Journal of the Korean Society for Marine Environment & Energy (한국해양환경ㆍ에너지학회지)

The Korean Society for Marine Environment and Energy (한국해양환경•에너지학회)
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A Linear Wave Equation Over Mild-Sloped Bed from Double Integration

이중적분을 이용한 완경사면에서의 선형파 방정식

  • Kim, Hyo-Seob (Department of Civil and Environmental Engineering, Kookmin University) ;
  • Jung, Byung-Soon (Department of Civil and Environmental Engineering, Kookmin University) ;
  • Lee, Ye-Won (Department of Civil and Environmental Engineering, Kookmin University)
  • 김효섭 (국민대학교 건설시스템공학부) ;
  • 정병순 (국민대학교 건설시스템공학부) ;
  • 이예원 (국민대학교 건설시스템공학부)
  • Received : 2009.05.21
  • Accepted : 2009.06.24
  • Published : 2009.08.25
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Abstract

A set of equations for description of transformation of harmonic waves is proposed here. Velocity potential function and separation of variables are introduced for the derivation. The continuity equation is in a vertical plane is integrated through the water so that a horizontal one-dimensional wave equation is produced. The new equation composed of the complex velocity potential function, further be modified into. A set up of equations composed of the wave amplitude and wave phase gradient. The horizontally one-dimensional equations on the wave amplitude and wave phase gradient are the first and second-order ordinary differential equations. They are solved in a one-way marching manner starting from a side where boundary values are supplied, i.e. the wave amplitude, the wave amplitude gradient, and the wave phase gradient. Simple spatially-centered finite difference schemes are adopted for the present set of equations. The equations set is applied to three test cases, Booij's inclined plane slope profile, Massel's smooth bed profile, and Bragg's wavy bed profile. The present equations set is satisfactorily verified against existing theories including Massel's modified mild-slope equation, Berkhoff's mild-slope equation, and the full linear equation.

연직 2차원 평면을 대상으로 하는 연속방정식을 수심방향으로 이중적분 하여 수평1차원 파랑방정식을 구하였다. 새 방정식은 복소수 포텐셜 함수로 구성되어 있으며, 파랑의 진폭과 위상경사함수를 도입하여 한 세트의 실수방정식으로도 변형되었다. 파랑진폭과 위상경사함수를 포함한 한 세트의 식은 각각 1차, 2차 상미분방정식이며, 한쪽 경계에서 경계조건을 적절히 지정하여 전 영역에서의 해를 한 방향으로 진행하면서 구할 수 있다. 이때 경계조건으로는 파랑진폭 값, 파랑진폭의 경사, 위상 경사 값이다. 단순한 중앙차분식을 이용하여 식을 차분화 하였다. 새 방정식을 Booij의 경사판, Massel의 부드러운 저면, Bragg의 싸인 함수의 저면에 대하여 적용하여 보았다. 본 방정식은 Massel의 수정완경사방정식, Berkhoff의 완경사방정식, 완전 선형방정식과 비교하여 유사한 결과를 나타내었으며, 유용함을 보였다.

Keywords

References

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