Changhoon Lee

Hyperbolic mild-slope equations extended to account for rapidly varying topography

Abstract In this paper, following the procedure outlined by Copeland [Copeland, G.J.M., 1985. A practical alternative to the mild-slope wave equation. Coastal Eng. 9, 125–149.] the elliptic extended refraction–diffraction equation of Massel [Massel, S.R., 1993. Extended refraction–diffraction equation for surface waves. Coastal Eng. 19, 97–126.] is recasted into the form of a pair of first-order equations, which constitute a hyperbolic system. The resultant model, which includes higher-order bottom effect terms proportional to the square of bottom slope and to the bottom curvature, is merely an extension of the Copelands model to account for a rapidly varying topography. The importance of the higher-order bottom effect terms is examined in terms of relative water depth. The model developed is verified against other numerical or experimental results related to wave reflection from a plane slope with different inclination, from a patch of periodic ripples, and from an arc-shaped bar with different front angle. The relative importance of the higher-order bottom effect terms is also examined for these problems.

Extended mild-slope equation for random waves

A time-dependent extended mild-slope equation is derived from the elliptic equation of Chamberlain and Porter [J. Fluid Mech. 291 (1995) 393] using the Taylor series technique. Numerical tests are made on a horizontally one-dimensional case for regular waves over sloping beds and for both regular and irregular waves over a ripple patch. Numerical results prove that the proposed model gives accurate results for both regular and irregular waves over rapidly varying topography.

Internal generation of waves for extended Boussinesq equations

It is studied whether the mass transport or energy transport is the proper viewpoint for internally generating waves in the extended Boussinesq equations of Nwogu (1993). The numerical solutions of the Boussinesq equations with the internal generation of sinusoidal waves show that the energy transport yields required wave energy properly while the mass transport yields wave energy different from the required one with the ratio of phase speed to energy velocity. The waves which pass through the wave generation point do not cause any numerical problems. It can be concluded that the proper viewpoint for the internal generation of waves in the hyperbolic wave equations is the energy transport rather than the mass transport.

Experimental verification of horizontal two-dimensional modified mild-slope equation model

In order to verify modified mild-slope equation models in a horizontal two-dimensional space, a hydraulic experiment is made for surface wave propagation over a circular shoal on which water depth varies substantially. A horizontal two-dimensional numerical model is also constructed based on the hyperbolic equations that have been developed from the modified mild-slope equation to account for the substantial depth variation. Comparison between experimental measurements and numerical results shows 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 gives large errors as the mild-slope assumption is violated.

Asymmetry in Directional Spreading Function of Random Waves due to Refraction

In this study, a more general directional spreading function is developed that allows for asymmetric directional distributions. For multidirectional random waves that approach the shore obliquely over a planar slope, we demonstrate that directional asymmetry is generated due to wave refraction. The asymmetry created by refraction increases with the offshore peak wave direction. The present spreading function is compared to a preexisting symmetric spreading function and is shown to better capture changes in the directional distribution that occur in a refracting, random wave field. Finally, the new asymmetric spreading function is compared to a long time series of wave directional spectra measured at a nearshore field site. The results demonstrate that refraction-induced asymmetry is common in the nearshore and the asymmetric spreading function gives an improved analytic representation of the overall directional distribution as compared to the symmetric function.

A note on linear dispersion and shoaling properties in extended Boussinesq equations

Abstract A set of optimum parameter α is obtained to evaluate the linear dispersion and shoaling properties in the extended Boussinesq equations of Madsen and Sorensen, 1992 , Nwogu, 1993 , and Chen and Liu (1995) . Optimum α values are determined to produce minimal errors in each wave property of phase velocity, group velocity, or shoaling coefficient relative to the analytical one given by the Stokes wave theory. Comparisons are made of the percent errors in phase velocity, group velocity, and shoaling coefficient produced by the Boussinesq equations with a different set of optimum α values. The case with a fixed value of α = −0.4 is also presented in the comparison. The comparisons reveal that the optimum α value tuned for a particular wave property gives in general poor results for other properties. Considering all the properties simultaneously, the fixed value of α = −0.4 may give overall accuracies in phase velocity and shoaling coefficient for all the types of Boussinesq equations selected in this study.

Prediction of Swell-like High Waves Using Observed Data on the East Coast of Korea

In this study, we develop an algorithm to predict swell-like high waves on the east coast of Korea using the directional wave gauge which was installed near Sokcho. Using the numerical wave model SWAN, we estimate wave data in open sea from the wave data collected by using the directional wave gauge. Then, using the wave ray method and SWAN model with the open-sea wave data as offshore boundary conditions, we predict the swell-like high waves at several major points on the east coast of Korea. We verify the prediction methods with the SWAN and wave ray methods by comparing predicted data against measured one at Wangdolcho. We can improve the prediction of the swell-like high waves in the east sea of Korea using both the real-time wave measurement system and the present prediction algorithm.

Prediction of Wave Force on a Long Structure of Semi-infinite Breakwater Type Considering Diffraction

In this study, the wave force distribution acting on a semi-infinite and vertical-type long structure is investigated considering diffraction. An analytical solution of the wave force acting on long structures is also suggested in this study. The wave forces on long structures are evaluated for monochromatic, uni-directional random, and multi-directional random waves. Diffraction effects in front of the breakwater and on the lee side of the breakwater are considered. The wave force on a long structure becomes zero when the relative length of the breakwater (1/L) is zero. The diffraction effects are relatively strong when the relative length of the breakwater is less than 1.0, and the wave forces decrease greatly for long structure when the relative length of the breakwater is larger than 0.5. Therefore, it is necessary to consider diffraction effects when the relative length of the breakwater is less than 1.0, and the relative length of the breakwater must be at least 0.5 in order to obtain a reduction of wave force on long structures.

Water surface resonance in the L-shaped channel of seawater exchange breakwater

Abstract The resonance period of the L-shaped channel in the caisson is predicted analytically for the seawater exchange breakwater of “Applicability Study of the Seawater Exchange Breakwater (1). Korea Ministry of Maritime Affairs and Fisheries (in Korean) (1999a)”. Hydraulic experiments are conducted for a composite breakwater with a rear reservoir that is one of the seawater exchange breakwaters developed by them. For regular waves, the water surface elevation in the channel and the flow rate through the breakwater are measured. For irregular waves, the flow rate through the breakwater and the reflection coefficient on the breakwater are measured. The resonant maximum values in both the surface elevation and the flow rate, and the resonant minimum values in the reflection coefficient are all at wave periods slightly longer than analytically predicted ones. The measured resonance period for irregular waves is closer to the predicted one than for regular waves. If the resonance period of the L-shaped channel is fitted to the dominant period of incident waves, there would be high efficiency of seawater exchange between inside and outside the harbor.

Short waves in a rotating, shallow tank with bathymetry: a model equation in the mild-slope approximation

A depth-integrated, mild-slope equation for dispersive linear wave motion in a domain under the combined effects of depth variation and rotation is derived. The model reproduces the usual finite-depth mild-slope equation in the absence of rotation and also reproduces the usual long wave approximations when the ratio of wavelength to water depth is large, either with or without the additional effect of rotation. The model could serve as the basis for numerical codes that could compute motions ranging from wind wave refraction to tidal oscillations, without restructuring the internal model coefficients. efraction and parabolic refraction-diffraction approximations are constructed, and a computational example for wave focusing by a shoal with the effect of rotation is included.