| Analytical Solution of Extended Mild Slope Equation for Waves Propagation over Axi-symmetric Pit |
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Taehwa Jung, Seung-Oh Lee, Sooyoung Kim, Yong-Uk Ryu |
| 축대칭 함몰지형 위를 통과하는 파에 대한 확장형 완경사 방정식의 해석해 |
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정태화, 이승오, 김수영, 류용욱 |
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| Abstract |
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Many analytical solutions to the extended mild slope equation were presented by adopting Hunt’s approximated solution of linear dispersion in order to investigate how to be affected water wave when wave propagating on a shallow water depth suddenly encounter a much deeper water depth. In this study, an analytical solution to the extended mild slope equation is derived for waves propagating over an axi-symmetric pit in which the water depth varies in proportion to a power of radial distance from the center of it. First, the governing equation is transformed into ordinary differential equation by using the method of separation of variables, and the coefficients expressed in terms of phase velocity and group velocity are transformed into explicit forms by using Hunt’s (1979) approximate solution for wave dispersion. The functions related to higher order terms such as bottom curvature and bottom slope-squared are expressed as power series. Finally, by using the Frobenius series, the analytical solution to the extended mild slope equation is derived. The validity of the analytical solution is demonstrated by comparison with the numerical solution computed by the finite element method. The present solution shows more accurate result than the solution of the mild slope equation while less accurate in intermediate depth region due to the feature of Hunt’s solution. |
| Key Words:
Longwave; Extended mild slope equation; Frobenius series; Analytical solution |
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