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# Loading and Constraints

is the elevation of the point considered, $\rho_{w}$ is the mass density of the fluid outside the pipe, $\rho_{f}$ is the mass density of the fluid inside the pipe, $z_{w}$ is the free surface elevation of the fluid outside the pipe, $z_{f}$ is the free surface elevation of the fluid inside the pipe, t is the outward normal to the exposed area, and g is the gravitational acceleration. In ABAQUS it is assumed that $\mathbf{g} = -g\mathbf{k}$ .

The buoyancy force is

$$
\mathbf {F} _ {p b} = - g \Delta A \left[ f _ {1} \rho_ {w} (z _ {w} - z) - f _ {2} \rho_ {f} (z _ {f} - z) \right] \mathbf {t},
$$

where

$$
f _ {1} = \left\{ \begin{array}{l l} 0 & \text { if   the   elevation   is   above   } z _ {w} \\ 1 & \text { otherwise } \end{array} \right.
$$

and

$$
f _ {2} = \left\{ \begin{array}{l l} 0 & \text {if the elevation is above z_{f}} \\ 1 & \text {otherwise.} \end{array} \right.
$$

# Load stiffness

To ensure the quadratic convergence of the Newton method in ABAQUS, it is necessary to calculate the changes in the above forces with respect to changes in the kinematic solution for the structure. Thus, a load stiffness is calculated for all of these load types.

# 6.2.2 Airy wave theory

This is a linearized wave theory based on irrotational flow of an inviscid incompressible fluid. The linearization is achieved by assuming the wave height a is small compared to the wavelength ¸ and the still water depth. It is also assumed that the fluid is of uniform depth (that is, the bottom is flat).

Since we have irrotational flow, there exists a flow potential, Á, obeying

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Equation 6.2.2-1

$$
\bigtriangledown^ {2} \phi = 0
$$

and giving the fluid particle velocities as

Equation 6.2.2-2

$$
\mathbf {v} = \frac {\partial \phi}{\partial \mathbf {x}}.
$$

Now assume there is a potential energy per unit mass, G, (in this case associated with the gravity field). Then, equilibrium is given by

Equation 6.2.2-3

$$
\rho \left[ \frac {\partial \mathbf {v}}{\partial t} + \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \cdot \mathbf {v} \right] = - \rho \frac {\partial G}{\partial \mathbf {x}} - \frac {\partial p}{\partial \mathbf {x}},
$$

where $\rho$ is the fluid density and p is the pressure. Substituting Equation 6.2.2-2 in Equation 6.2.2-3, we obtain

$$
\rho \left[ \frac {\partial^ {2} \phi}{\partial \mathbf {x} \partial t} + \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \cdot \mathbf {v} \right] = - \rho \frac {\partial G}{\partial \mathbf {x}} - \frac {\partial p}{\partial \mathbf {x}}.
$$

This equation can be integrated with respect to position to give

Equation 6.2.2-4

$$
\rho \left[ \frac {\partial \phi}{\partial t} + \frac {1}{2} \mathbf {v} \cdot \mathbf {v} \right] = - \rho G - (p - p _ {0}) + F (t),
$$

since the fluid is assumed to be incompressible; thus, $\rho$ is constant. Here F is an arbitrary function of t and $p _ { 0 }$ is the pressure in the air just above the free surface. For convenience we choose $F ( t ) = 0$ :

The term $1 / 2 { \bf v } \cdot { \bf v }$ can be neglected compared to the other terms (this can be shown, from the resulting solution, to be consistent with the order of approximation that the wave height is small compared to the wavelength). By choosing the z-coordinate to point vertically upward, the gravity potential is conveniently chosen as

$$
G = g (z - z _ {s}),
$$

where $z _ { s }$ is the undisturbed surface level. Equation 6.2.2-4 then becomes

Equation 6.2.2-5

$$
\rho \frac {\partial \phi}{\partial t} + \rho g (z - z _ {s}) + p - p _ {0} = 0.
$$

From this equation the total pressure at a point below the instantaneous fluid surface is

