KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
권호 표지
DOI QR코드

DOI QR Code

Derivation of Extended Mild-Slope Equation Using Euler-Lagrange Equation

Euler-Lagrange 식을 사용한 확장형 완경사방정식 유도

  • 이창훈 (세종대학교 토목환경공학과) ;
  • 김규한 (관동대학교 토목환경공학과)
  • Received : 2009.05.11
  • Accepted : 2009.08.04
  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we derive the extended mild-slope equation in terms of the velocity potential using the Euler-Lagrange equation. First, we follow Kim and Bai (2004) who derive the complementary mild-slope equation in terms of the stream function using the Euler-Lagrange equation and we compare their equation to the existing extended mild-slope equations of the velocity potential. Second, we derive the extended mild-slope equation in terms of the velocity potential using the Euler-Lagrange equation. In the developed equation, the higher-order bottom variation terms are newly developed and found to be the same as those of Massel (1993) and Chamberlain and Porter (1995). The present study makes wide the area of coastal engineering by developing the extended mild-slope equation with a way which has never been used before.

본 연구에서 Euler-Lagrange 식을 사용하여 속도포텐셜로 표현되는 확장형 완경사방정식을 유도하였다. 먼저, Euler-Lagrange 식을 사용하여 흐름함수로 표현된 확장형 완경사방정식을 유도한 Kim과 Bai(2004)의 유도과정을 따라가면서 속도 표텐셜로 표현된 확장형 완경사방정식과의 관계를 찾았다. 속도포텐셜로 표현된 Euler-Lagrange 식을 찾아낸 다음 고차의 수심변화 항을 유도하였다. 본 연구에서 유도된 확장형 완경사방정식은 기존의 식인 Massel(1993)의 식과 Chamberlain과 Porter(1995)의 식과 정확히 일치하였다. 본 연구의 연구 성과는 확장형 완경사방정식의 유도 방법을 새로 제시하여 해안공학의 영역을 넓히는데 의의가 있다.

Keywords

References

  1. 이창훈, 조대희, 조용식(2003) 선형파 이론을 사용하여 파랑변형 예측 시 소멸파 성분의 중요성 검토: 2. 수치실험. 한국해안해양공학회지, 한국해안해양공학회, 제15권 제1호, pp. 51-58.
  2. Berkhoff, J.C.W. (1972) Computation of combined refraction-diffraction. Proc. 13th Int. Conf. Coastal Engineering, ASCE, pp. 471-490.
  3. Chamberlain, P.G. and Porter, D. (1995) The modified mild-slope equation. J. Fluid Mech., Vol. 291, pp. 393-407. https://doi.org/10.1017/S0022112095002758
  4. http://en.wikipedia.org/wiki/Euler-Lagrange_equation.
  5. Kim, J.W. and Bai, K.J. (2004) A new complementary mild-slope equation. J. Fluid Mech., Vol. 511, pp. 25-40. https://doi.org/10.1017/S0022112004007840
  6. Lee, C., Kim, G., and Suh, K.D. (2003) Extended mild-slope equation for random waves. Coastal Engineering, Vol. 48, pp. 277-287. https://doi.org/10.1016/S0378-3839(03)00033-4
  7. Lee, C., Park, W.S., Cho, Y.-S., and Suh, K.D. (1998) Hyperbolic mild-slope equations extended to account for rapidly varying topography. Coastal Engineering, Vol. 34, pp. 243-257. https://doi.org/10.1016/S0378-3839(98)00028-3
  8. Massel, S.R. (1993) Extended refraction-diffraction equation for surface waves, Coastal Engineering, Vol. 19, pp. 97-126. https://doi.org/10.1016/0378-3839(93)90020-9
  9. Radder, A.C. and Dingemans, M.W. (1985) Canonical equations for almost periodic, weakly nonlinear gravity waves. Wave Motion, Vol. 7, pp. 473-485. https://doi.org/10.1016/0165-2125(85)90021-6
  10. Smith, R. and Sprinks, T. (1975) Scattering of surface waves by a conical island. J. Fluid Mech., Vol. 72, pp. 373-384. https://doi.org/10.1017/S0022112075003424
  11. Suh, K.D., Lee, C., and Park, W.S. (1997) Time-dependent equations for wave propagation on rapidly varying topography. Coastal Engineering, Vol. 32, pp. 91-117. https://doi.org/10.1016/S0378-3839(97)81745-0