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A finite difference approach for extended mild slope equation
Abstract
In this study, the numerical model for the determination of transformations of waves while propagating has been presented. This numerical model was developed to solve the extended mild slope equation that is applicable to the rapidly varying topographies. It includes the effects of wave refraction, diffraction, shoaling, reflection, harbor resonance, higher order bottom configurations; dissipative terms due to wave breaking and bottom friction. Nonlinear wave celerity and group velocity were introduced in the solution to obtain results that are more accurate. Mac Cormack Method and Point Gauss Seidel Method were applied together in the proposed new solution approach. The numerical model was tested on the semicircular shoaling area [1] and shoreparallel breakwater [2]. The comparison of the numerical model in the current study and the physical experiments that are present in the literature shows the reliability of the model for wave transformations and dissipations over uneven bottoms.
... Bu modelin veri tabanında Türkiye kıyı meteoroloji istasyonlarının yaklaşık 40 yıllık saatlik verileri bulunmaktadır [4]. İncelenen parametrelerin gerek Türk Standartları gerekse dünya standartlarındaki değerleri dikkate alınmalıdır [5]. ...
... K8 istasyonu marina girişini temsil etmektedir. K9 istasyonu Bucak körfezi en dar boğaz noktasını temsil eder ve K10 istasyonu Bucak Körfezi açık deniz ortamını temsil eder.Journal of the Faculty of Engineering and Architecture of Gazi UniversityDeniz sularından numune alma TS ISO 5667-9 standardına göre gerçekleştirilmiştir[4,15]. İstasyonlarda (K1-K10) deniz su kalitesini ifade eden su kalitesi parametrelerine ait başlıca fiziksel, kimyasal ve biyokimyasal değerler su yüzeyinden aşağıya doğru -0.5m mesafeden alınan numunelerden ölçülmüştür. ...
... İstasyonlarda (K1-K10) deniz su kalitesini ifade eden su kalitesi parametrelerine ait başlıca fiziksel, kimyasal ve biyokimyasal değerler su yüzeyinden aşağıya doğru -0.5m mesafeden alınan numunelerden ölçülmüştür. Bulanıklık (TS 5091 EN ISO 7027), pH (TS 3263 ISO 10523(1999)) , çözünmüş oksijen (DO) (TS 5677), nitrat (TS ISO 7890-3), askıda katı madde (TS EN 872), fekal koliform (TS ISO 9308-2 ), toplam koliform (TS EN ISO 9308-2) tayinleri belirtilen Türk Standartlarına uygun olarak gerçekleştirilmiştir. Şekil 1. Ölçüm alma noktalarının şematik gösterimi (Schematic representation of sampling points ) Tablo 1.Örnek alma noktalarının koordinatları ve su derinlikleri (Water depths and properties of sampling points)[4,15] ...
With phase-plane analysis in Matlab, the relations between the bacterial indicators and dissolved oxygen (DO), pH and temperature, was examined by considering the dynamic during 20 September 2013 to 8 February 2014 for water quality monitoring in enclosed coastal waters at the edge of the Kaş region of Turkey. It has been suggested that the bacterial indicator phase plane analysis provides a good method of monitoring. It was shown that temperature (significance value, p=4.9e-05), DO (p=1.0e-04) and pH (p=4.6e-04) values have a significant effect on bacterial indicators. The relationship between the bacterial indicators was examined by comparing the same scale “cusp” diagrams, and it was demonstrated that total coliforms concentration were closely related with faecal coliforms (FC) concentration. In presenting the enclosed coastal water quality information, weighting average water quality index (WQIMP) has been used by selecting the sets of appropriate parameters. To keep track of and analyse changes over time by calculating the index numbers, the set of seven variables which are FC, DO, pH, temperature, turbidity, nitrate, total suspended solids and the set of three variables which are FC, temperature, DO were utilized successfully (p=8.3e-05). Parameter quality values were obtained from the software developed using cubic hermite polynomial in Matlab. © 2018 Gazi Universitesi Muhendislik-Mimarlik. All Rights Reserved.
