On the Use of Linear and Weakly Nonlinear Wave Theory in Continuous Ocean Wave Spectra: Convergence With Respect to Frequency

The basic principle of wave superposition in the analysis of irregular linear wave problems such as the analysis of the wave surface itself or forces on large volume or slender structures, has been used for more than fifty years. Also the extension of the perturbation solution to second and in some cases third order is well known. In the nonlinear extension of the perturbation scheme, the basic idea is that nonlinear interaction between two or, in the third order case, three wave components can take place. The solution may therefore be found for two or three wave components and extended to the interaction of all wave components without further calculation. Whereas this superposition idea is an efficient and accurate method to determine wave loads and properties, the convergence properties with respect to the high frequency tail of the spectrum is often neglected. Many of the terms arising in practical applications increase rapidly as the frequency increases so that their convergence properties in a continuous wave spectrum are strongly dependent on the tail of the wave spectrum. The lack of convergence with respect to frequency will typically lead to a choice of: • Using an equivalent regular wave to represent the problem knowing that a regular wave cannot represent all the relevant physical and statistical properties of the wave field; • Make a sensible truncation of the wave spectrum knowing that the chosen truncation frequency affects the results; • Resort to an engineering solution such as the Wheeler (1970) stretching technique for crest kinematics above the crest. It is the object of the present paper to investigate the requirements to linear and second order problems to converge with respect to frequency. Using the Lindgren (1970) properties of a wave crest in a linear wave field and linear Monte Carlo simulations, it is found that requirements to convergence in a spectrum with an ω⁻⁴ and ω⁻⁵ tail is very strict indeed. It is further found that it is convenient to distinguish between problems where the linear component itself is not defined and problems where the linear component is defined but where the higher order component is not defined. It is shown that the latter problem may be overcome and an example of this is given.