Diffraction Of Random Waves Research Articles

Wave spectral modeling of multidirectional random waves in a harbor through combination of boundary integral of Helmholtz equation with Chebyshev point discretization

A mathematical model is presented to predict the wave spectrum in a complex geometry harbor domain due to the diffraction of multidirectional random waves propagating through entrance of the harbor from the open sea. An accurate description of the harbor geometry, bathymetry and an incident wave spectrum are required. The solution of Helmholtz equation in the bounded and unbounded region is obtained by 2-D boundary integral method with the consideration of partial reflection and corner point approximation. Moreover, the Chebyshev point discretization is applied on boundary of the harbor to improve the accuracy of the numerical scheme. The principle of superposition is adopted to simulate the directional random waves based on the solution of monochromatic incident waves. The form of Mitsuyasu’s frequency spectrum was utilized for the random wave simulation. The wave spectra for multidirectional random waves are analyzed at various recorder stations in the Pohang New Harbor (PNH). Comparison of simulation results with in situ measurement data recorded at various recorder observatories in the PNH demonstrates the feasibility and accuracy of our proposed method. After this study, we analyzed the wave height distribution in the whole PNH domain. Based on the simulation results, some tactic has been introduced to reduce the wave amplitude in the PNH. Our primary results confirm that present numerical model is accurate and flexible to implement on various real harbors to predict the multidirectional random wave diffraction.

Multidirectional random wave diffraction in harbours by the placing of submarine pit: A case study

The purpose of this study was to estimate the decreasing effects of diffracted wave fields in-and around selected harbours (Kashima and Soma Harbours) including the region of outer breakwater, when a navigation channel is dredged at the vicinity of the harbour entrance. The wave field in the problem is considered to be in a two dimensional plane, and the configuration of the submarine pit breakwater in the sea bed is designated by a single rectangular type. The numerical simulation is performed by using the solution of the boundary integral equation based on the Green function. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. Incident wave height is to be decreased in-and behind a pit region by depth discontinuity. It is observed that the wave energy is decreased when a submarine pit has been placed. Key words: Green function, harbour, navigation channel, Mitsuyasu directional spectrum, pit.

A three-dimensional modeling of multidirectional random wave diffraction by rectangular submarine pits

A three-dimensional modeling of multidirectional random-wave diffraction by a group of rectangular submarine pits is presented in this paper. The fluid domain is divided into N interior regions representing the pit area and an overall exterior region separated by the imaginary pit boundaries. In the interior region, the analytical expressions of the Fourier series expansion for velocity potentials in the pit regions have been derived with the unknown coefficients determined from a series of Green's function based boundary integral equations. The boundary integral approach has also been applied to obtain the velocity potential and free-surface elevation in the exterior region. The Pierson–Moskowitz (P–M) frequency spectrum was selected for the random wave simulation using the superposition of solutions of a finite number of decomposed wave components. Numerical results for the cases of regular waves and random waves are presented to examine the influences of the pit geometry and incident wave condition on the overall wave field. The general diffraction pattern of alternate bands of increase and decrease of relative wave height in front of the pit system can be observed. It is found that, in the shadow region, the relative wave height is reduced. As the number of pit increases, the effectiveness of reducing the relative wave height behind the multiple-pit system increases. However, the relative wave height within the pit area and in front of the leading pit shows increasing trend. It is noticed that under the random-wave condition, the level of increase and decrease of the relative wave height due to the existence of submarine pits is less pronounced than that observed from results in regular-wave condition.

A comparison of several wave spectra for the random wave diffraction by a semi-infinite breakwater