... Yıllık, mevsimlik ve/veya istenilen zaman aralığı için rüzgar ve dalga gülleri hazırlanmaktadır. Derin denizden kıyıya yaklaşan dalgaların, düzensiz taban topoğrafyasına ve değişen su derinliklerine bağlı olarak sığlaşma, sapma, dönme, yansıma, taban sürtünmesi ve kırılma etkileri sonucunda sahip olacakları dalga yükseklikleri, geliştirilmiş yumuşak eğim denklemleri çözülerek benzeştirilmektedir [10,11]. Dalga ilerleme alt modelinden elde edilen kıyı bölgesi dalga yükseklikleri, yaklaşım açıları ve gerilme akıları, dalga etkisiyle oluşan akıntı düzeninin, kıyı boyu sediman taşınımının ve morfolojik değişimlerin benzeştirilmesinde kullanılmaktadır [12,13]. ...
Wind climate, wave climate and current pattern numerical modeling studies that are crucial in the determination of hydromorphological properties of coastal waters, have been performed. Modeling system has been applied to Samsun Bay coastal waters. In the study, hourly wind measurements of Samsun Regional Meteorological Station between 1970-2016; measurements of Marine Automatic Meteorological Observation Station between January 2014-March 2016; measurements of wave buoy located at deep coastal waters of the bay between August 2015-March 2016; predictions with six hours interval of European Centre for Medium-Range Weather Forecasts (ECMWF) operational archive at the coordinates of 41.3°N-36.4°E, 41.4°N-36.4°E and 41.5°N-36.4°E between 2000-2016; regular monthly measurements carried out for the physical parameters and currents in Samsun Bay coastal waters between May 2015-May 2016; and geographic information system (GIS) and cloud computing based, three dimensional hydrodynamic, turbulence and transport model system HYDROTAM-3D predictions, have been used. The verification studies performed by the comparisons of predictions and measurements have proved the success of modeling for the wind climate, wave climate and current patterns in the Samsun Bay coastal waters and have shown that model system can be used as a powerful coastal areas management tool. © 2018 Gazi Universitesi Muhendislik-Mimarlik. All rights reserved.
... Bu iki akarsu Mut ilçesi civarında "Suçatı" mevkiinde birleşerek Göksu nehrine bağlanır ve Silifke yakınlarında denize dökülür. Yaklaşık 10 000 km 2 Pek çok parametrenin göz önüne alınması gereken fiziksel problemlerin çözümü için geliştirilen modeller kısıtlamaları ile sunulmaktadır [28][29][30]. Parametrik modellerde mevcut olan kısıtlamalar parametrik olmayan yöntemlerde giderilmeye çalışılsa dahi parametrik olmayan yöntemlerin veri sayısı ile ilişkili olması yadsınamaz. Bu bakımdan, "Parametrik olmayan yöntemlerden 50, 100 ve 500 yıl görülme sıklığında güvenilir sonuçlar elde edilip edilemeyeceği?" akla gelebilecek bir sorudur. ...
Goksu Nehrinin mansabi olan Dogu Akdeniz kiyilari gecmis donemlerde bircok kez taskin felaketine maruz kalmistir. Olusabilecek zararlari onleme ve akarsudan faydalanabilmek icin havza uzerinde pek cok calisma yapilagelmektedir. Bunlardan bir tanesi de otuz yili askin suredir planlama asamasinda olan Kayraktepe Barajidir. Bu calismada, Kayraktepe Barajinin tarihsel sureci hakkinda bilgi verilecek ve barajin amaclarindan biri olan taskin kontrolune katki saglamak amaciyla taskin frekans egrilerinin parametrik olmayan K en yakin komsu yontemi (KNN) ile elde edilmesi uzerinde durulacaktir. Onerilen yontem ile bulunan noktasal taskin frekans egrileri literaturde yer alan parametrik ve halen kullanilan tum Goksu havzasinin kesif ve planlamasini yapan Elektrik Isleri Etut Idaresi tarafindan verilen sonuclar ile karsilastirilacaktir. Parametrik olmayan yaklasimin parametrik yaklasimlara kiyasla gozlenmis veriye daha iyi uydugu saptanmistir. Elde edilen sonuclar yontemin, Kayraktepe barajinin tasarim debisinin hesaplanmasinda uygulanabilir cazip bir alternatif oldugunu gostermektedir.