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# Loading and Constraints

$$
p = p _ {0} + \rho g (z _ {s} - z) + p _ {d y n}.
$$

Hence, the total pressure is the air pressure plus the hydrostatic pressure plus the dynamic pressure, $p _ { d y n }$ , where $p _ { d y n }$ is given by

$$
p _ {d y n} = - \rho \frac {\partial \phi}{\partial t}.
$$

Let ´ be the elevation of the fluid surface at time t above the mean (undisturbed) fluid surface level, $z _ { s }$ . Since the position of the free surface is a part of the solution, there are two boundary conditions that must be applied on $z = \eta$ . The first is a dynamic equilibrium condition at the interface between the fluid and the air. Since the interface is assumed to have no mass, the forces normal to the interface in the fluid and the air must be equal. If the surface tension of the interface is neglected, the pressure in the water and the air must be equal at the interface. Assuming that the pressure due to the motion of the air is negligible (which can be shown to be reasonable), the air pressure can be approximated by its undisturbed value (Whitham, 1974). The dynamic boundary condition then implies $p = p _ { 0 }$ , where p is the pressure in the water at the free surface and $p _ { 0 }$ is the pressure in the undisturbed air. Since ´ is assumed to be small with respect to the depth of fluid, the boundary condition can be made linear by applying it at $z = z _ { s }$ instead of at $z = z _ { s } + \eta$ .

With these assumptions Equation 6.2.2-5 provides the boundary term

Equation 6.2.2-6

$$
\eta = - \frac {1}{g} \frac {\partial \phi}{\partial t} \bigg | _ {z = z _ {s}}.
$$

The second boundary condition on the free surface comes from the kinematics of the free surface. Let the free surface be given by

$$
f (x _ {1}, x _ {2}, z, t) = \eta (x _ {1}, x _ {2}, t) - (z - z _ {s}) = 0.
$$

The velocity of the fluid normal to the surface must be equal to the velocity of the surface normal to itself.

Differentiating this expression with respect to time yields

$$
\frac {\partial \eta}{\partial t} - \frac {\partial z}{\partial t} = 0.
$$

If we assume that the wave height is small compared to the wavelength $( \eta _ { m a x } < < \lambda )$ ; then we can approximate $\frac { \partial z } { \partial t }$ by the velocity $v _ { z }$ , so that

Equation 6.2.2-7

$$
\frac {\partial \eta}{\partial t} - v _ {z} = \frac {\partial \eta}{\partial t} - \frac {\partial \phi}{\partial z} = 0 \mathrm{at} z = z _ {s}.
$$

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Eliminating ´ between Equation 6.2.2-6 and Equation 6.2.2-7 gives

Equation 6.2.2-8

$$
\frac {\partial^ {2} \phi}{\partial t ^ {2}} + g \frac {\partial \phi}{\partial z} = 0 \mathrm{at} z = z _ {s}.
$$

The boundary condition on the bottom, $z = z _ { b }$ , is

Equation 6.2.2-9

$$
v _ {z} = \frac {\partial \phi}{\partial z} = 0 \mathrm{at} z = z _ {b}.
$$

The problem is now defined by Equation 6.2.2-1, Equation 6.2.2-8, Equation 6.2.2-9, and the requirement that the solution be a plane wave periodic in the horizontal plane, such that

$$
\phi (x _ {1}, x _ {2}, z, t) = \phi (\mathbf {d} \cdot \mathbf {x} - c t, z),
$$

where d is the direction of wave propagation and c is the wave speed.

We solve the problem by assuming that $\phi = P ( z ) \Phi ( x _ { 1 } , x _ { 2 } , t )$ . Since P and © are independent functions, Equation 6.2.2-1 provides the two equations

$$
\frac {d ^ {2} P}{d z ^ {2}} - k ^ {2} P = 0,
$$

$$
\frac {\partial^ {2} \Phi}{\partial x _ {1} ^ {2}} + \frac {\partial^ {2} \Phi}{\partial x _ {2} ^ {2}} + k ^ {2} \Phi = 0,
$$

where k is a constant.

