scholarly journals CLOSED-FORM SOLUTIONS FOR THE PROBABILITY DENSITY OF WAVE HEIGHT IN THE SURF ZONE

1988 ◽  
Vol 1 (21) ◽  
pp. 60 ◽  
Author(s):  
William R. Dally ◽  
Robert G. Dean
Keyword(s):  
Probability Density Function ◽  
Shallow Water ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Wave Height ◽  
Surf Zone ◽  
Wave Steepness ◽  
Bottom Slope ◽  
Closed Form Solutions

By invoking the assumption that in the surf zone, random waves behave as a collection of individual regular waves, two closed-form solutions for the probability density function of wave height on planar beaches are derived. The first uses shallow water linear theory for wave shoaling, assumes a uniform incipient condition, and prescribes breaking with a regular wave model that includes both bottom slope and wave steepness effects on the rate of decay. In the second model, the shallow water assumption is removed, and a distribution in wave period (incipient condition) is included. Preliminary results indicate that the models exhibit much of the behavior noted for random wave transformation reported in the literature, including bottom slope and wave steepness effects on the shape of the probability density function.

scholarly journals THE ENERGY SPECTRA OF SURF WAVES ON A CORAL REEF

1978 ◽  
Vol 1 (16) ◽  
pp. 33 ◽  
Author(s):  
Theodore T. Lee ◽  
Kerry P. Black
Keyword(s):  
Time Series ◽  
Coral Reef ◽  
Probability Density Function ◽  
Weibull Distribution ◽  
Shallow Water ◽  
Probability Density ◽  
Density Function ◽  
Linear Wave ◽  
Gain Function ◽  
Fourier Spectra

The transformation of waves crossing a coral reef in Hawaii including the probability density function of the wave heights and periods and the shape of the spectrum is discussed. The energy attenuation and the change of height and period statistics is examined using spectral analysis and the zero up-crossing procedure. Measurements of waves at seven points along a 1650 ft transect in depths from 1 to 3.5 ft on the reef and 35 ft offshore were made. The heights were tested for Rayleigh, truncated Rayleigh and Wei bull distributions. A symmetrical distribution presented by Longuet-Higgins (1975) and the Weibull distribution were compared to the wave period density function. In both cases the Weibull probability density function fitted with a high degree of correlation. Simple procedures to obtain Weibull coefficients are given. Fourier spectra were generated and contours of cumulative energy against each position on the reef show the shifting of energy from the peak as the waves move into shallow water. A design spectrum, with the shape of the Weibull distribution, is presented with procedures given to obtain the coefficients which govern the distribution peakedness. Normalized non-dimensional frequency and period spectra were recommended for engineering applications for both reef and offshore locations. A zero up-crossing spectrum (ZUS) constructed from the zero upcrossing heights and periods is defined and compared with the Fourier spectrum. Also discussed are the benefits and disadvantages of the ZUS, particularly for non-linear wave environments in shallow water. Both the ZUS and Fourier spectra are used to test the adequacy of formulae which estimate individual wave parameters. Cross spectra analysis was made to obtain gain function and squared coherency for time series between two adjacent positions. It was found that the squared coherency is close to unity near the peak frequency. This means that the output time series can be predicted from the input by applying the gain function. However, the squared coherency was extremely small for other frequencies above 0.25 H2.

scholarly journals On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

A CLOSED FORM FOR THE DENSITY FUNCTIONS OF RANDOM WALKS IN ODD DIMENSIONS

2015 ◽  
Vol 93 (2) ◽  
pp. 330-339 ◽  
Author(s):  
JONATHAN M. BORWEIN ◽  
CORWIN W. SINNAMON
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Random Walks ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Dimensional Space ◽  
Density Functions ◽  
Piecewise Polynomial ◽  
Odd Dimensions

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

scholarly journals On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (01) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

scholarly journals A NOTE CONCERNING THE DISTANCES OF UNIFORMLY DISTRIBUTED POINTS FROM THE CENTRE OF A RECTANGLE

2012 ◽  
Vol 87 (1) ◽  
pp. 115-119 ◽  
Author(s):  
ROBERT STEWART ◽  
HONG ZHANG
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Average Distance ◽  
Cumulative Distribution

AbstractGiven a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

scholarly journals Conceptual Limitations of the Probability Density Function Method for Modeling Turbulent Premixed Combustion and Closed-Form Description of Chemical Reactions’ Effects

Fluids ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 142
Author(s):  
Vladimir Zimont
Keyword(s):  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Inverse Modeling ◽  
Turbulent Combustion ◽  
Laminar Flame ◽  
Chemical Effects ◽  
Instantaneous Combustion ◽  
Modeling Strategy

In this paper, we critically analyzed possibilities of probability density function (PDF) methods for the closed-form description of combustion chemical effects in turbulent premixed flames. We came to the conclusion that the concept of a closed-form description of chemical effects in the classical modeling strategy in the PDF method based on the use of reaction-independent mixing models is not applicable to turbulent flames. The reason for this is the strong dependence of mixing on the combustion reactions due to the thin-reaction-zone nature of turbulent combustion confirmed in recent optical studies and direct numerical simulations. In this case, the chemical effect is caused by coupled reaction–diffusion processes that take place in thin zones of instantaneous combustion. We considered possible alternative modeling strategies in the PDF method that would allow the chemical effects to be described in a closed form and came to the conclusion that this is possible only in a hypothetical case where instantaneous combustion occurs in reaction zones identical to the reaction zone of the undisturbed laminar flame. For turbulent combustion in the laminar flamelet regime, we use an inverse modeling strategy where the model PDF directly contains the characteristics of the laminar flame. For turbulent combustion in the distributed preheat zone regime, we offer an original joint direct/inverse modeling strategy. For turbulent combustion in the thickened flamelet regime, we combine the joint direct/inverse and inverse modeling strategies correspondingly for simulation of the thickened flamelet structure and for the determination of the global characteristics of the turbulent flame.

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