Second-Order Directional Seas and Associated Wave Forces
Authors: Sharma J. , Dean R.G. 
Published: October 2008

ABSTRACT
Most available methods for wave force computation incorporate either the nonlinearities of the ocean surface for a single fundamental component or the random and/or directional characteristics using the superposition of linear wave moments. An exception is the intuitive "hybrid" method1 which combines elements of linear and nonlinear waves. The present paper describes and applies a method which is correct to second order in wave height, for calculating waves and wave forces on an offshore structure due to a directional wave spectrum.
Starting with a prescribed linear spectrum of directional waves, a set of random phases is generated and the second-order spectrum computed with phases defined by all contributing pairs of first-order components. With one realization of the spectrum thus complete up to the second order, the wave profile and water particle kinematics can be simulated in the time domain. The wave forces are also computed in the time domain taking full account of their nonlinear and directional properties. The resulting wave forces at any level vary in direction and magnitude. The total wave forces summed over all piling of a structure are less than those for a unidirectional train of waves with the same one-dimensional spectrum.
Several examples are presented to illustrate reductions in maximum wave forces due to the directional distribution of the waves. It is found that for a single piling the maximum force decreases by a factor ranging from 1.0 to 0.6l as the directional spread increases from unidirectional to omnidirectional. For a four-pile group on a square array of 300 ft. spacing, the corresponding decrease in the factor is from 1.0 to 0.51 for a Bretschneider spectrum with a peak period of a approximately 12 seconds. The results of this complete model are compared with the more intuitive and approximate hybrid method and are found to agree quite well. Force spectra are presented and discussed for the inline and transverse directions.

INTRODUCTION
The nonlinearity, randomness and directionality of a real sea preclude a simplified yet realistic determination of wave loading on a single or multiple pile group. At present there are two essentially different but complementary methods for computing wave loadings. One method represents nonlinearities of a single wave composed of a characteristic fundamental period and its higher harmonics. A number of such theories have been developed Dalrymple extended the stream function approach of Dean to waves on a shear current. Some of these theories can be shown to account for the nonlinearities adequately; however, they avoid the random and directional characteristics of the sea surface. The second method utilizes the principle of linear superposition of infinitely many waves having given frequencies, amplitudes and directions of propagation, but independent phases; the total energy is distributed over a continuum of frequencies and directions. In this manner, a three-dimensional Gaussian sea can be represented fully. However, ignoring the nonlinearities makes the random Gaussian model unrealistic, especially for large waves.