A comparison of the diffraction of multidirectional random waves using several selected wave spectrum models is presented in this paper. Six wave spectrum models, Bretschneider, Pierson–Moskowitz, ISSC, ITTC, Mitsuyasu, and JONSWAP spectrum, are considered. A discrete form for each of the given spectrum models is used to specify the incident wave conditions. Analytical solutions based on both the Fresnel integrals and polynomial approximations of the Fresnel integrals and numerical solutions of a boundary integral approach have been used to obtain the two-dimensional wave diffraction by a semi-infinite breakwater at uniform water depth. The diffraction of random waves is based on the cumulative superposition of linear diffraction solution. The results of predicted random wave diffraction for each of the given spectrum models are compared with those of the published physical model presented by Briggs et al. [1995. Wave diffraction around breakwater. Journal of Waterway, Port, Coastal and Ocean Engineering—ASCE 121(1), 23–35]. Reasonable agreement is obtained in all cases. The effect of the directional spreading function is also examined from the results of the random wave diffraction. Based on these comparisons, the present model for the analysis of various wave spectra is found to be an accurate and efficient tool for predicting the random wave field around a semi-infinite breakwater or inside a harbor of arbitrary geometry in practical applications.

Comparison of time-dependent extended mild-slope equations for random waves

The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O( ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O( ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom.

A three-dimensional numerical modeling of multidirectional random wave diffraction by rectangular submarine pits

A three-dimensional numerical model of multidirectional random wave diffraction by rectangular submarine pits has been investigated theoretically in the uniform water depth. In the exterior region, the three-dimensional Green function has been us ed for the solution of the velocity potential and its derivative in fluid interface between regions. The form of the Fourier expansion has also been utilized for the solution of the velocity potential in the interior region. The Mitsuyasu directional spectrum has been di scretized to specify the incident wave conditions for obtaining linear diffraction solutions, and then subsequent cumulative superposition is performed to obtain the random wave results. The results of the present model have been compared with those of previous theoretical studies for both regular and random wave diffraction by single or multiple pits. reasonable agreement was consistently obtained in all cases. In accordance with good agreement from these comparisons, the present numerical model may accurately be utilized to predict the three-dimensional wave field around multiple submarine pits or navigation channels.

Boundary element modeling of multidirectional random wave diffraction by multiple rectangular submarine pits

The diffraction of multidirectional random surface waves by one or more rectangular submarine pits is investigated theoretically. The present method is based on the principle of cumulative superposition of diffraction solutions for linear Airy wave theory obtained by a two-dimensional boundary integral equation in water of uniform depth. The constant segment mode has been utilized to obtain the velocity potentials on the interface of each pit, and a specified discrete form of the Mitsuyasu directional spectrum is used for the incident wave conditions. The results of the present model have been validated through the comparison with those obtained by previous works ( kh<1, long wave) for random wave diffraction by single or multiple pits. A further investigation for random wave diffraction has been also studied by considering short wave range kh>1. In accordance with reasonable agreement from these comparisons, it is concluded that the present BEM numerical model may accurately be utilized to predict the wave field around multiple submarine pits or navigation channels in many practical applications.

Multidirectional random wave interactions with rectangular submarine pits

The diffraction of multidirectional random surface waves with one or more rectangular submarine pits is investigated theoretically. A specified discrete form of the Mitsuyasu directional spectrum is used for the incident wave conditions. The present method is based on the cumulative superposition of diffraction solutions for linear Airy wave theory obtained by a two-dimensional boundary integral approach in water of uniform depth. The results of the present model have been validated through the comparison with those obtained by previous works for both regular and random wave diffraction by single or multiple pits. In accordance with good agreement from these comparisons, it is concluded that the present numerical model may accurately be utilized to predict the wave field around multiple submarine pits or navigation channels in many practical applications.

Numerical modeling of multi-directional irregular waves incorporating 2-D numerical wave absorber and subgrid turbulence

In this paper, a finite difference scheme with an efficient 2-D numerical wave absorber for solving the extended Boussinesq equations as derived by Nwogu (Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coastal and Ocean Engineering, ASCE 119, 618–638) is proposed. The alternate direction iterative method combined with an efficient predictor-corrector scheme are adopted for the numerical solution of the governing differential equations. To parameterize the contribution of unresolved small-scale motions, the philosophy of the large eddy simulation is applied on the horizontal plane. The proposed method is verified by two test cases where experimental data are available for comparison. The first case is wave diffraction around a semi-infinite breakwater studied by Briggs et al. (Briggs, M.J., Thompson, E.F., Vincent, C.L., 1995. Wave diffraction around breakwater. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE 121, 23–35). The other case is wave concentration by a navigation channel as reported by Yu et al. (Yu, Y.-X., Liu, S.-X., Li, Y.S., Wai, O.W.H., 2000. Refraction and diffraction of random waves through breakwater. Ocean Engineering 27, 489–509). Numerical results agree very well with the corresponding experimental data in both cases.