Let d ¢ $\mathbf { x } = s$ (so that s measures distance in the direction of travel of the wave). The solution to these equations is $P = ( A _ { 1 } \cosh k z + A _ { 2 }$ sinh kz¢ and $\begin{array} { r } { \Phi = \sin \big [ k ( s - c t ) + \frac { 2 \pi } { 3 6 0 } \alpha \big ] } \end{array}$ ; hence,

Equation 6.2.2-10

$$
\phi = \left(A _ {1} \cosh k z + A _ {2} \sinh k z\right) \sin \left[ k (s - c t) + \frac {2 \pi}{3 6 0} \alpha \right],
$$

where $A _ { 1 }$ and $A _ { 2 }$ are constants and ® is the phase angle of the wave in degrees (® provides an arbitrary choice of origin in time and is chosen so that the vertical displacement of a fluid particle is a minimum when $s = 0 , t = 0$ , and ® = 0).

There is no motion at the bottom of the fluid in the vertical direction, so by Equation 6.2.2-9 we find

$$
A _ {1} = - \frac {A _ {2}}{\tanh k z _ {b}}.
$$

Substituting this into Equation 6.2.2-10 gives

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# Loading and Constraints

Equation 6.2.2-11

$$
\phi = C \cosh \left[ k (z - z _ {b}) \right] \sin \left[ k (s - c t) + \frac {2 \pi}{3 6 0} \alpha \right],
$$

where C is a constant.

The dispersion relation can be obtained by substituting Equation 6.2.2-11 into Equation 6.2.2-8 and setting $z = z _ { s } ,$ , giving

Equation 6.2.2-12

$$
c ^ {2} = \frac {g}{k} \tanh \left[ k (z _ {s} - z _ {b}) \right].
$$

The wave frequency $\omega$ is related to the wave period ¿ by $\omega = 2 \pi / \tau$ . The constant k is called the wave number and is related to the wavelength ¸ by $k = 2 \pi / \lambda$ , so that $\omega _ { } ^ { } = c k$ .

The free surface elevation above the undisturbed fluid surface, ´, is given by Equation 6.2.2-6:

Equation 6.2.2-13

$$
\eta = C \frac {k c}{g} \cosh \bigl [ k (z _ {s} - z _ {b}) \bigr ] \cos \bigl [ k (s - c t) + \frac {2 \pi}{3 6 0} \alpha \bigr ].
$$

Writing the wave amplitude (half the wave height) as $^ { a , }$ this defines

$$
a = - C \frac {k c}{g} \cosh \left[ k (z _ {s} - z _ {b}) \right]
$$

so that Equation 6.2.2-11 can be rewritten

Equation 6.2.2-14

$$
\begin{array}{l} \phi = - \frac {g a}{k c} \frac {\cosh [ k (z - z _ {b}) ]}{\cosh [ k (z _ {s} - z _ {b}) ]} \sin [ k (s - c t) + \frac {2 \pi}{3 6 0} \alpha ] \\ = - \frac {g a \tau}{2 \pi} \frac {\cosh [ (2 \pi / \lambda) (z - z _ {b}) ]}{\cosh [ (2 \pi / \lambda) (z _ {s} - z _ {b}) ]} \sin 2 \pi \big (\frac {s - c t}{\lambda} + \frac {\alpha}{3 6 0} \big). \\ \end{array}
$$

This solution provides fluid particle velocities $\mathbf { v } = \partial \phi / \partial \mathbf { x }$ and accelerations throughout the fluid for this one wave component. The term $\frac { 1 } { 2 } \left( { \bf v } \cdot { \bf v } \right)$ in Equation 6.2.2-4 was neglected because the wave amplitude a is small compared to the wavelength ¸. This implies, from Equation 6.2.2-3, that the fluid particle acceleration is approximated as $\mathbf { a } = \partial \mathbf { v } / \partial t$ ; that is, the convective part of the acceleration, $\partial \mathbf { v } / \partial \mathbf { x } \cdot \mathbf { v }$ , is neglected.

Since the theory is linear, any set of waves can be superposed by linear superposition of its components:

$$
\phi = \sum_ {\mathrm{components}} \phi_ {N},
$$

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# Loading and Constraints

where $\phi _ { N }$ is the potential Equation 6.2.2-10 of the $N ^ { \mathrm { t h } }$ wave component. Hence, the theory can be summarized as follows.