... Seçilen denizel alanda, her yön için, uzun dönem dalga istatistiğinden elde edilen dalgaların, değişen dalga yükseklikleri aralıklarındaki, oluşma olasılıkları da kıyı boyu sediman taşınımı modelinde dikkate alınmaktadır. Denizel alanda, net ve toplam (gross) kıyı boyu sediman taşınım miktarları (m 3 /yr), dünyada en yaygın olarak kullanılan CERC metodu [27][28][29][30]ile hesaplanmaktadır. gözlemlenmektedir. ...
The extended mild slope equation has been solved numerically to simulate wave propagation. Refraction, diffraction, shoaling, reflection, bottom friction, breaking energy dissipation and resonance with nonlinear wave celerity and group velocity have been considered. Mac Cormack Method and Point Gauss Seidel Method are applied together on an irregular mesh. In the predictor step, forward finite difference approximations are applied to first order derivatives and central finite difference approximations are applied to second order derivatives. In the corrector step, backward finite diffirence approximations are used for first order derivatives and central finite difference approximations are applied to second order derivatives. The developed numerical model has been applied to the Fethiye Bay located in the Mediterranen coast of Turkey.
A mild slope wave equation is derived which governs the propagation of
linear surface waves in the presence of large ambient currents. The
equation is shown to differ from two previously derived models, and
arguments for the validity of the new version in comparison to previous
versions are presented. A linearized evolution equation and parabolic
equation approximation are constructed in order to show the
correspondence between the present corrected version and a previously
derived version of the time-dependent Schrödinger equation.
An intuitive expression for the spatial change in energy flux associated with waves breaking in the surf zone is developed. Using shallow water linear wave theory, analytical solutions for wave height transformation due to shoaling and breaking on a flat shelf, a plane slope, and an “equilibrium” beach profile are derived and then compared to laboratory data with favorable results. The effect of beach slope on wave decay is included explicitly, while wave steepness effects are included implicitly by specification of the incipient conditions. Set-down/set-up in the mean water level, bottom friction losses, and bottom profiles of arbitrary shape are introduced, and solutions are obtained numerically. The model is calibrated and verified using laboratory data with very good results for the wave decay but not so favorable results for set-up. A test run on a prototype scale profile containing two bar and trough systems demonstrates the model's ability to describe the shoaling, breaking, and wave re-forming process commonly observed in nature. Bottom friction is found to play a negligible role in wave decay in the surf zone when compared to shoaling and breaking.
In this paper a complementary mild-slope equation CMSE is derived in order to investigate the transformation of progressive waves obliquely propagating over the sloping bottom more realistically. We introduce a new depth function which includes the wave refraction and the influence of the bottom slope , perturbed to the second-order in the integral equation. A new depth-integrated mild-slope equation is derived, by using the above mentioned depth function, to model a time-harmonic motion of small amplitude waves in varying water depth. The simulated results reveal that the proposed model provides a significant improvement in the calculation of the wavenumber and the group velocity at different bottom slopes. With the increasing bottom slope, the discrepancies in the reflection coefficient of Bragg scattering between the analytical solution and the one calculated from the conventional mild-slope equation MSE and the modified MSE MMSE are seen to steadily increase. The group velocity of the waves, when compared with the conventional MSE and MMSE, also shows its dependence on the bottom slope and wave propagating angle. The present model is observed to be quite efficient in taking into account the effect of steeper bottom slope. © 2006 American Institute of Physics.
