Mild-slope Wave Equation Research Articles

Wave energy reflection by the Bragg breakwater composed of rectangular bars on the horizontal permeable seabed

Wave energy is a kind of renewable and clean energy. When waves meet the breakwaters, wave energy reflection will happen. In this paper, the modified mild slope equation for waves propagating over the permeable seabed is introduced, and the finite difference model to solve the equation is set up. The accuracy and applicability of the model is verified with Zeng et al’s analytical solution for wave reflection by the rectangular Bragg breakwaters on the impermeable seabed. Furthermore, in case of a permeable seabed, the effects of the seabed permeability, the bar width, the bar number and the submergence of the bars on wave reflection coefficient, as well as the difference with the case of an impermeable seabed, is studies in details. The results show the reflection coefficient of the Bragg resonant reflection increases with the increase in the bar number, but decreases with the increase in the seabed permeability and the submergence of the bars. When the bar number increases from 1 to 8, the Bragg resonance reflection coefficient increases from 0.146 to 0.772. When the permeability parameter of the seabed increases from 0.005 s to 0.03 s, the coefficient decreases from 0.403 to 0.347. When the bar submergence increases from 0.5 to 0.875, the coefficient decreases from 0.842 to 0.195. Moreover, there exists a particular value of the bar width, that is half of the distance between two adjacent rectangular bars that maximizes the wave energy reflection.

Submerged breakwater of a flexible porous membrane with a vertical flexible porous wall over variable bottom topography

An analytical model of oblique wave interaction with a breakwater consisting of a submerged horizontal flexible porous membrane near a vertical porous barrier over variable bottom topography is developed based on linear water wave theory. Using Green's second identity, a mild-slope equation for oblique waves propagating over a submerged porous membrane and a finite variable bottom topography is derived. The analytical solution of the referred problem is obtained by using the eigenfunction expansion method along with the mild-slope equation and Darcy's law is used for wave past horizontal and vertical porous structure. The present results are compared with existing published analytical results and experimental data and observed that they have a good level of agreement. The effect of different design parameters on the wave quantities and wave forces on the porous wall in variable bottom depth is analyzed. It is found that full-wave reflection is drastically reduced and the wave energy is dissipated by the submerged horizontal flexible porous membrane and porous barrier. The analysis of the reflection and dissipation coefficients as well as wave forces on the vertical wall in terms of wave energy dissipation and wave loads by the proposed submerged flexible porous breakwater will be helpful for better understanding in the design of various types of protecting infrastructures for marine operations.

A Harbor Resonance Numerical Model with Reflecting, Absorbing and Transmitting Boundaries

Background:A very important aspect in the planning and design of a harbor is to determine the response of the harbor basin to incident waves. Many previous investigators have studied various aspects of the harbor resonance problem, though correct to a certain extent, have some disadvantages.Objective:To calculate wave response in an offshore or coastal harbor of arbitrary shape, this research develops a two-dimensional linear, inviscid, dispersive, hybrid finite element harbor resonance model using conservation of energy approach. Based on the mild-slope wave equation, the numerical model includes wave refraction, diffraction, and reflection. The model also incorporates the effects of variable bathymetry, bottom friction, variable, full or partial absorbing boundaries, and wave transmission through permeable breakwaters.Methods:Based on the mild-slope wave equation, the numerical model includes wave refraction, diffraction, and reflection. The model also incorporates the effects of variable bathymetry, bottom friction, variable, full or partial absorbing boundaries, and wave transmission through permeable breakwaters. The Galerkin finite element method is used to solve the functional which was obtained using the governing equations. This model solves both long-waves as well as short-wave problems. The accuracy and efficiency of the present model are verified by comparing different cases of rectangular harbor numerical results with analytical and experimental results.Results:There said results indicate that reduction in wave amplitude inside a harbor caused by energy dissipation due to water depth, linearly sloping bottom, and bottom friction is quite small for a deep harbor. But for a shallow harbor, these factors are critical. They also show that reduction in wave amplitude inside a harbor due to boundary absorption, permeable transmission, harbor entrance width, and horizontal dimensions.Conclusion:Those factors are very important for both deep and shallow harbors as proven by accurate agreement with the prediction of this numerical model. The model presented herein is a realistic method for solving harbor resonance problems.