For $z \leq z _ { s }$

Velocity potential:

$$
\phi = - \frac {g}{2 \pi} \sum_ {\mathrm{components}} a _ {N} \tau_ {N} \frac {\cosh [ (2 \pi / \lambda_ {N}) (z - z _ {b}) ]}{\cosh [ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) ]} \sin 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right).
$$

Fluid particle displacements:

horizontally,

$$
u _ {i} = \frac {g}{2 \pi} \sum_ {\mathrm{components}} d _ {N i} \frac {a _ {N} \tau_ {N} ^ {2}}{\lambda_ {N}} \frac {\cosh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \sin 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right),
$$

and vertically,

$$
u _ {z} = - \frac {g}{2 \pi} \sum_ {\mathrm{components}} \frac {a _ {N} \tau_ {N} ^ {2}}{\lambda_ {N}} \frac {\sinh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right).
$$

Fluid particle velocities:

horizontally,

$$
v _ {i} = - g \sum_ {\mathrm{components}} d _ {N i} \frac {a _ {N} \tau_ {N}}{\lambda_ {N}} \frac {\cosh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right),
$$

and vertically,

$$
v _ {z} = - g \sum_ {\mathrm{components}} \frac {a _ {N} \tau_ {N}}{\lambda_ {N}} \frac {\sinh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \sin 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right).
$$

Fluid particle accelerations:

horizontally,

$$
a _ {i} = - 2 \pi g \sum_ {\mathrm{components}} d _ {N i} \frac {a _ {N}}{\lambda_ {N}} \frac {\cosh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \sin 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right),
$$

and vertically,

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# Loading and Constraints

$$
a _ {z} = 2 \pi g \sum_ {\mathrm{components}} \frac {a _ {N}}{\lambda_ {N}} \frac {\sinh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right).
$$

Free surface profile:

$$
z = - \sum_ {\mathrm{components}} a _ {N} \cos 2 \pi \bigg (\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0} \bigg).
$$

Dispersion relation for each mode:

$$
\frac {2 \pi}{\tau_ {N} ^ {2}} = g \frac {1}{\lambda_ {N}} \tanh \bigl [ \frac {2 \pi}{\lambda_ {N}} (z _ {s} - z _ {b}) \bigr ].
$$

Dynamic pressure:

$$
p _ {d y n} = - \rho g \sum_ {\mathrm{components}} a _ {N} \frac {\cosh [ (2 \pi / \lambda_ {N}) (z - z _ {b}) ]}{\cosh [ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) ]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right),
$$

$$
\frac {\partial p _ {d y n}}{\partial z} = - 2 \pi \rho g \sum_ {\mathrm{components}} \frac {a _ {N}}{\lambda_ {N}} \frac {\sinh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right),
$$

and

$$
\frac {\partial^ {2} p _ {d y n}}{\partial z ^ {2}} = - 4 \pi^ {2} \rho g \sum_ {\mathrm{components}} \frac {a _ {N}}{\lambda_ {N} ^ {2}} \frac {\cosh \left[ (2 \pi / \lambda_ {N}) (z - z _ {b}) \right]}{\cosh \left[ (2 \pi / \lambda_ {N}) (z _ {s} - z _ {b}) \right]} \cos 2 \pi \left(\frac {s _ {N}}{\lambda_ {N}} - \frac {t}{\tau_ {N}} + \frac {\alpha_ {N}}{3 6 0}\right).
$$

Airy wave theory is a linearized theory; however, the wave amplitude can be large compared with the size of a structure. We, therefore, must make an assumption about the wave kinematics below a crest and above the mean water level. The assumption used here follows modified Airy wave theory as described in Hansen (1988) and Faltinsen (1990). The free surface boundary condition has been made linear in Equation 6.2.2-6. Above the mean or undisturbed surface level $z _ { s }$ the velocity, acceleration, and dynamic pressure are extrapolated from their values at the mean surface level. Hence, for

$$
z _ {s} <   z \leq \eta
$$

$$
\mathbf {v} = \left. \mathbf {v} \right| _ {z _ {s}}, \left. \mathbf {a} = \mathbf {a} \right| _ {z _ {s}}, \mathrm{and} p _ {d y n} = p _ {d y n} \left. \right| _ {z _ {s}}.
$$

Accordingly,

$$
\frac {\partial p _ {d y n}}{\partial z} = 0 \quad \mathrm{and} \quad \frac {\partial^ {2} p _ {d y n}}{\partial z ^ {2}} = 0.
$$

When the \*WAVE option is used, the penetration of the structure into the fluid must be calculated. Although the Airy wave theory assumes that the fluid displacements are small with respect to the wavelength and the fluid depth, they cannot be small with respect to the dimensions of the structure

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immersed in the fluid. Hence, the instantaneous fluid surface is used to determine if a point on the structure sees loads due to the presence of the fluid.

