The wave may be input using one of four wave theories in the following table (input as KWAVE via TB,WATER).

The free surface of the wave is defined by

(4–1)

where:

ηs = total wave height

Kw = wave theory key (input as KWAVE with TB,WATER)

ηi = wave height of component i

R = radial distance to point on element from origin in the X-Y plane in the direction of the wave

λi = wave length = input as WL(i) if WL(i) > 0.0 and if Kw = 0 or 1 otherwise derived from Equation 4–2

t = time elapsed (input as TIME on TIME command) (Note that the default value of TIME is usually not desired. If zero is desired, 10-12 can be used).

ϕi = phase shift = input as ϕ(i)

If λ_i is not input (set to zero) and K_w < 2, λ_i is computed iteratively from:

(4–2)

where:

λi = output quantity small amplitude wave length

g = acceleration due to gravity (Z direction) (input on ACEL command)

d = water depth (input as DEPTH via TB,WATER)

Each component of wave height is checked that it satisfies the "Miche criterion" if Kw ≠3. This is to ensure that the wave is not a breaking wave, which the included wave theories do not cover. A breaking wave is one that spills over its crest, normally in shallow water. A warning message is issued if:

(4–3)

where:

When using wave loading, there is an error check to ensure that the input acceleration does not change after the first load step, as this would imply a change in the wave behavior between load steps.

For K_w = 0 or 1, the particle velocities at integration points are computed as a function of depth from:

(4–4)

(4–5)

where:

= radial particle velocity

= vertical particle velocity

ki = 2π/λi

= height of integration point above the ocean floor = d+Z

= time derivative of ηi

= drift velocity (input via TB,WATER)

The particle accelerations are computed by differentiating and with respect to time. Thus:

(4–6)

(4–7)

where:

Expanding equation 2.29 of the Shore Protection Manual for a multiple component wave, the wave hydrodynamic pressure is:

(4–8)

However, use of this equation leads to nonzero total pressure at the surface at the crest or trough of the wave. Thus, Equation 4–8 is modified to be:

(4–9)

which does result in a total pressure of zero at all points of the free surface. This dynamic pressure, which is calculated at the integration points during the stiffness pass, is extrapolated to the nodes for the stress pass. The hydrodynamic pressure for Stokes fifth order wave theory is:

(4–10)

Other aspects of the Stokes fifth order wave theory are discussed by Skjelbreia et al.. The modification as suggested by Nishimura et al.has been included. The stream function wave theory is described by Dean.

If both waves and current are present, the question of wave-current interaction must be dealt with. Three options are made available through K_cr (input as KCRC via TB,WATER):

For K_cr = 0, the current velocity at all points above the mean sea level is simply set equal to W_o, where W_o is the input current velocity at Z = 0.0. All points below the mean sea level have velocities selected as though there were no wave.

For K_cr = 1, the current velocity profile is "stretched" or "compressed" to fit the wave. In equation form, the Z coordinate location of current measurement is adjusted by

(4–11)

where:

Z(j) = Z coordinate location of current measurement (input as Z(j))

= adjusted value of Z(j)

For K_cr = 2, the same adjustment as for K_cr = 1 is used, as well as a second change that accounts for "continuity." That is,

(4–12)

where:

W(j) = velocity of current at this location (input as W(j))

= adjusted value of W(j)

These three options are shown pictorially in Figure 4.1: Velocity Profiles for Wave-Current Interactions.

Figure 4.1: Velocity Profiles for Wave-Current Interactions

To compute the relative velocities ( , ), both the fluid particle velocity and the structure velocity must be available so that one can be subtracted from the other. The fluid particle velocity is computed using relationships such as Equation 4–4 and Equation 4–5 as well as current effects. The structure velocity is available through the Newmark time integration logic (see Transient Analysis).

Finally, a generalized Morison's equation is used to compute a distributed load on the element to account for the hydrodynamic effects:

(4–13)

where:

{F/L}d = vector of loads per unit length due to hydrodynamic effects

CD = coefficient of normal drag (see below)

ρw = water density (mass/length3) (input as DENSW on MP command with TB,WATER)

De = outside diameter of the pipe with insulation (length)

= normal relative particle velocity vector (length/time)

CM = coefficient of inertia (input as CM on the R command)

= normal particle acceleration vector (length/time2)

CT = coefficient of tangential drag (see below)

= tangential relative particle velocity vector (length/time)

Two integration points along the length of the element are used to generate the load vector. Integration points below the mud line are simply bypassed. For elements intersecting the free surface, the integration points are distributed along the wet length only.

The coefficients of drag (C_D,C_T) may be defined in one of two ways:

The dependency on Reynolds number (Re) may be expressed as:

(4–14)

where:

fD = functional relationship (input on the water motion table as RE, CDy, and CDz via TB,WATER)

μ = viscosity (input as VISC on MP command)

and

(4–15)

where:

fT = functional relationship (input on the water motion table as RE and CT via TB,WATER)

Temperature-dependent quantity may be input as μ, where the temperatures used are those given by input quantities T(i) of the water motion table.

When the MacCamy-Fuchs corrections are requested to account for diffraction effects, especially for large diameter objects with shorter wave lengths, two things occur: