ICCE 2014 Cover Image


Lagrangian solution
partial standing wave
sloping bottom
mass transport
particle trajectory
nonlinear waves.

How to Cite

Li, M.-S., Zou, Q., Chen, Y.-Y., & Hsu, H.-C. (2014). SECOND-ORDER PARTIAL STANDING WAVE SOLUTION FOR A SLOPING BOTTOM. Coastal Engineering Proceedings, 1(34), waves.46.


This paper presents a second-order asymptotic solution in Lagrangian description for a nonlinear partial standing wave over a sloping bottom. The particle trajectories are obtained as a function of the nonlinear ordering parameters, wave steepness and the bottom slope to the second order. The analytical Lagrangian solution assumes irrotational flow and satisfies the boundary condition of constant pressure p = 0 at the free surface. This solution is applicable to progressive, standing and partial standing waves, shoaling from deep to shallow water. Mass transport and particle trajectory nonlinear partial standing waves on a sloping bottom are investigated using the closed form Lagrangian wave solution


Buldakov, EV, Taylor, PH, and Taylor, RE, 2006. New asymptotic description of nonlinear water waves in Lagrangian Coordinates. J. Fluid Mech., 562, 431-444.

Brunone, B. and Tomasicchio, G.R. 1997. Wave kinematics at steep slopes: Second-order model. Journal of Waterway, Port, Coastal and Ocean Engineering, 123, 223-232.

Chen, Y.Y., Hwung, H.H., Hsu, H.C. 2005. Theoretical analysis of surface waves propagation on sloping bottoms, Part 1, Wave Motion, 42, 335-351.

Chen, Y.Y., Hsu, H.C., Chen, G.Y., Hwung, H.H. 2006. Theoretical analysis for surface waves propagation on sloping bottoms, Part 2, Wave Motion, 43, 339-356.

Chen, Y.Y., Hsu, H.C., 2009. A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates,Chinese Physics B, 18, 861-871. (SCI)

Chen, Y.Y., Hsu, H.C., Chen, G. Y., 2010. Lagrangian experiment and solution for irrotational finiteamplitude progressive gravity waves at uniform depth. Fluid Dynamics Research, 42, 045501 (doi:10.1088/0169-5983/42/4045511).

Chen, Y.Y., Li, M.S., Hsu, H.C., Ng, C.O. 2012 Theoretical and experimental study of particle trajectories for nonlinear water waves propagating on a sloping bottom, Phil. Trans. Roy. Soc. A., 370, 1543-1571.

Hsu, H. C., Chen, Y. Y., Wang, C. F., 2010. Perturbation analysis of the short-crested waves in Lagrangian coordinates. Nonlinear Analysis Series B: Real World Applications 11, 1522-1536.

Hughes, S. A., Fowler, J.E. 1995. Estimating wave-induced kinematics at sloping structures. Journal of Waterway, Port, Coastal and Ocean Engineering, 121(4), 209-215.

Kobayashi, N., Tomasicchio, G. R., Brunone, B. 2000 Partial Standing Waves on a Steep Slope. Journal of Coastal Research 16, 379-384.

Lamb, H., 1932. Hydrodynamics. 6th edn. Cambridge University Press.

Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. Roy. Soc. A. 245, 533-581.

Longuet-Higgins, M. S., Stewart, R. W., 1964 Radiation stress in water waves-a physical dicussion with applications. Deep sea Res., 11, 529-562.

Mei, C.C., 1985. Resonant reflection of surface water waves by periodic sandbars, J. Fluid Mech., 152, 315-335.

Miche, A., 1944. Mouvements ondulatoires de la mer en profondeur constante ou décroissante. Annales des ponts et chaussees , 25-78, 131-164, 270-292, 369-406.

Naciri, M., Mei, C.C., 1993. Evolution of short gravity waves on long gravity waves. Phys. Fluids A 5, 1869-1878.

Ng, C.O., 2004a. Mass transport in a layer of power-law fluid forced by periodic surface pressure. Wave Motion 39, 241-259.

Ng, C.O., 2004b. Mass transport and set-ups due to partial standing surface waves in a two layer viscous system. Journal of Fluid Mechanics 520, 297-325.

Pierson, W.J., 1962. Perturbation analysis of the Navier-Stokes equations in Lagrangian form with selected linear solution. J. Geophys Res. 67, 3151-3160.

Rayleigh, L. 1883 On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems. Philos. Trans. Roy. Meteor. Soc.,125,1-21.

Sumer, B. M., Fredsoe, J. 2000 Experimental study of 2D scour and its protection at a rubble-mound breakwater, Coastal Engineering, 40, 59-87.

Tsai, C. P., Jeng D. S., 1994 Numerical Fourier solutions of standing waves in finite water depth 16, 185-193.

Ünluata, Ü., Mei, . C. 1970 Mass transport in water waves. J. Geophys. Res., 75, 7611-7618.

Yakubovich, E.I., Zenkovich, D.A., 2001. Matrix approach to Lagrangian fluid dynamics. J. Fluid Mech. 443, 167-196.

Zou, Q. P., Hay, A. E., Bowen, A. J. 2003 The vertical structure of surface gravity waves propagating over a sloping sea bed: theory and field measurements. J. Geophys. Res. 108, 3265, doi:10.1029/2002JC001432.

Zou, Q. P., Hay, A. E. 2003 The vertical structure of the wave bottom boundary layer over a sloping bed: theory and field measurements. J. Phys. Oceanogr. 33, 1380-1400.

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