The Airy wave field is a spatial description of the wave field. The wave field defines velocity, acceleration, and dynamic pressure at spatial locations for all values of time. Hence, the velocity, acceleration, and dynamic pressure are determined by using the current (for geometrically nonlinear analysis) or reference (for geometrically linear analysis) location of the structure at the current time in the appropriate equations. The time used in the wave field equations is the total time for the analysis, which accumulates over all steps in the analysis (\*STATIC, \*DYNAMIC, etc.).

# 6.2.3 Stokes wave theory

Assume that an infinite series of plane, uniform waves travels through the fluid in the positive S-direction. The z-coordinate is chosen to be positive in the vertical direction, so the gravity potential is $G = g ( z - z _ { 0 } )$ , where $z _ { 0 }$ is an arbitrary datum.

Assume that the fluid is inviscid and incompressible. The fluid particle velocities are derivable from a flow potential

Equation 6.2.3-1

$$
\bigtriangledown^ {2} \phi = 0
$$

$$
\mathbf {v} = - \frac {\partial \phi}{\partial \mathbf {x}}.
$$

Equilibrium is

$$
\rho \left(\frac {\partial \mathbf {v}}{\partial t} + \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \cdot \mathbf {v}\right) = - \rho \frac {\partial G}{\partial \mathbf {x}} - \frac {\partial p}{\partial \mathbf {x}},
$$

where $\rho$ is the fluid density and $p$ is the pressure. Writing ${ \partial \mathbf { v } } / { \partial t }$ in terms of the flow potential then gives

$$
\rho \left(\frac {\partial^ {2} \phi}{\partial \mathbf {x} \partial t} - \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \cdot \mathbf {v}\right) = \rho \frac {\partial G}{\partial \mathbf {x}} + \frac {\partial p}{\partial \mathbf {x}}.
$$

Integrating with respect to $\mathbf { x }$ (note that $\rho$ is constant since the fluid is incompressible) gives the Bernoulli equation

$$
\frac {\partial \phi}{\partial t} - \frac {1}{2} \mathbf {v} \cdot \mathbf {v} = G + \frac {p - p _ {0}}{\rho} + g F (t),
$$

where $F ( t )$ is an arbitrary function (which for convenience is set to zero) and $p _ { 0 }$ is the atmospheric pressure. Substituting in the gravity potential, this is

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# Loading and Constraints

$$
\frac {\partial \phi}{\partial t} - \frac {1}{2} \mathbf {v} \cdot \mathbf {v} = g (z - z _ {s}) + \frac {p - p _ {0}}{\rho},
$$

where $z _ { s }$ is the undisturbed surface level. From this equation the total pressure at a point below the instantaneous fluid surface is

$$
p = p _ {0} + \rho g (z _ {s} - z) + p _ {d y n}.
$$

Hence, the total pressure is the air pressure plus the hydrostatic pressure plus the dynamic pressure, $p _ { d y n }$ , where $p _ { d y n }$ is given by

$$
p _ {d y n} = \rho \left(\frac {\partial \phi}{\partial t} - \frac {1}{2} \mathbf {v} \cdot \mathbf {v}\right).
$$

Let ´ be the elevation of the free surface above this level. At the free surface the Bernoulli equation is

$$
\left(\frac {\partial \phi}{\partial t} - \frac {1}{2} \mathbf {v} \cdot \mathbf {v}\right) \bigg | _ {z = \eta + z _ {s}} = g \eta ,
$$

assuming the pressure at the surface is negligible.

