On the Diffraction and Radiation of Water Waves by a Rectangular Buoy

On the diffraction and radiation of water waves by a rectangular buoy

1. General formulation

In this report, a two-dimensional problem of interaction between linear waves and a rectangular buoy at the water surface is investigated. As shown in Fig. 1, the water depth is . The half width of the buoy is and the drought is . Initially, Cartesian coordinate system is adopted in which the is set at the still water level and measured upwardly which passes through the center of the buoy.

Based on the assumptions that the fluid is incompressible, inviscid and the flow is irrotational, the wave motion can be described by the velocity potential , which satisfies the Laplace equation:

(1)

where is the gradient operator.

In the present problem, consider the case that a train of linear wave propagates to the negative x direction. The corresponding velocity potential of the incident wave can be expressed as

(2)

where, Re is the real part of the complex function; ; is the gravitational acceleration; is the incident wave amplitude; is the frequency, which must satisfy the dispersion relation

(3)

where K is the wave number.

Furthermore, the velocity potential can be assumed to be time harmonic with angular frequency , and is represented as:

(4)

where,

(5)

Meanwhile, the velocity potential can be expressed as

(6)

where is the spatial velocity potential.

It is assumed that the amplitude of the wave motion is small, hence, the linear wave theory is valid and can be decomposed into three parts, the incident wave potential , the diffraction potential and the radiated potential , respectively. So can be expressed as

(7)

where, stands for heave, 2 for sway and 3 for roll.

1.1. General formulation for diffracted potential

The diffracted potential satisfies the following governing equation and boundary conditions at the free surface, the bed, the buoy surface as well as infinity which can be expressed, respectively, as:

(8)

(9)

(10)

(11)

(12)

(14)

1.2. General formulation for radiated potentials

Small amplitude of the rectangular is assumed too. So, If the amplitude of the motion of the buoy is denoted by , the radiated potential can be expressed as

(15)

where, is the spatial velocity potential too, which satisfies the following governing equation and the boundary conditions

(16)

(17)

(18)

(19)

(20)

(21)

where, is assumed as the center of rotation and the is given by

(22)

2. Analytic Solutions

2.1. expression for the potentials

To solve the above problem analytically, the fluid domain can be divided into three regions as

depicted in Fig. 1, which are , and , respectively.

The corresponding diffracted potentials are assumed as , which can be obtained from the method of separation of variables and expressed as:

(23)

(24)

(25)

where , and are given by

(26)

(27)

(28)

Following Lee (1995), the diffracted potential in region ? can be divided into two parts, the general solution which satisfies the equation (8) and the homogenous condition in region ?, and the particular solution which satisfies the equation (8) and the