9] proposed the ordinary differential equations about depth Green function and its derivative, and a rapid Green function calculation method combining solving ordinary differential and interpolation between nodes. John [10,11] showed a variety of representations for Green function with finite and infinite water depth. Other expressions for the free-surface Green’s function, in two dimensions and for infinite water depth, have been improved by Liu [12], Thorne [13], Kim [14], Greenberg [15], Macaskill [16], and Dautray and Lions [17]. Some representations for the Green’s function are also discussed in [18-21]. A more general two-dimensional water-wave problem that considers surface tension is treated in [22-24].

This paper will discuss mathematic model of Green function with two dimension free water surface. In Section 1, the Green function is represented by using two parameters. In Section 2, intrinsic properties of Green function are discussed. In Section 3, special value of Green function is given for vertical line and horizontal line. In Section 4, the derivation of Green function is obtained for two dimension free water surface.

Suppose velocity potential φ satisfies Laplace equation:

Here P is field point,. Q is source point,. The right of equation is delta function. The boundary condition is:

Here F is complex potential. On the free surface y = 0, velocity potential satisfies linear condition. We easy find its solution [12]:

Here is called green function, I is principle value integration of below

where the integral lower limit is 0, The integral upper limit is ¥, is zero point of denominator, upper subscribe (¥, K) shows that it is principle value integration at point. From the expression of integration, the value I is determined by the parameters of K, ,. Let unit transfer as

The principle value integration may be written as

where is called H-function with two parameters:

here parameters take value as.

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Let Re(H) and Im(H) are real part and imaginary part of respectively. We will discuss Re(H) and Im(H) with parameters a and d.

By using expression of H-function, we have below theorem:

Theorem 3. Consider field point and source point are below free surface, the Green function may be expressed as:

Here

where constant C_a = –0.577215.

We know, the derivation of potential are:

It is easy to obtain the derivation of Green function as:

Consider field point and source point ζ = ξ + iη take value at vertical line S:, , we have formula of two parameter:

In the paper, the Green function is simplified from integral formula by using two parameters. The intrinsic properties of Green function are discussed on vertical line and horizontal line. The derivation of Green function is obtained by using complex theory.