Boundary element modeling of multidirectional random waves in a harbor with partially reflecting boundaries

A numerical model is presented for the prediction of the wave field due to the diffraction of directional random waves in a harbor of arbitrary shape with partially reflecting boundaries. The water depth is assumed uniform and the method is based upon the superposition of diffraction solutions for monochromatic waves obtained by a two-dimensional boundary integral equation approach. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. The present numerical model has been validated through comparisons with previous experimental data and theoretical results for both regular and random wave diffraction by offshore breakwaters and in harbors. Good agreement was obtained in all cases. Based on these comparisons it is concluded that the present numerical model is an accurate and efficient tool to predict the wave field inside a harbor or around a breakwater in many practical applications.

Refraction and diffraction of random waves through breakwater

A physical model study of combined refraction and diffraction of waves through a breakwater gap at different incident angles was conducted. Both regular and random waves with narrow and broad frequency and direction spreading were studied. Besides the presence of a mild bottom slope in the lee of the breakwater, the distribution of wave heights across the width of a navigation channel inside the model harbor was also simulated. In addition to contributing to an understanding of the phenomenon of refraction and diffraction of random waves, the relatively complete set of data obtained can serve as a benchmark for testing of numerical models.

Diffraction Of Wave Spectra By Two Breakwaters AtAn Angle

The present study tackles the problem of random wave diffraction through an opening between two breakwaters forming an arbitrary angle. The numerical scheme used is result-oriented and employs the solution regarding the diffraction coefficient k for the monochromatic case. The study comprises two parts, the first dealing with uni-directional and the second with multi-directional incoming spectra. A simple energy conservation principle applied to a number of components associated with the monochromatic solution of the diffraction problem, produced one-dimensional spectra of the diffracted waves at any point in the lee of the breakwaters. A compound diffraction coefficient K could thus be established in terms of the k's and the characteristics of the incoming spectrum. Graphs of equal K-values are presented for the various cases examined. The second part of the study deals with waves presenting spatial distribution of energy. A directional spreading function of the Mitsuyasu-type was assumed. The diffracted spectra yielded a composite diffraction coefficient by integrating through the directions as well as through the frequencies.

Diffraction of nonlinear random waves by a vertical cylinder in deep water

Abstract A solution, exact to second-order, is presented for the nonlinear diffraction of random waves by a fixed, surface-piercing vertical circular cylinder in deep water. The incident wave field is considered as a stationary random process, and the nonlinear diffraction problem is analyzed utilizing the Stokes perturbation expansion procedure combined with a Fourier-Stieltjes spectral representation of the stationary random wave kinematics. The second-order velocity potential is explicitly obtained by applying a modified form of Weber's Integral Theorem to invert the inhomogeneous second-order free-surface condition. Particular attention is directed towards the second-order diffraction forces on the cylinder. The spectral description of the second-order diffraction forces involves a complicated integral expression with highly oscillatory wave-wave interaction kernels and multiple convolutions of the linear wave spectrum. The present approach provides a complete spectral description of the second-order diffraction forces, and yields the spectral densities of the diffraction forces at the sum and difference frequencies. Numerical results are presented which illustrate the spectral content of the diffraction force due to an incident wave field represented by a superposition of waves described by band-limited white noise processes centered at different frequencies.

DIFFRACTION DIAGRAMS FOR DIRECTIONAL RANDOM WAVES

Conventional wave diffraction diagrams often yield erroneous estimation of wave heights behind breakwaters in the sea, because they are prepared for monochromatic waves while actual waves in the sea are random with directional spectral characteristics. A proposal is made for the standard form of directional wave spectrum on the basis of Mitsuyasu's formula for directional spreading function. A new set of diffraction diagrams have been constructed for random waves with the proposed directional spectrum. Problems of multi-diffraction and multi-reflection within a harbour can also be solved with serial applications of random wave diffraction.