Ch2-2.doc

2.2 Linear Solution for A Periodic Wave Train If the amplitude of a wave train is much smaller than its wavelength, then the free-surface boundary conditions can be linearized. The linearization involves two major steps. First, Equations (2.4) and (2.5) are non-dimensionalized. Secondly, they are expanded at the still water surface, z = 0 . The nonlinear terms in these two boundary conditions are neglected based on the assumption of small wave steepness. The linearized free-surface boundary conditions are given below. � f + gh = C (t ) � t � f � h =0 � z � t

at z = 0

(2.6)

at z = 0

(2.7)

The linear solution for the potential of a periodic wave train can be derived using the method of separation of variables, and the corresponding solution for wave elevation can be obtained using Equation (2.6). ag cosh [ k ( z + h) ] sin q , (2.8) s cosh kh h = a cos q , (2.9) C (t ) = 0, where q = k (cos c x + sin c y ) - s t + b , r k = �hq and k denotes the magnitude of wavenumber vector, c the direction of the q /� t , the radian wave frequency, g the wave train with respect to the x-axis, s = -� gravitational acceleration, and b the initial phase. The wavenumber and frequency are related through the dispersion relation.

f=

s 2 = gk tanh kh.

(2.10)

Since the Laplace equation and linearized boundary conditions constitute a linear problem, its solution can be derived in an alternative manner, by first ‘guessing’ a solution for the wave potential and then proving it satisfies the Laplace equation and boundary conditions. The solution obtained through ‘guessing’ and ‘proof’ is the same as that obtained through lengthy and rigorous derivation based on the method of separation of variables but much simpler. When a periodic wave train advances in the x-axis direction, the phase in the above solution reduces to,

q = kx - s t + b .

(2.11)

Knowing the potential and elevation of a wave train, a linear solution for wave induced kinematics and dynamic pressure can be calculated using Equations (2.1), and the following two equations. ur r � V � �f a= = , � t � t pd 1� f =. rg g � t

(2.12) (2.13)

It is noticed that only linear terms are retained in the computation of wave induced pressures and accelerations for the consistency with linear wave theory.