• Journal of the Korean Mathematical Society
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• v.15 no.1
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• pp.39-57
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• 1978

It is given in the Lecture Note [1] of Berger, Gauduchon and Mazet that, if ${\pi}$n: (${\tilde{M}}$, ${\tilde{g}}$)${\rightarrow}$(${\tilde{M}}$, ${\tilde{g}}$) is a Riemannian submersion with totally geodesic fibers, ${\Delta}$ and ${\tilde{\Delta}}$ are Laplace operators on (${\tilde{M}}$, ${\tilde{g}}$) and (M, g) respectively and f is an eigenfunction of ${\Delta}$, then its lift $f^L$ is also an eigenfunction of ${\tilde{\Delta}}$ with the common eigenvalue. But such a simple relation does not hold for an eigenform of the Laplace-Beltrami operator ${\Delta}=d{\delta}+{\delta}d$. If ${\omega}$ is an eigenform of ${\Delta}$ and ${\omega}^L$ is the horizontal lift of ${\omega}$, ${\omega}^L$ is not in genera an eigenform of the Laplace-Beltrami operator ${\tilde{\Delta}}$ of (${\tilde{M}}$, ${\tilde{g}}$). The present author has obtained a set of formulas which gives the relation between ${\tilde{\Delta}}{\omega}^L$ and ${\Delta}{\omega}$ in another paper [7]. In the present paper a Sasakian submersion is treated. A Sasakian manifold (${\tilde{M}}$, ${\tilde{g}}$, ${\tilde{\xi}}$) considered in this paper is such a one which admits a Riemannian submersion where the base manifold is a Kaehler manifold (M, g, J) and the fibers are geodesics generated by the unit Killing vector field ${\tilde{\xi}}$. Then the submersion is called a Sasakian submersion. If ${\omega}$ is a eigenform of ${\Delta}$ on (M, g, J) and its lift ${\omega}^L$ is an eigenform of ${\tilde{\Delta}}$ on (${\tilde{M}}$, ${\tilde{g}}$, ${\tilde{\xi}}$), then ${\omega}$ is called an eigenform of the first kind. We define a relative eigenform of ${\tilde{\Delta}}$. If the lift ${\omega}^L$ of an eigenform ${\omega}$ of ${\Delta}$ is a relative eigenform of ${\tilde{\Delta}}$ we call ${\omega}$ an eigenform of the second kind. Such objects are studied.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.24 no.2
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• pp.89-96
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• 2012

Based on the hydroelastic theory and the matched eigenfunction expansion method(MEEM), the dynamic behavior of the porous flexible net sheet and wave forces have been investigated in monochromatic waves. The net sheet is installed vertically with the submergence depth. Top end of a net sheet is fixed and its lower end is attached by a clump weight. It is assumed that the initial tension is sufficiently large so that the effects of dynamictension variation can be neglected. The boundary condition on the porous flexible net sheet is derived based on Darcy's fine-pore model and body boundary condition. The developed analytic model can be extended to the impermeable/permeable vertical plate and the impermeable flexible membrane. The analytical model was used to study the influence of design parameters(wave characteristics, porosity, submergence depth, initial tension) on the response characteristics and wave load of the net sheet.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.18 no.2
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• pp.97-111
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• 2006

Using the eigenfunction expansion method, a mathematical model has been developed to calculate the reflection and transmission of regular waves from a multiple-row curtainwall-pile breakwater. In addition, hydraulic model experiments have been conducted with different values of porosities between the piles, drafts of the curtain walls, and distances between the rows of the breakwater. It is found that the reflection and transmission coefficients decrease and increase, respectively, with decreasing relative water depth, but they bounce to increase and decrease, respectively, as the relative water depth decreases further. When either the porosity between the piles or the draft of the curtain wall is changed with other parameters fixed, the relative magnitudes of the reflection and transmission coefficients have been changed, but the general trend remained the same. When the wavelength is the same as the distance between the rows of the breakwater, a rapid change was observed for the reflection and transmission coefficients. A good agreement between the measurement and prediction was also founded for three-row breakwaters.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.26 no.2
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• pp.103-111
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• 2014

The wave power, extracted from a circular array of small power buoys, is investigated under the potential theory. It is assumed that the buoy's radius, the draft, and the separation distance are much smaller than the water depth, the wave length, and the radius of a circular deployment area. The boundary value problem involving the macro-scale boundary condition on the mean surface covered by buoys is solved using the eigenfunction expansion method. The capture width, which is defined as the ratio of the extracted power to the wave power per unit length of the incident wave crest, is assessed for various combinations of packing ratio, radius of a circular array, and PTO damping coefficient. It is found that the circular array deployment is more effective in the viewpoint of efficiency than the single large buoy of the same total displaced volume.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.32 no.4
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• pp.242-251
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• 2020