Assuming the waves are uniform, of wavelength ¸ and period $\tau ,$ and that they travel in the positive S-direction means that the solution as a function of $S$ and t must appear in terms of a phase angle

$$
\theta = 2 \pi \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg) = \frac {2 \pi}{\lambda} \bigg (S - \bar {c} t + \frac {\lambda \alpha}{3 6 0} \bigg),
$$

where $\begin{array} { r } { \bar { c } = \frac { \lambda } { \tau } } \end{array}$ is the wave celerity. This means that, for any function in the solution,

$$
\frac {\partial}{\partial t} = - \bar {c} \frac {\partial}{\partial s}.
$$

Thus, at the free surface boundary

$$
\frac {\partial \phi}{\partial t} = - \bar {c} \frac {\partial \phi}{\partial s} = \bar {c} v _ {s},
$$

and the Bernoulli equation at the free surface is

$$
\left. (- \bar {c} v _ {s} + 1 / 2 (\mathbf {v} \cdot \mathbf {v})) \right| _ {z = \eta + z _ {S}} = - g \eta
$$

or

$$
\left. (v _ {z} ^ {2} + (v _ {s} - \bar {c}) ^ {2}) \right| _ {z = \eta + z _ {S}} = \bar {c} ^ {2} - z g \eta .
$$

Equation 6.2.3-2

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A further boundary condition at the free surface is that the fluid particle velocity relative to the wave celerity must be tangential to the slope of the wave:

Equation 6.2.3-3

$$
\frac {v _ {z}}{v _ {s} - \overline {{c}}} | _ {z = \eta + z _ {s}} = \frac {\partial \eta}{\partial s}.
$$

At the seabed (z = zb), there is no fluid motion in the vertical direction:

$$
- v _ {z} = \frac {\partial \phi}{\partial z} | _ {z = z _ {b}} = 0.
$$

The problem now consists of finding a potential function, Á, that satisfies Equation 6.2.3-1--the boundary condition at the seabed--as well as the boundary conditions at the surface-- Equation 6.2.3-2 and Equation 6.2.3-3.

Stokes proposed a power series solution to this problem, and Skjelbreia and Hendrickson (1960) have obtained that solution to fifth-order. The potential function is assumed to be

Equation 6.2.3-4

$$
\begin{array}{l} \frac {\beta \phi}{\bar {c}} = \frac {2 \pi \phi}{\lambda \bar {c}} = (\mu A _ {1 1} + \mu^ {3} A _ {1 3} + \mu^ {5} A _ {1 5}) \cosh [ \beta (z - z _ {b}) ] \sin \theta \\ - \left(\mu^ {2} A _ {2 2} + \mu^ {4} A _ {2 4}\right) \cosh [ 2 \beta (z - z _ {b}) ] \sin 2 \theta \\ + \left(\mu^ {3} A _ {3 3} + \mu^ {5} A _ {3 5}\right) \cosh \left[ 3 \beta \left(z - z _ {b}\right) \right] \sin 3 \theta \\ - \mu^ {4} A _ {4 4} \cosh [ 4 \beta (z - z _ {b}) ] \sin 4 \theta \\ + \mu^ {5} A _ {5 5} \cosh [ 5 \beta (z - z _ {b}) ] \sin 5 \theta , \\ \end{array}
$$

where $\beta = 2 \pi / \lambda$ , the $A _ { i j }$ are constants that depend on the ratio of water depth to wavelength $\left( z _ { s } - z _ { b } \right) / \lambda$ , and $\mu$ is a parameter. The wave profile, ´(µ), is assumed to be

Equation 6.2.3-5

$$
\begin{array}{l} \beta \eta = - \mu \cos \theta \\ + \left(\mu^ {2} B _ {2 2} + \mu^ {4} B _ {2 4}\right) \cos 2 \theta \\ - \left(\mu^ {3} B _ {3 3} + \mu^ {5} B _ {3 5}\right) \cos 3 \theta \\ + \mu^ {4} B _ {4 4} \cos 4 \theta \\ - \mu^ {5} B _ {5 5} \cos 5 \theta , \\ \end{array}
$$

where the $B _ { i j }$ are constants for a given water depth and wavelength. Finally, it is assumed that

Equation 6.2.3-6

$$
\beta \bar {c} ^ {2} = c _ {0} ^ {2} (1 + \mu^ {2} C _ {1} + \mu^ {4} C _ {2})
$$

and that