SHORELINE SOLUTION FOR TAPERED BEACH FILL By Todd L. Walton Jr., ~ Member, ASCE

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1 SHORELINE SOLUTION FOR TAPERED BEACH FILL By Todd L. Walton Jr., ~ Member, ASCE ABSTRACT: An analytical solution for the evolution of the planform for a beach fill with tapered ends is presented, and solution charts are given for two cases of taper-section length to straight fill-section length. An analytical solution for the proportion of fill remaining after a given time is also presented as a function of a dimensionless time constant. INTRODUCTION If the angle of the shoreline is small with respect to the waves breaking along it, a linearized analytical shoreline change partial-differential equation can be derived from the longshore sand transport equation coupled with the continuity of sand equation [see for example Walton and Chiu (979)]. The solution is a function of time t, and is given as Oy_ E.OEy Ot - Ox 2 () where x = alongshore coordinate; y = offshore coordinate of the shoreline in planform; and E = so-called shoreline diffusion coefficient, which is given by E- 8 where K = dimensionless constant relating immersed-weight sand transport to alongshore energy flux; Hb = breaking wave height; Cgb = breaking wave celerity; p = density of seawater; p~ = density of sediment; n = porosity of sediment; D~ = berm height of beach above still water; and Dc = limiting depth of appreciable sand transport in the alongshore direction. Numerous investigators have taken advantage of the linearized shoreline change equation, (), and solved various cases of shoreline change, assuming constant wave conditions with different boundary and initial conditions. Dean (984) reviews analytical solutions of shoreline change applicable to beach nourishment projects, along with a new solution for the shoreline change between two groins initially filled with sand. Larson et al. (987) provide a review of a number of analytical solutions to the one-line model, as well as additional solutions for a number of cases such as a trapezoidal fill area, river-point source of sand input to shoreline, river-line source of sand input to shoreline, and others. The most widely utilized shoreline solution in practice appears to be that of the rectangular fill case. This particular solution has been utilized in design and calculation of loss percentages to various beach nourishment XHydr. Engr., U.S. Army Corps of Engrs., Coast. Engrg. Res. Ctr., U.S. Army Engr. Wtrwy. Experiment Station, 3909 Halls Ferry Road, Vicksburg, MS Note. Discussion open until May, 995. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this technical note was submitted for review and possible publication on May 24, 993. This technical note is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 20, No. 6, November/December, ISSN X/ 94/ /$ $.25 per page. Technical Note No

2 ~'.'~ I / o! ";~ ~,~ I / I ~ //////////////////////////////~////////////////////// -b -o 0 Q b o. o o FIG.. Plan View of Tapered Beach Fill (at t = 0).5~ Z Qe~ff" t I I _ x/a FIG. 2. Shoreline Evolution for Tapered Fill with b/a =. projects in Florida (General 990, 99). Typically, beach fills are not built rectangularly, but have tapers at the ends of the projects to provide for smooth transitions to the original shoreline. These tapers have practical value in the prevention of lagoonal enclosure of dead-water space at the ends of the projects, which might otherwise occur if a sharp transition from the new (fill) shoreline to the original one exists. This technical note presents a solution to the shoreline change equation, (), in the case of a tapered fill shoreline and provides nondimensionalized solution graphs that can be utilized in evaluating shoreline changes. METHODOLOGY AND RESULTS The basic partial-differential equation is provided in (). For an infinitely long straight beach, where the superimposed (fill) shoreline shape at time t = 0 is described by a function f(x), the solution of () is given by the integral (Carslaw and Jaeger 959) 652

