• KSCE Journal of Civil and Environmental Engineering Research
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• v.28 no.4B
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• pp.413-419
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• 2008

In this study, we develop analytical solutions for long waves propagating over several types of axis-symmetric topographies where the water depth varies in an arbitrary power of radial distance. The first type is a cylindrical island mounted on a shoal. The second type is a circular island. To get the solution, the methods of separation of variables, Taylor series expansion and Frobenius series are used. Developed analytical solutions are validated by comparing with previously developed analytical solutions. We also investigate various cases with different incident wave periods, radii of the shoal, and the powers of radial distance.

• The Journal of the Korea Contents Association
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• v.15 no.6
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• pp.539-546
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• 2015

In this study, we develop numerical model using the elliptic mild-slope equation for waves propagating around axis-symmetric topographies where the water depth varies arbitrarily having zero at the coastline. The entire region is divided into three regions. In the both of inner and outer regions, an existing analytical solutions are used. In the middle region, the finite element technique is applied to the governing equation. To get the solution, the methods of separation of variables, Frobenius series are used. Developed solution is validated by comparing with previously developed analytical solution. We also investigate various cases with different bottom topographies.

• Journal of Korean Association for Spatial Structures
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• v.6 no.4 s.22
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• pp.81-92
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• 2006

This paper presents exact solutions for the free vibrations and buckling of rectangular plates having two opposite, simply supported edges subjected to linearly varying normal stresses causing pure in-plane moments, the other two edges being free. Assuming displacement functions which are sinusoidal in the direction of loading (x), the simply supported edge conditions are satisfied exactly. With this the differential equation of motion for the plate is reduced to an ordinary one having variable coefficients (in y). This equation is solved exactly by assuming power series in y and obtaining its proper coefficients (the method of Frobenius). Applying the free edge boundary conditions at y=0, b yields a fourth order characteristic determinant for the critical buckling moments and vibration frequencies. Convergence of the series is studied carefully. Numerical results are obtained for the critical buckling moments and some of their associated mode shapes. Comparisons are made with known results from less accurate one-dimensional beam theory. Free vibration frequency and mode shape results are also presented. Because the buckling and frequency parameters depend upon Poisson's ratio ( V ), results are shown for $0{\leq}v{\leq}0.5$, valid for isotropic materials.

• 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.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.18 no.4
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• pp.308-320
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• 2006

An analytic solution to the mild-slope equation is derived for waves propagating over an axi-symmetric pit. The water depth inside the pit varies in proportion to a power of radial distance from the pit center. The governing equation is transformed into ordinary differential equations by using separation of variables, and the coefficients of the equations are transformed into explicit forms by using Hunt's (1979) approximate solution. Finally, by using the Frobenius series, the analytic solution is derived. Due to the feature of Hunt's equation, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth water. The validity of the analytic solution is demonstrated by comparison with numerical solutions. The analytic solution is also used to examine the effects of pit geometry and relative depth on wave transformation.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.26 no.1
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• pp.75-81
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• 2016

In this study, a transfer matrix method in which can produce an infinite number of accurate natural frequencies using a single element for the bending vibration of rotating Bernoulli-Euler beam with linearly reduced width, is developed. The roots of the differential equation in the proposed method are calculated using the Frobenius method in the power series solution. To demonstrate the accuracy of the method, the calculated natural frequencies are compared with the results given by using the commercial finite element analysis program(ANSYS), and the comparison results between these two methods show the excellent agreement. Based on the comparison results, a parametric study is performed to investigate the effect of the centrifugal forces on the non-dimensional natural frequencies for rotating beam with the variable width.