• Journal of Ocean Engineering and Technology
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• v.18 no.2
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• pp.18-24
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• 2004

In this study, the Mild slope equation is extended to both rapidly varying topography and nonlinear waves, using the Hamiltonian principle. It is shown that this equation is equivalent to the modified mild-slope equation (Kirby and Misra, 1998) for small amplitude wave, and it is the same form with the nonlinear mild-slope equation (Isobe, 1994) for slowly varying bottom topography. Comparing its numerical solutions with the results of some hydraulic experiments, there is good agreement between them.

• Journal of Ocean Engineering and Technology
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• v.18 no.4 s.59
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• pp.40-45
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• 2004

An efficient numerical model of the modified mild slope equation, based on the robust iterative method is presented. The model developed is verified against other numerical experimental results, related to wave reflection from an arc-shaped bar and wave transformation over a circular shoal. The results show that the modified mild slope equation model is capable of producing accurate results for wave propagation in a region where water depth varies substantially, while the conventional mild slope equation model yeilds large errors, as the mild slope assumption is violated.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.19 no.6
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• pp.521-532
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• 2007

In the derivation of mild slope equation, a Galerkin method is used to rigorously form the Sturm-Liouville problem of depth dependent functions. By use of the canonical transformation to the dependent variable of the equation a reduced Helmholtz equation is obtained which exclusively consists of terms proportional to wave number, bottom slope and bottom curvature. Through numerical studies the behavior of terms is shown to play an important role in wave transformations over variable depth and it is proved that their relative magnitudes limit applicability of the mild slope equation(MSE) against the modified mild slope equation(MMSE).

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.10 no.2
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• pp.55-63
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• 1998

Recently the modified mild-slope equation has been developed by several researchers using different approaches, which, compared to the Berkhoff's mild-slope equation, includes additional terms proportional to the square of bottom slope and to the bottom curvature. By examining this equation, it is shown that both terms are equally important in intermediate-depth water, but in shallow water the influence of the bottom curvature term diminishes while that of the bottom slope square term remains significant. In order to examine the importance of these terms in more detail, the modified mild-slope equation and the Berkhoff's mild-slope equation are tested for the problems of wave reflection from a plane slope, a non-plane slope, and periodic ripples. It is shown that, when only the bottom slope is concerned, the mild-slope equation can give accurate results up to a slope of 1 in 1 rather than 1 in 3, which, until now, has been known as the limiting bottom slope for its proper application. It is also shown that the bottom curvature term plays an important role in modeling wave propagation over a bottom topography with relatively mild variation, but, where the bottom slope is not small, the bottom slope square term should also be included for more accurate results.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.29 no.5B
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• pp.493-496
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• 2009

In this study, we derive the extended mild-slope equation in terms of the velocity potential using the Euler-Lagrange equation. First, we follow Kim and Bai (2004) who derive the complementary mild-slope equation in terms of the stream function using the Euler-Lagrange equation and we compare their equation to the existing extended mild-slope equations of the velocity potential. Second, we derive the extended mild-slope equation in terms of the velocity potential using the Euler-Lagrange equation. In the developed equation, the higher-order bottom variation terms are newly developed and found to be the same as those of Massel (1993) and Chamberlain and Porter (1995). The present study makes wide the area of coastal engineering by developing the extended mild-slope equation with a way which has never been used before.

• Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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• 2003.05a
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• pp.72-77
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• 2003

In this study, Mild slope equation is extended to both of rapidly varying topography and nonlinear waves in a Hamiltonian formulation. It is shown that its linearzed form is the same as the modified mild-slope equation proposed by Kirby and Misra(1998) And assuming that the bottom slopes are very slowly, it is the equivalent with nonlinear mild-slope equation proposed by Isobe(]994) for the monochromatic wave. Using finite-difference method, it is solved numerically and verified, comparing with the results of some hydraulic experiments. A good agreement between them is shown.

• The Journal of the Korea Contents Association
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• v.12 no.3
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• pp.434-441
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• 2012

An analytic solution to the extended mild-slope equation was derived for waves propagating over an axi-symmetric pit. The water depth inside the pit was in proportion to a power of radial distance from the center of pit. The equation was transformed into the ordinary differential equation using the method of separation of variables. The coefficients of differential terms were expressed as an explicit form composing of the phase and group velocities. The bottom curvature and the square of bottom slope terms, which were added to the extended mild-slope equation, were expressed as power series. Finally, using the Frobenius series, the analytic solution to the extended mild-slope equation was derived. The present analytic solution was validated by comparing with the numerical solution obtained from FEM.

• Proceedings of the KSME Conference
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• 2002.08a
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• pp.209-212
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• 2002

Since Berkhoff proposed the mild-slope equation in 1972, it has widely been used for calculation of shallow water wave transformation. Recently, it was extended to give an extended mild-slope equation, which includes the bottom slope squared term and bottom curvature term so as to be capable of modeling wave transformation on rapidly varying topography. These equations were derived by integrating the Laplace equation vertically. In the present study, we develop a finite element model to solve the Laplace equation directly while keeping the same computational efficiency as the mild-slope equation. This model assumes the vertical variation of wave potential as a cosine hyperbolic function as done in the derivation of the mild-slope equation, and the Galerkin method is used to discretize . The computational domain was discretized with proper finite elements, while the radiation condition at infinity was treated by introducing the concept of an infinite element. The upper boundary condition can be either free surface or a solid structure. The applicability of the developed model was verified through example analyses of two-dimensional wave reflection and transmission. .

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.18 no.4
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• pp.360-371
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• 2006

In order to calculate waves propagating into the shallow water region, a generalized parabolic approximate model is presented. The model is derived from the modified mild slope equation and includes all the existing parabolic models presented in the paper. Numerical results are presented in comparison to laboratory data of Berkhoff et al.(1982). The existing parabolic model shows almost same accuracy against the modified parabolic model and both results of models stand in closer agreement to the laboratory data. Therefore the existing parabolic model based on mild slope equation is a useful tool to compute shallow water waves which turns out to be more fast and stable in computational aspect.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.6 no.1
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• pp.81-87
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• 1994

The mild slope equation has been directly derived from the energy equation, and the relation between energy equation and Green's first and second identities was also clarified. It is shown here that the mild slope equation of elliptic type in the wave-current interaction has to have the same form as the one derived by Berkhoff (1972), and its physical meaning was investigated through analytical solutions.