3 0.9 " " " I FIG. 3. Shoreline Evolution for Tapered Fill with b/a =.5 fo which is applicable for t > 0 and - oo < x < ~. For the case (see Fig. ) in which the beach fill is described (at time t = 0) by a rectangle extending from the original beach, a distance y = Yo, offshore and alongshore between the limits -a <- x <- a, and then tapering linearily from y = Y0 at x = +-a to y = 0 at x = -+ b, the aforementioned integral can be solved as follows: x/a y(x, t)! - - = : [erf(ax + A.) - erf(ax - A)] Yo 2 + ~ [erf(ax - A) - erf(ax - B)] + ~ [erf(ax + B) - erf(ax + A)] + + {exp[ - (AX - B) 2] - exp[- (AX - A)2]} 2v~(B - A) 2V"~(B - A) {exp[- (AX + B) z] - exp[- (AX + A)2]} where A = a/2x/-ett; B = b/2w~t; and X = x/a. This solution is presented in Figs. 2 and 3 for taper length ratios of b/a =. and.5. As the solution is symmetric around the y axis, only its positive (x >-- 0) portion is graphed. (4) 653

4 I 0.9 o.s I I I I.5 i. I I I I FIG. 4. Proportion of Fill Remaining for b/a =.0,., and.5 On comparison with the rectangular fill solution [see for example Walton and Chiu (979)], the solution is very similar, as would be expected. For the case where b/a =, the solution (not shown here) reduces to the rectangular fill solution, exactly. The proportion of fill remaining within project limits (- b <-- x --- b) can be found via the integration of (4) to give M(t) = ~ + eff(b + A) + ~exp[-(b + A)2I - (AB---- ) erf(b -A) A~/~exp[-(B-A)E]} + exp( - 4B 2) + (2A 2 - nab - 6B 2 - )erf(a + B) 4(B - A) 2 2r _ (2A 2-4AB + 2B 2 - )erf(a - B) + (8B2 + ) erf(2b) 4(B - A) 2 4(B - A) 2 { 3. - exp[-(a - B)2 + exp[-(a + B) 2] ~2---B- J + ;V~(B - A) (5) where M(t) = proportion of fill remaining in the project area as a function 654

5 of time. This solution has been provided in Fig. 4 for taper length ratios of b/a =.0 (rectangular fill),., and.5. Fig. 4 shows that a short length of taper (b/a <-.) does not modify the "proportional" loss of the fill significantly, with time, from that of a rectangular fill. A longer taper section (b/a ->.5) would significantly alter the proportional loss of the fill, with time, and show consequent improved economic benefits for the project. In large beach nourishment projects constructed to date, the length of the taper section has been sufficiently short that the consideration of a rectangular fill was a reasonable assumption. ACKNOWLEDGMENTS The writer wishes to thank R. G. Dean for his constructive comments on an earlier version of this technical note. The work presented here was conducted under the Coastal Engineering Manual Work Unit of the Shore Protection and Restoration Research Program, United States Army Corps of Engineers, Coastal Engineering Research Center, and the U.S.A.E. Waterways Experiment Station. Permission was granted by the Office of Chief of Engineers, U.S. Army Corps of Engineers, to publish this information. APPENDIX. REFERENCES Carslaw, H., and Jaeger, J. (959). Conduction of heat in solids. Clarendon Press, Oxford, U.K. Dean, R. G. (984). "Principles of beach nourishment." CRC handbook of coastal processes and erosion. P. D. Komar, ed., CRC Press, Boca Raton, Fla.,, General design memorandum, Broward County Florida; Shore Protection Project. (990). U.S. Army Corps of Engrs., Jacksonville Dist. Ofc., Jacksonville, Fla. General design memorandum, Palm Beach County Florida; Shore Protection Project. (99). U.S. Army Corps of Engrs., Jacksonville Dist. Ofc., Jacksonville, Fla. Larson, M., Hanson, H., and Kraus, N. C. (987). "Analytical solutions of the one line model of shoreline change." TR-CERC Coast. Engrg. Res. Ctr., U.S. Army Wtrwy. Experiment Station, Vicksburg, Miss. Walton Jr., T. L., and Chiu, T. Y. (979). "A review of analytical techniques to solve the sand transport equation and some simplified solutions." Proc., Coast. Struct. '79, ASCE, New York, N.Y.,

Comparison of Numerical Method for Forward. and Backward Time Centered Space for. Long - Term Simulation of Shoreline Evolution

Applied Mathematical Sciences, Vol. 7, 13, no. 14, 16-173 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.13.3736 Comparison of Numerical Method for Forward and Backward Time Centered Space for