In this study, the extraction efficiency and reflection coefficient by a two-dimensional OWC (Oscillating Water Column) WEC (wave energy converter) installed in front of a seawall was investigated for regular/irregular waves. The matched eigenfunction expansion method (MEEM) based on the linear potential theory was applied as an analytical tool. The diffraction problem by the incident wave in the open-chamber and the radiation problem by the oscillating pressure in the closed-chamber were solved to obtain the volume fluxes at the internal free-surface. Applying the volume fluxes into the continuity equation for the airflow in a chamber, we got the oscillating air pressure. The maximum extracted power and corresponding reflection coefficient were determined at the optimal turbine coefficient that maximizes the extracted power. OWC device designed for a high extracted efficiency simultaneously contributes to reduce reflected waves.

• Journal of Sensor Science and Technology
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• v.8 no.5
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• pp.407-413
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• 1999

The resonant frequency of dielectric rod resonator can be calculated very accurately by means of the concept of electric and magnetic walls from the symmetry. But in a real situation, when the supporter is placed in the cavity, asymmetry appears due to the supporter and its dielectric constant. Then we need research into the resonance characteristic of asymmetric dielectric resonator. In this paper, for the dielectric rod resonator which is placed in the asymmetric position in the conducting cavity, the equation for resonance characteristic was derived and the resonant frequency was calculated with the eigenfunction expansion method. We found that the calculated resonant frequency is very accurate when it was compared with the experimental result and that asymmetry affects the resonant frequency more in TM mode than in TE or in hybrid mode.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.32 no.6
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• pp.396-410
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• 2020

The design and the construction are carried out by installation of new caissons on the back or the front of existing caissons to increase the stability of existing caisson breakwater. In this study, we use the eigenfunction expansion method to analyze the effects of wave structure interaction when new porous dual circular caissons are installed on the back or the front of existing breakwater. The porous dual circular caisson which consisting of a porous outer cylinder circumscribing an impermeable inner cylinder is one type of seawater exchanging breakwater. The comparison of numerical results between present method and Sankarbabu et al. is made, and the wave force and the wave run-up acting on each porous dual circular caisson are calculated for various parameters by considering the wave structure interaction.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.32 no.6
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• pp.420-433
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• 2020

In order to increase the stability of existing breakwater, new caissons are installed on the back or the front of existing caissons. It is very important to evaluate wave force and wave run-up according to the change of porosity among caissons and the energy loss due to separation effects. In this study, we use the eigenfunction expansion method with Darcy's law, which describes the flow of a fluid through a porous plate, to analyze the characteristics of wave on circular breakwater of double array for various porous coefficients. To verify the numerical method, the comparison between present results and Sankarbabu et al. (2008) is made. The wave force and the wave run-up acting on each dual cylindrical caisson are calculated for various parameters by considering the energy loss and the change of porosity.

• Journal of Mechanical Science and Technology
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• v.20 no.1
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• pp.158-166
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• 2006

Extended Graetz problem in microtube is analyzed by using eigenfunction expansion to solve the energy equation. For the eigenvalue problem we applied the shooting method and Galerkin method. The hydrodynamically isothermal developed flow is assumed to enter the microtube with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microtube wall, axial conduction and viscous dissipation are included. From the temperature field obtained, the local Nusselt number distributions on the tube wall are obtained as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.

• Structural Engineering and Mechanics
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• v.48 no.1
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• pp.103-124
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• 2013

This paper studies the static and dynamic characteristics of composite plates subjected to an arbitrary periodic load in hygrothermal environments. The material properties of composite plates are depended on the temperature and moisture. The governing equations of motion of Mathieu-type are established by using the Galerkin method with reduced eigenfunction transforms. A periodic load is taken to be a combination of axial pulsating load and bending stress in the example problem. The regions of dynamic instability of laminated composite plates are determined by solving the eigenvalue problems based on Bolotin's method. The effects of temperature rise and moisture concentration on the dynamic instability of laminated composite plates are investigated and discussed. The influences of various parameters on the instability region and dynamic instability index are also investigated. The numerical results reveal that the influences of hygrothermal effect on the dynamic instability of laminated plates are significant.