Three irregular wave tests were conducted on a 1:16 smooth impermeable slope to investigate the detailed cross-shore variations of the probability distributions and statistics of the free surface elevations and middepth horizontal velocities in the shoaling, surf, and swash zones. The exponential gamma distribution with the measured mean, standard deviation, and skewness is shown to be capable of describing the measured probability distributions in a unified manner, although the agreement becomes worse in the lower swash zone. The probability distribution of the free surface elevation whose lower limit is imposed by the beach face in the swash zone becomes exponential with the skewness s= 2 and the standard deviation σ=h¯ with h¯= mean water depth. These upper limits of s and σ/h¯ in the lower swash zone are in qualitative agreement with the data in the region of h¯≳ 0.4 cm in these small-scale tests. The cross-shore variations of the mean (undertow) and standard deviation of the middepth horizontal veloc...

Probability distribution of surface elevation in surf and swash zones

The purpose of this paper is twofold. The first is to clarify the transition of stochastic characteristics of waves; in particular, the transition from a Gaussian to non-Gaussian random process in the nearshore zone as a function of water depth and sea severity, and the second is to develop a method to estimate the non- Gaussian properties of waves at a specified water depth in the nearshore zone from knowledge of the sea severity in deep water. The probability density function applicable to non-Gaussian waves in finite water depth is applied to more than 1,000 samples of wave data obtained by the Coastal Engineering Research Center, U.S. Army, during the ARSLOE Project. From analysis of the data, the limiting water depth is defined below which wind-generated waves can no longer be considered Gaussian for a given sea severity. Furthermore, by presenting the parameters of the probability density function as a function of water depth and sea severity, it is possible to estimate the sea condition at a specified water depth in the nearshore zone from knowledge of sea severity in deep water.

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Transition of stochastic characteristics of waves in the nearshore zone

The Edgeworth expansion with the measured skewness and kurtosis is shown to be capable to describe the measured nonlinear distributions of the surface elevation and horizontal velocity in the shoaling and surf zones. The moments involved in the energetics-based model are expressed in terms of the skewness and kurtosis and shown to be in agreement with available data. Stokes wave theory is applied to obtain a relationship between skewness and kurtosis. The relationship is adjusted empirically because of the limitation of Stokes wave theory in the shoaling and surf zones. The relative simple relationship for the higher moments obtained here may be applied for cross-shore sediment transport analysis.

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Nonlinear distribution of nearshore free surface and velocity

The numerical computation of expectations for (nearly) singular multivariate normal distribution is a difficult problem, which frequently occurs in widely varying statistical contexts. In this article we discuss several strategies to improve the algorithm proposed by Genz and Kwong (2000) when either a moderate accuracy is requested, the correlation structure is strong, and, most importantly, the dimension of the integral is large. Test results for typical problems show an average speedup of 10 using the modified algorithm, but even more is gained as the dimension of the problem increases. We apply the modified algorithm to compute long-run distributions of Gaussian wave characteristics, a difficult problem where previous algorithms fail to compute accurate values in reasonable time.

Evaluating Nearly Singular Multinormal Expectations with Application to Wave Distributions

In this paper, a simple evolution formulation, based on polar coordinate, is proposed to efficiently and accurately simulate the dynamic movement of sandy shoreline in a crenulate-shaped bay. The consistent actions by seasonal waves and swell usually result in severe erosion or deposition along sandy shoreline, so some mathematical formulations and numerical models, based on the concept of headland control, have been recently proposed to forecast and protect the shoreline planform. A simple and general formulation is derived in this study by considering the balance of longshore sediment transport, since the accurate prediction of dynamic movements of high planform-curvature shoreline in the shadow zone behind a headland is essential and critical to the headland control. The proposed formulation can be directly adopted to accurately and simply simulate the temporal variations of shoreline in both of the hooked zone, where is protected by a headland, and the unhooked zone, where is straight attacked by incident waves. Numerical results and comparisons of temporal variation of shoreline between two headlands are provided in this paper to demonstrate the accuracy and efficiency of the proposed evolution formulation. Besides, different time increments and different numbers of control volume are adopted to examine the merits of the proposed numerical model.

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Developments of Dynamic Shoreline Planform of Crenulate-Shaped Bay by a Novel Evolution Formulation

This paper presents the development of a probability density function applicable to waves in finite water depth (which can be considered to be a nonlinear, non-Gaussian random process) in closed form. The derivation of the density function is based on the Kac- Siegert solution developed for a nonlinear mechanical system, but the parameters involved in the solution are evaluated from the wave record only. Further, the probability density function is asymptotically expressed in closed form. Comparisons between the presently developed probability density function and histograms constructed from wave records show good agreement.

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K. Ahn

Papers

Non-Gaussian Probability Distribution of Coastal Waves