Product: ABAQUS/Aqua
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 , where is an arbitrary datum.
Assume that the fluid is inviscid and incompressible. The fluid particle velocities are derivable from a flow potential
Let be the elevation of the free surface above this level. At the free surface the Bernoulli equation is
Assuming the waves are uniform, of wavelength and period , 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
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
where , the are constants that depend on the ratio of water depth to wavelength , and is a parameter. The wave profile, , is assumed to be where the are constants for a given water depth and wavelength. Finally, it is assumed that and thatSkjelbreia and Hendrickson obtain the 18 constants , , and from matching terms in equal powers of and in the free surface boundary conditions, Equation 6.2.3–2 and Equation 6.2.3–3. They give the constants as functions of as
Skjelbreia and Hendrickson (1960) have a factor +2592 multiplying in the equation for . This was corrected to 2592 by Nishimura et al. (1970).
They then obtain equations for and . The wave height is
The flow potential has been approximated as
The Stokes wave field is a spatial description of the wave field. All wave field quantities are calculated up to the instantaneous fluid level. 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.).