In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation
with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic
equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. (2003), the
numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present
model succeeds the merits in Song et al. (2007)’s model because of the introduced dissipation terms. For the purpose of verifying
its performance on wave nonlinearity, rapidly varying topography and wave breaking, the present model is applied to study:
(1) wave refraction and diffraction over a submerged elliptic shoal on a slope (Berkhoff et al., 1982); (2) Bragg reflection
of monochromatic waves from the sinusoidal ripples (Davies and Heathershaw, 1985); (3) wave transformation near a shore attached
breakwater (Watanabe and Maruyama, 1986). Comparisons of the numerical solutions with the experimental or theoretical ones
or with those of other models (REF/DIF model and FUNWAVE model) show good results, which indicate that the present model is
capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy
dissipation and weak nonlinearity in the near shore zone.
Key wordstime-dependent-mild-slope equation-nonlinear amplitude dispersion-steep or rapidly varying topography-bottom friction-wave breaking
The purpose of this paper is to extend the validity of Li's parabolic model (1994) by incorporating a combined energy factor in the mild-slope equation and by improving the traditional radiation boundary conditions. With wave breaking and energy dissipation expressed in a direct form in the equation, the proposed model could provide an efficient numerical scheme and accurate predictions of wave transformation across the surf zone. The radiation boundary conditions are iterated in the model without use of approximations. The numerical predictions for wave height distributions across the surf zone are compared with experimental data over typical beach profiles. In addition, tests of waves scattering around a circular pile show that the proposed model could also provide reasonable improvement on the radiation boundary conditions for large incident angles of waves.
The "Halloween' Northeaster of October 29-31, 1991 generated severe seas; the maximum wave height and peak wave period reached 8 m and 20 s at an offshore wave station located at the continental shelf break about 100 km off the Virginia coast. Storm waves were also measured shoreward at a nearshore wave station located at the Chesapeake Bay Light Tower. We calculated the representative wave heights, periods, and deepwater wave directions for each band. We then used this offshore wave information, a numerical scheme for calculating wave height attenuation caused by bottom friction, and the given bathymetry to evaluate the following four wave transformation processes: refraction, diffraction, shoaling, and bottom friction. -from Author
A numerical model is developed for calculating the transformation of irregular waves due to refraction, diffraction and breaking. The model is based on a parabolic equation which is derived from the mild slope equation¹⁾ with an additional term of energy dissipation.
In the present model, irregular waves are treated as a superposition of component regular waves with different frequencies and directions. In order to calculate the transformation accurately for a wide range of propagation directions of component waves, a new parabolic equation is derived by taking into account the difference between directions of wave propagation and a prechosen coordinate. As a first step in modeling the breaking transformation of irregular waves, the breaking criterion and the energy dissipation coefficient are formulated as a function of a parameter corresponding to the significant wave.
Results of numerical calculation are compared with available analytical solutions and show a fairy good agreement. Sample results of applications are also shown.
A numerical model based on newly-derived mild slope equations is presented. Its accuracy is verified through comparisons with analytical solutions of shoaling, refraction and diffraction. Breaking and wave decay in the surf zone are also incorporated in the model. Good agreements are shown between calculations and hydraulic experiments of nearshore waves around a detached breakwater and a jetty. Formulas for radiation stresses of the compound wave field are also presented.
The flooding event occurs when the discharge of a river is more than the river capacity. With high and rough topographic structure, Turkey is located in a semi-arid climate zone and both spatial and seasonal distribution of precipitation is quite irregular. These irregular precipitations create the flooding events with landform, topographic structure, faulty land use, unplanned urbanization and destruction of forest areas. Since floods are characterized by discharge velocity, discharge level and high water levels, these flood characteristics should be known and preventive actions must be taken for all buildings to be built in river basins. The solutions which are made for determining flood characteristics are called as flood routing and developed by means of St. Venant equations. St. Venant equations can be solved in different wave approaches and named hydraulic methods in flood routing phenomenon. In addition to hydraulic methods, hydrological methods that based only mass conservation can also be used in flood routing phenomenon. St. Venant equations can be linearized mathematically with some assumptions, however different wave approaches can be used, it can be denoted as diffusion wave approach. The diffusion wave equation can be solved by different methods like finite difference and finite element methods. In this study, the differential quadrature method (DQM) is used for the numerical solution of diffusion wave equation and it is employed to real flood events data obtained from Sivapalan(1997) and Ozdogan(2010). The DQM results are compared with finite difference results [1,2]. As seen from the examples, for the solution in DQM it is enough to use fewer solution points.