A coupled finite and boundary spectral element method for linear water-wave propagation problems

• A coupled finite and boundary spectral element method for linear water-wave propagation problems is proposed. • Boundary spectral element method (BSEM) is a new technique that combines the advantages of the spectral approach and the BEM. • BSEM has been applied to the mild-slope equation with variable bathymetry in one direction. • A convergence study has been made for the BSEM alone and coupled with finite spectral elements. • The proposed formulation has been validated by solving classical water-wave propagation problems. A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p -convergence of spectral methods.

Extended mild-slope equation for surface waves interacting with a vertically sheared current

Propagation of water waves in coastal zones is mainly affected by the influence of currents and bathymetry variations. Models describing wave propagation in coastal zones are often based on the numerical solution of the Mild Slope equation (Kirby, 1984). In this work, an extension of this equation is derived, taking into account the linear variation of the current with depth, which results in a constant horizontal vorticity, slowly varying horizontally, within the background current field. The present approach is based on the asymptotic expansion of the depth-integrated lagrangian, assuming the linear variation of the background current with depth. With the aid of selected examples the role of this horizontal vorticity, associated with the assumed background current velocity profile, is then illustrated and emphasized, demonstrating its effect on the propagation of water waves in coastal areas.

A finite difference method for effective treatment of mild-slope wave equation subject to non-reflecting boundary conditions

A numerical model for coastal water wave motion that includes an effective method for treatment of non-reflecting boundaries is presented. The second-order one-way wave equation to approximate the non-reflecting boundary condition is found to be excellent and it ensures a very low level of reflection for waves approaching the boundary at a fairly wide range of the incidence angle. If the Newman approximation is adopted, the resulting boundary condition has a unique property to allow the free propagation of wave components along the boundary. The study is also based on a newly derived mild-slope wave equation system that can be easily made compatible to the one-way wave equation. The equation system is theoretically more accurate than the previous equations in terms of the mild-slope assumption. The finite difference method defined on a staggered grid is employed to solve the basic equations and to implement the non-reflecting boundary condition. For verification, the numerical model is then applied to three coastal water wave problems including the classical problem of plane wave diffraction by a vertical circular cylinder, the problem of combined wave diffraction and refraction over a submerged hump in the open sea, and the wave deformation around a detached breakwater. In all cases, the numerical results are demonstrated to agree very well with the relevant analytical solutions or with experimental data. It is thus concluded that the numerical model proposed in this study is effective and advantageous.

A new model for wave transformation around axis-symmetric islands without vertical wall along the coastline

In this study, we develop a numerical model using the elliptic mild-slope equation for waves propagating around axis-symmetric topographies where the water depth varies arbitrarily, with the depth being zero at the coastline. The entire area is divided into three regions. In both the inner and outer regions, existing analytical solutions are used. In the middle region occupying most computational domain, the finite element technique is applied to the governing equation. To obtain the solution, a method of the separation of variables and the Frobenius series are used. The developed solution is validated by comparing it with previously developed analytical solutions.

Numerical study on water waves and wave-induced longshore currents in Obaköy coastal water

In this paper, the water waves and wave-induced longshore currents in Obakoy coastal water which is located at the Mediterranean coast of Turkey were numerically studied. The numerical model is based on the parabolic mild-slope equation for coastal water waves and the nonlinear shallow water equation for the wave-induced currents. The wave transformation under the effects of shoaling, refraction, diffraction and breaking is considered, and the wave provides radiation stresses for driving currents in the model. The numerical results for the water wave-induced longshore currents were validated by the measured data to demonstrate the efficiency of the numerical model. Then the water waves and longshore currents induced by the waves from main directions were numerically simulated and analyzed based on the numerical results. The numerical results show that the movement of the longshore currents was different while the wave propagated to a coastal zone from different directions.

