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The only work to date to consider the case o f wave interaction with a current possessing arbitrarily wide horizontal shear is that of Thomas (2001). The model has been developed for linear or near-hnear waves under the moderate current approximation. This requires that the wave slope be small, e = a k « 1, that the characteristic current magnitude, Û, be small compared to the wave phase velocity, Ô = Ûlc

« 1 and that the two parameters satisfy the condition that e « 0 « \ . N o constraint is placed on the scale of the horizontal variation o f the current nor on the nature o f the vertical variation in the formulation of the model. However, the recovered transmission coefficient is dependent on the total mass flux o f the current and requiring the transmission coefficient to be physically meaningful, imposes a limitation on the current magnitude and horizontal extent.

The model is formulated for the interaction between a jet-like current and waves in the shallow water, finite depth and deep water regimes. For each regime solutions to the governing equations were obtained for the local wave kinematics in terms o f the Fourier transform o f the arbitrary current profile

U(y,z). Expressions for the reflection and transmission coefficients are obtained from consideration o f the far-field asymptotic behaviour. The difficulty o f establishing analytical forms o f the Fourier transform, and its subsequent inversion, Umits the forms o f the current profile for which analytical solutions, expressed directly in terms of the current profile U(y,z), are possible. However, numerical evaluation o f the solution is possible for arbitrary profiles.

The formulation presented here is that for finite depth. A s in previous sections a coordinate system is chosen such that the x- and y-axis lie on the undisturbed free surface where the z-axis, defined as positive upward, has its origin. The current is assumed to be jet-like and o f the form U.=(U(y,z), 0,0), that is flowing in the positive x-direction but varying in both the span-wise and vertical directions (see figure 2.4). The bed is taken to be horizontal and located at z=-d. Regular waves o f frequency co and wavenumber vector k={l,m,0) are generated on still water and meet the current at an arbitrary angle a. It is assumed that caustics are not formed in the flow.

For an in viscid, incompressible fluid of constant density and negligible surface tension, the governing equations of motion are the Euler equation (equation [2.1]) and the continuity equation (equation [2.2]) together with the usual boundary conditions at the bed and the free-surface. Solutions for the velocity, pressure and surface elevation are sought in terms o f a perturbation expansion in the two parameters

e and â o f the form

U = ô u ^ + 6 ^ ( m , q + . . . ) + . . . ) + . . .

p = - p g z + (^^02 + % ] + g(Pio+ •••) + - [2.111]

77 = J ^ 7 7 o 2 + ^ ^ 0 1 + ^ ( ^ 1 0 + ^ ^ 1 1 + • • • ) + ^ ( ^ 2 0 + ^ ^ 2 1 + ” • ) + -

The wave phase speed is also expanded in a similar manner,

c — + Cqq + £t(C]q+ <5cj j +. . . ) + ... [2.112]

Substituting these expansions into the equations o f motion and boundary conditions and grouping terms of similar order results in a hierarchy o f equations governing the motion at each order. At order

Ô the motion is that of the current alone. The order e equations govern the wave alone, or incident wave, motion. The primary interaction between the wave and the current, that is the transmitted and reflected wave, is governed by the order eô equations.

For the wave alone motion, the solution to the governing equations at order e for the surface elevation and pressure correspond to the usual linear Stokes’ theory,

t/jq = k~^cos(lx + m y - cot)

P .11 3]

C(J) = gk~^ tanh W

This represents a physical wave propagating in the positive y-direction with amplitude a. For the primary interaction between wave and current, the Euler equation and the continuity equation at order

eô are respectively

^ [2,114]

[2.115]

In the Euler equation the temporal derivative, d/dt has been replaced by where 0 = I x - cot is

the phase function. Applying the operator V to the Euler equation at this order yields

[ ( f / g. V ) + (M]o 5 )1 /^ ] [2.116]

after making use o f the continuity equation. The form o f the solutions for interaction quantities and 77^ 2 are assumed to be

/7jj(x,y,z,r) = F (y ,z )e '‘" and ri^^{x,y,t) = N { y ) è ^ [2.117]

where the real parts are to be taken. The constant factor to the amplitude function P{y,z) has been chosen to simplify the following equations, and consequently P{y,z) has dimensions o f velocity. Introducing the proposed solution for and the order e solutions, given in equation [2.113], into equation [2.116] yields

dy^ dz^

im cosh [/:(z+J)] sinh[/:(z+J)] , i m y

[2.118]

k cosh kd dy cosh kd dz

to be solved for Piy,z). At order eô, the bed boundary condition can be written as

- ^ = 0 at z = -^ [2.119]

while the dynamic and kinematic free surface boundary conditions yield

M(y) = at Z = 0 [2.120]

A C Cq o

- AtanhM P = -llU^idirhkd e'"*^ at z = 0 [2.121]

respectively. In the above, the wavenumber magnitudes, k, I and m, are constants taking the values for the incident wave on still water. Any variation in the pressure due to the presence o f the current Uq,

is contained in amplitude function P(y,z). Solving equation [2.118] for P(y,z) subject to the bed boundary condition, the kinematic free surface boundary condition and the application o f a radiation condition to ensure that the transmitted and reflected waves propagate away from the interaction zone yields

P ( y , z ) = [A*e™^ + /l-e-"">']C(fcz) + - ^ j L ^ ( a , z ) e - “‘’' d a [2.122]

The details o f the solution are presented in appendix 2C. Here, and A' are complex constants,

C{k,z) =cosh[k(z+d)] andLg(a,z) is defined in equation [2C.16]. Given this solution for P(y,z), the surface elevation perturbation, is obtained directly from the dynamic free surface boundary condition. However, evaluating the integral term in the solution for P(y,z) is not straightforward. The function L ^ a , z ) itself depends on several integrals involving the Fourier transform of the function

V{y,z) = f/Q(y,z)e""^. The analytical evaluation of these integrals is only possible if the current profile belongs to a small number of profiles with particular forms. For an arbitrary current profile it will be necessary to resort to a numerical evaluation.

Fortunately, regardless of whether analytical or numerical solutions are being sought, the integrals can be sim plified if it is assumed that the horizontal and vertical variation o f the current profile are separable - that the profile can be expressed in the form

% (y,z) = U J { y ) Z { z ) [2.123]

for some constant such that the functions Y{y) and Z(z) are o f order one. The simplest situation is to assume Z(z) = 1, in other words that current has no depth variation. Then the function L ^ a , z )

evaluated at the free surface, z=0, becomes

m T(k)

L J a , 0 ) = 21 V J a ) _[2.124]