Introduction of an effective wave elevation function, the simplest time-dependent hyperbolic mild-slope equation has been presented and an effective numerical model for the water wave propagation has been established combined with different boundary conditions in this paper. Through computing the effective wave elevation and transforming into the real transient wave motion, then related wave heights are computed. Because the truncation errors of the presented model only induced by the dissipation terms, but those of Lin’s model (2004) contributed by the convection terms, dissipation terms and source terms, the error analysis shows that calculation stability of this model is enhanced obviously compared with Lin’s one. The tests show that this model succeeds to the merit in Lin’s one and the computer program simpler, computational time shorter because of calculation stability enhanced efficiently and computer memory decreased obviously. The presented model has the capability of simulating exactly the location of transient wave front by the speed of wave propagation in the first test, which is important for the real-time prediction of the arrival time of water waves generated in the deep sea. The model is validated against experimental data for combined wave refraction and diffraction over submerged circular shoal on a flat bottom in the second test. Good agreements are gained. The model can be applied to the theory research and engineering applications about the wave propagation in the coastal waters.
Based on the linear wave theory, the mild-slope equation (MSE) is a preferred mathematical model to simulate nearshore wave propagation. A numerical model to solve the MSE is developed here on the basis of a self-adaptive finite element model (FEM) combined with an iterative method to determine the wave direction angle to the boundary and thus to improve the treatment of the boundary conditions. The numerical resolution of the waves into ideal domains and multidirectional waves through a breakwater gap shows that the numerical model developed here is effective in representing wave absorption at the absorbing boundaries and can be used to simulate multidirectional wave propagation. Finally, the simulated wave distribution in a real harbor shows that the numerical model can be used for engineering practice.
A parabolic approximation formulation is presented herein to study the wave-current interactions on a slowly varying topography. Model equations are first derived from the mild-slope assumption. A parabolic approximation is then applied to convert a boundary value problem to an initial value problem. Several limiting cases are discussed. Numerical examples are given for the case concerning the interactions between jet-like currents and normal incident waves on a sloping beach (Arthur, 1950).
The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland's hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit (ADI) method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.
The oscillatory flow near the sea bed under a wave motion is always rough turbulent in a coastal zone. This type of an oscillatory boundary layer (or “wave boundary layer”) was therefore chosen as a subject for detailed velocity measurements, from which characteristics such as shear stresses, eddy viscosities, energy loss, and boundary layer thickness were determined.
This paper reports the results of a new mathematical derivation for the transformation of a progressive wave propagating obliquely on a gentle slope. On the basis of the conservation principle of wave motion and in a wave-ray coordinate system, an explicit expression for the velocity potential of the wave field is derived as a function of the bottom slope a perturbed to a second order in an Eulerian coordinate system. Wave profile is then obtained in the Lagrangian system. Together, these enable the description of the features of wave shoaling and refraction in the direction of wave propagation from deep to shallow water, particularly, the process of successive deformation of a wave profile.
A numerical model based on the mild-slope equation is applied to reproduce the propagation of small-amplitude transient waves. The model makes use of the Fourier Transform to convert the time-dependent hyperbolic equation into a set of elliptic equations in the frequency domain. The results of two available experimental studies on tsunamis generated by landslides are used to validate the model, which appears to be able of carefully reproducing the effects of the frequency dispersion. An example application of tsunamis propagating around the Stromboli island is also presented to show the applicability of the present approach to real life scenarios. It is finally discussed how this model could be applied as support to a tsunami early warning system.