An exact analytic solution to the modified mild-slope equation for waves propagating over a trench with various shapes

An exact analytic solution to the modified mild-slope equation (MMSE) in terms of Taylor series for waves propagating over an asymmetrical trench with various shapes is given. Because of the use of the MMSE, on one hand, the present analytic solution can be valid in the whole wave range from long waves to short waves, which is clearly superior to all previous long-wave analytic solutions; on the other hand, the present analytic solution can get rid of the limitation of the ‘mild slope’ assumption and be valid for bottom slope as high as 1:1. It is clarified that the improvement in solution accuracy by using the mass-conserving matching condition against the conventional matching condition mainly depends upon the jump quantities at all common boundaries. In addition, in comparison with previous approximate analytic model based on the approximate mild-slope equation, the present model is more accurate and can converge in the whole trench region without any restriction to trench depth. Based on the present MMSE solution, influence of trench dimensions to reflection effect is analyzed, which shows that total reflection effect increases when trench wall becomes steep and the phenomenon of zero reflection mainly occurs for symmetrical trenches.

An analytic solution for combined wave diffraction and refraction around a vertical cylinder with idealized scour pit

An analytic solution of the mild slope wave equation for wave propagating over a vertical cylinder with a scour pit has been derived. The scour pit is idealized to be geometrically axisymmetrical, and the water depth inside the pit is assumed to decrease in proportion to an arbitrary power of the radial distance from the center of the cylinder. By separation of variables, solution of the mild slope wave equation is reduced to find the general solution of an ordinary differential equation, of which the coefficients are functions of the wavenumber or the wave celerity, and can eventually be transformed into explicit functions of the independent variable by means of Hunt's approximate formula for the dispersion relation of small amplitude waves. By a tactical mapping of the coefficients into polynomials, the ordinary differential equation is successfully solved in the form of Frobenius series. The solution obtained is a significant extension to the long wave solution previously presented by the same authors. The long wave assumption is now removed and the shape of the scour pit is also generalized to a large extent. Based on the solution obtained, the wave run up around the cylinder is carefully studied, and the effects of the incident wave conditions, the extent, the depth, and the shape of the scour pit on wave transformation are also discussed.

Long-Wave Scattering by a Vertical Cylinder with Idealized Scour Pit

The combined diffraction and refraction of long waves around a vertical cylinder with scour pit is studied analytically on the basis of a newly derived analytic solution of the mild-slope wave equation. The geometry of the scour pit around the cylinder is idealized to be axisymmetrical, and a simple power function is employed to represent its radial profile. The problem is solved in the polar coordinate system by separation of variables, and the resulting ordinary differential equation in the radial direction over the scour pit in this case can be transformed into the Bessel equation of real order. On the basis of the analytic solution obtained, the effect of the scour pit on wave scattering is investigated by comparing the cases with and without the presence of the scour pit. The scour pit contributes to reduce the wave run-up against the cylinder. In addition, the effects of the scour pit on wave run-up against the cylinder are more significant when the dimensions of the scour pit are larger or the incident wave is relatively shorter.

Analytical study on long wave refraction over a dredge excavation pit

An analytical solution of the mild-slope wave equation is derived to describe long wave propagating over the idealized dredge excavation pit. The pit is assumed to be axisymmetrical and composed of a flat bottom and a convex slope. The convex slope is expressed by a simple power function. The problem is solved in the polar coordination system by the separation of variables. By the obtained solution, the characteristics of the wave refraction and reflection over the dredge excavation pit are discussed. The results show that wave amplitude is attenuated within and in the lee side of the pit and amplified at the rear flank of the pit due to wave refraction. The effects of the incident wave length and the shape of the pit on wave refraction are also discussed.

Analytic solution of long wave propagation over a submerged hump

A new analytical solution of the long wave refraction by a submerged circular hump is presented. The geometry of the hump is assumed to be axisymmetric and be described by a power function in the radial direction with arbitrary values of both the exponent and the scaling factor. The submergence of the hump is also variable. The water surface elevation governed by the long wave version of the mild slope wave equation is solved by separation of variables, and a series solution of the Frobenius type is obtained. The solution is shown to be valid when the hump is sufficiently submerged or is of a relatively small height. Matching method is employed to illustrate the refraction of long waves under given conditions of incidence. Effects of the shape, the scale, and the submergence of the hump on wave refraction are discussed.

