KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
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Explicit Solution of Wave Dispersion Equation Using Recursive Relation

순환 관계에 의한 파랑분산식의 양해

  • 이창훈 (세종대학교 토목환경공학과) ;
  • 장호철 (세종대학교 토목환경공학과)
  • Received : 2007.08.29
  • Accepted : 2007.11.13
  • Published : 2008.01.31
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Abstract

Explicit solutions of the wave dispersion equation are developed using the recursive relation in terms of the relative water depth. We use the solutions of Eckart (1951), Hunt (1979), and the deep-water and shallow-water solutions for initial values of the solution. All the recursive solutions converge to the exact one except that with the initial value of deep-water solution. The solution with the initial value by Hunt converged much faster than the others. The recursive solutions may be obtained quickly and simply by a hand calculator. For the transformation of linear water waves in whole water depth, the use of the recursive solutions will yield more accurate analytical solutions than use of previously developed explicit solutions.

파랑의 분산관계식에서 상대수심이 반복적으로 표현되는 순환 관계를 이용하여 파랑분산식의 양해를 개발하였다. 순환 관계를 이용한 해의 초기값으로 Eckart(1951), Hunt(1979)의 해와 심해와 천해에서의 해를 사용했다. 순환해 가운데 심해에서의 해를 초기값으로 사용한 해를 제외하고는 모두 참값으로 수렴하였다. 특히, Hunt의 해를 초기값으로 사용한 해는 다른 해보다 더 빠르게 수렴했다. 순환해는 휴대용 계산기를 사용하여 손 쉽고 빠르게 구할 수 있는 장점이 있다. 선형파의 변형을 예측하기 위하여 해석 해를 사용할 경우 파랑분산식의 양해를 사용해야 하는 데 본 연구에서 개발한 순환해를 사용하면 기존의 양해를 사용한 것보다 더 정확한 해석 해를 도출할 것이다.

Keywords

References

  1. Berkhoff, J.C.W. (1972) Computation of combined refraction-diffraction, Proc. 13th Coastal Engineering Conference, ASCE, pp. 471-490
  2. Chamberlain, P.G. and Porter, D. (1995) The modified mild-slope equation, J. Fluid Mechanics, Vol. 291, pp. 393-407 https://doi.org/10.1017/S0022112095002758
  3. Eckart, C. (1951) Surface waves on water of variable depth, Scripps Institute of Oceanography, University of California, La Jolla
  4. Fenton, J. and McKee, W.D. (1990) On the calculating lengths of water waves, Coastal Engineering, Vol. 14, pp. 499-513 https://doi.org/10.1016/0378-3839(90)90032-R
  5. Guo, J. (2002) Simple and explicit solution of wave dispersion equation, Coastal Engineering, Vol. 45, pp. 71-74 https://doi.org/10.1016/S0378-3839(02)00039-X
  6. Homma, S. (1950) On the behaviour of seismic sea waves around circular island, Geophys. Mag., Vol. 21, pp. 199-208
  7. Hunt, J.N. (1979) Direct solution of wave dispersion equation, J. Waterway, Port, Coastal and Ocean Engineering, Vol. 105, pp. 457-459
  8. 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
  9. 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
  10. Liu, H.-W., Lin, P., and Shankar, N.J. (2004) An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal, Coastal Engineering, Vol. 51, pp. 421-437 https://doi.org/10.1016/j.coastaleng.2004.04.005
  11. 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
  12. Nielsen, P. (1982) Explicit formulae for practical wave calculations, Coastal Engineering, Vol. 6, pp. 389-398 https://doi.org/10.1016/0378-3839(82)90008-4
  13. Nielsen, P. (2002) Teaching Notes on Coastal and Estuarine Processes, Department of Civil Engineering, University of Queensland, Australia
  14. Radder, A.C. (1979) On the parabolic equation method for waterwave propagation, J. Fluid Mechanics, Vol. 95, pp. 159-176 https://doi.org/10.1017/S0022112079001397
  15. Smith, R. and Sprinks, T. (1975) Scattering of surface waves by a conical island, J. Fluid Mechanics, Vol. 72, pp. 373-384 https://doi.org/10.1017/S0022112075003424
  16. 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
  17. Svendsen, I.A. (2006) Introduction to Nearshore Hydrodynamics, World Scientific
  18. You, Z.-J. (2003) Discussion of "Simple and explicit solution to the wave dispersion equation" [Coastal Engineering 45 (2002) 71-74], Coastal Engineering, Vol. 48, pp. 133-135 https://doi.org/10.1016/S0378-3839(02)00170-9
  19. Zhang, Y.L. and Zhu, S.-P. (1994) New solutions for the propagation of long water waves over variable depth, J. Fluid Mechanics, Vol. 278, pp. 391-406 https://doi.org/10.1017/S0022112094003769
  20. Zhu, S.-P. and Zhang, Y.L. (1996) Scattering of long waves around a circular island mounted on a conical shoal, Wave Motion, Vol. 23, pp. 353-362 https://doi.org/10.1016/0165-2125(95)00057-7