The multigrid method is used to solve very efficiently the elliptic form of the mild-slope equation for water wave propagation over large areas in the presence of currents, taking into account the combined effects of shoaling, refraction, diffraction and wave breaking. Wave reflections may also be taken into account but in this case additional computational resources are required. The present scheme offers significant advantages over other existing elliptic and hyperbolic solution techniques for the mild-slope equation with regard to computational efficiency and speed. The original equation as well as its variant which includes the effects of wave-current interaction are first recast into forms which can be readily handled by the multigrid method. Solutions of the governing equations are successively obtained for a number of increasingly coarser grid meshes, using the Gauss-Seidel Iterative Method for all grid meshes apart from the coarsest for which the Gauss Elimination Method may also be used. The main advantage of this approach is that, while traditional elliptic and hyperbolic solution schemes for the mild-slope equation require a large number of grid points per wavelength, the present scheme requires no more than two to three points, thus reducing overall computational effort. Moreover, the solution procedure is as efficiently computationally as the parabolic approximations for the mild-slope equation, without imposing any of the constraints of those schemes. Verification of the model for a number of test cases confirms that it is stable, highly accurate and economical to use.
The standard implementation of the GMRES method for solving large nonsymmetric linear systems involves a Gram-Schmidt process which is a potential source of significant numerical error. An alternative implementation is outlined here in which orthogonalization by Householder transformations replaces the Gram-Schmidt process. This implementation requires slightly less storage but somewhat more arithmetic than the standard one; however, numerical experiments suggest that it is more stable, especially as the limits of residual reduction are reached. The extra arithmetic required may be less significant when products of the coefficient matrix with vectors are expensive or on vector and, in particular, parallel machines.
A method is proposed for smoothly matching an approximate, shallow-water dispersion relation to an analytically obtained relation for intermediate and deep water. The method provides a correct limit for increasing water depth in the case of weakly non-linear waves, and provides a smooth prediction of wave parameters for the entire range of water depth. The model is applied to a parabolic equation form of the combined refraction-diffraction model, and numerical results are presented in comparison to published data.
A model equation is derived for calculating transformation and propagation of Stokes waves. With the assumption that the water depth is slowly varying, the model equation, which is a nonlinear Schrödinger equation with variable coefficients, describes the forward-scattering wavefield. The model equation is used to investigate the wave convergence over a semicircular shoal. Numerical results are compared with experimental data (Whalin 1971). Nonlinear effects, which generate higher-harmonic wave components, are definitely important in the focusing zone. Mean free-surface set-downs over the shoal are also computed.
The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.
A modified version of the mild-slope equation is derived and its predictions of wave scattering by two-dimensional topography compared with those of other equations and with experimental data. In particular, the modified mild-slope equation is shown to be capable of describing known scattering properties of singly and doubly periodic ripple beds, for which the mild-slope equation fails. The new equation compares favourably with other models of scattering which improve on the mild-slope equation, in that it is widely applicable and computationally cheap.
A parabolic approximation to the reduced wave equation is investigated for the propagation of periodic surface waves in shoaling water. The approximation is derived from splitting the wave field into transmitted and reflected components.
In the case of an area with straight and parallel bottom contour lines, the asymptotic form of the solution for high frequencies is compared with the geometrical optics approximation.
Two numerical solution techniques are applied to the propagation of an incident plane wave over a circular shoal.
A new form of the Boussinesq equations applicable to irregular wave propagation on a slowly varying bathymetry from deep to shallow water is introduced. The equations incorporate excellent linear dispersion characteristics, and are formulated and solved in two horizontal dimensions. In an earlier paper we concentrated on wave propagation and diffraction on a horizontal bottom in deep water. In this paper these principles are generalized and the Boussinesq equations are extended to include terms proportional to the bottom slope, which are essential for the shoaling properties of the equations. The paper contains a linear shoaling analysis of the new equations and a verification of the numerical model with respect to shoaling and refraction-diffraction in deep and shallow water.