An analytical solution to the extended mild-slope equation for long waves propagating over an axi-symmetric pit

In this study, an analytic solution to the extended mild-slope equation is derived for long waves propagating over an axi-symmetric pit, where the water depth decreases in proportion to a power of radial distance from the pit center. The solution is obtained using the separation of variables method and the Frobenius method. By comparing the extended and conventional mild-slope equations for waves propagating over conical pits with different bottom slopes, it is shown that for long waves the conventional mild-slope equation is reasonably accurate for bottom slopes less than 1:3 in horizontal two-dimensional domains. The effects of the pit shape on wave scattering are discussed based on the analytic solutions for different powers. Comparison is also made with an analytic solution for a cylindrical pit with a vertical sidewall. Finally, wave attenuation in the region over the pit is discussed.

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.

Modelling linear wave transformation induced by dissipative structures—Regular waves

A numerical model of the modified time-independent mild-slope equation for linear waves over a rapidly changing finite porous bed is presented. In this solution the reflection and phase coefficient shift are solved implicitly. Boundaries are assumed to be open, partially reflecting, or fully absorbing through the second-order parabolic approximation. Discretisation of the governing equation and boundary conditions is by means of a second-order accurate central difference scheme. The resulting sparse-banded matrix is solved using an inexpensive banded solver with Gaussian elimination. The model has been validated and the numerical predictions are in excellent agreement with analytical solutions.

Generation of random waves in time-dependent extended mild-slope equations using a source function method

We develop techniques of numerical wave generation in the time-dependent extended mild-slope equations of Suh et al. [1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Engineering 32, 91–117] and Lee et al. [2003. Extended mild-slope equation for random waves. Coastal Engineering 48, 277–287] for random waves using a source function method. Numerical results for both regular and irregular waves in one and two horizontal dimensions show that the wave heights and the frequency spectra are properly reproduced. The waves that pass through the wave generation region do not cause any numerical disturbances, showing usefulness of the source function method in avoiding re-reflection problems at the offshore boundary.

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.

On the modeling of wave–current interaction using the elliptic mild-slope wave equation

Methods to incorporate the effect of ambient currents in the prediction of nearshore wave transformation are developed. This is accomplished through the construction of a finite-element coastal/harbor wave model based on an extended mild-slope wave–current equation that includes wave breaking. Improved boundary conditions are used to provide more accurate forcing and to minimize spurious wave reflections from the boundaries. Multiple nonlinear mechanisms, appearing both in the governing equations and in the boundary conditions, are handled successfully and efficiently with iterative techniques. The methods are tested against results from other types of models based on parabolic approximations or Boussinesq equations for three wave–current problems of common interest and varying complexity. While indicating good agreement in general, the analysis also highlights the limitations of parabolic approximation models in case of strong local currents and velocity shear. We also consider the harbor engineering problem pertaining to waves approaching an inlet with a jettied entrance, where wave–current interaction can create a complex wave pattern that adversely affects small craft navigation and causes scouring. The role of ebb and flood currents on wave transformation and on breaking in the vicinity of the inlet is investigated using the model in conjunction with hydraulic laboratory data. It is found that although the ebb currents cause larger waves outside the inlet, much of the wave energy is soon dissipated due to breaking; during the flood tide, in contrast, more wave energy can penetrate into the inlet throat.

Long waves propagating over a circular bowl pit

An analytic solution to the mild slope wave equation is derived for long waves propagating over a circular, bowl-shaped pit located in an otherwise constant depth region. The analytic solution is shown to reduce to a previously derived analytic solution for the case of a bowl-shaped enclosed basin and to agree well with a numerical solution of the hyperbolic mild-slope equations. The effects of the pit dimensions on wave scattering are discussed based on the analytic solution. This analytic solution can also be applied to pits of different general shapes. Finally, wave attenuation in the region over the pit is discussed.