Two time-dependent equations for wave propagation on rapidly varying topography are developed using different theoretical approaches and are shown to be identical. The developed equations include higher-order bottom effect terms proportional to the square of bottom slope and to the bottom curvature. Without these higher-order terms, the equations developed are reduced to the time-dependent mild-slope equations of Smith and Sprinks and Radder and Dingemans, respectively. For a monochromatic wave, the equation reduces to the extended refraction-diffraction equation of Massel or the modified mild-slope equation of Chamberlain and Porter, which in turn, without the higher-order terms, reduces to the Berkhoff's mild-slope equation. For a monochromatic wave, the theory is verified against other theoretical and experimental results related to the waves propagating over a plane slope with different inclination and over a patch of periodic ripples. For random waves, numerical tests are made for the transmission of unidirectional random waves normally incident on a finite ripple patch.
A wave transformation model (RIDE) was enhanced to include the process of wave breaking energy dissipation in addition to water wave refraction, diffraction, reflection, shoaling, bottom friction, and harbor resonance. The Gaussian Elimination with partial Pivoting (GEP) method for a banded matrix equation and a newly developed bookkeeping procedure were used to solve the elliptic equation. Because the bookkeeping procedure changes the large computer memory requirements into a large hard-disk-size requirement with a minimum number of disk I/O, the simple and robust GEP method can be used in personal computers to handle realistic applications. The computing time is roughly proportional to N1.7, where N is the number of grid points in the computing domain. Because the GEP method is capable of solving many wave conditions together (limited by having the same wave period, no bottom friction and no breaking), this model is very efficient compared to iteration methods when simulating some of the wave transformation process.
The “mild-slope” equation which describes wave propagation in shoaling water is normally expressed in an elliptic form. The resulting computational effort involved in the solution of the boundary value problem renders the method suitable only for small sea areas. The parabolic approximation to this equation considerably reduces the computation involved but must omit the reflected wave. Hence this method is not suited to the modelling of harbour systems or areas near to sea walls where reflections are considerable. This paper expresses the “mild-slope” equation in the form of a pair of first-order equations, which constitute a hyperbolic system, without the loss of the reflected wave. A finite-difference numerical scheme is described for the efficient solution of the equations which includes boundaries of arbitrary reflecting power.
The present paper describes a method based on linear diffraction theory for predicting the wave field in a harbour containing partially reflecting boundaries. The method utilizes a point source representation of the harbour boundaries and a matching boundary which separates regions interior and exterior to the harbour, and involves the application of a partial reflection boundary condition. Numerical results are presented for the wave field within a rectangular harbour with a pair of symmetrical breakwaters, for cases of fully absorbing, fully reflecting, and partially reflecting boundaries. The method appears to be able to account adequately for the effects of wave diffraction and partial reflections, and to predict the wave field realistically.
Iterative solution procedures for solving the complete mild-slope wave (combined refraction-diffraction) equation are developed. Existing models for investigating wave refraction-diffraction in coastal areas have one of two main problems; (i) Some of the physics is lost as they resort to approximate solutions (e.g. parabolic approximations). Thus they are inappropriate in many situations. (ii) If all of the physics is to be incorporated, the problem defies computer solution except for extremely small domains (approximately 10 wavelengths), chiefly because the matrix equation associated with numerical discretization of the complete problem does not normally lend itself to solution by iteration. This paper describes the construction of iterative models that overcome both problems. First, a modified equation with an identical solution but which lends itself to iterative procedures is formulated, and the conjugate gradient method is used. A second, more rapidly converging algorithm is obtained by preconditioning.It is shown that the algorithms can be conveniently implemented on regions much larger thanthose handled by conventional models, without compromising the physics of the equation. Further, they can be efficiently run in either the linear or nonlinear mode. Comparisons of model results with laboratory data and other numerical and analytical solutions are found to be excellent for several cases. The procedures thus enable the engineer to expand the scope of the mild-slope equation. As an example, an experiment is performed to demonstrate the sensitivity of the wavefield to the location of a breakwater in a region with complex bathymetry.
A system of differential equations, the stationary part of which can be reduced to the elliptic mild-slope equation, is derived. The transient terms make the system of equations hyperbolic and similar to the system of equations governing nearly horizontal flow. The highly efficient ADI algorithm for the latter is used iteratively to find the stationary solution. By extracting the time-harmonic part and using a varying time step in the iterations the computational time is reduced greatly as compared with previous techniques.