Analysis of Bayed Beaches in Static Equilibrium

Abstract

Ille FIG. 6. R/Ro versus 13/0 for Platja de Sant Pol 0.10+.5-.--0'.6-~0..,.1-~-a.r8~-a.r9~---lt.a tive to describe the bay planform. However, the discussers have found in their field analysis that linear expressions can also give reasonable results to reproduce the planform of some hayed beaches. In the example just presented, the fit using a first-order polynomial (linear fit) gives r 2 = 0.99. This linear approach, however, can only he used when the upcoast headland does not permit the development of the most curved part of the bayed heach, which appears to be a very common case in nature, at least for the stretch of Mediterranean coast considered. Moreover, the authors, like many others researchers [e.g., Hsu and Evans (1989)], suggest that their adjusted coefficients have an universal value hecause the pattern of wave energy redistribution on the beach of any hay is nearly independent of its dimensions. At the same time, they consider that variations in the material forming the heach do not affect the shape of the heach, and that the same coefficients can he applied. With respect to this point, Fig. 6 shows the application of the authors' method andHsu and Evans' method to predict the planform of another bayed beach (Sant Pol, Catalonian coast). Also included in the figure is the empirical fit to a parabolic curve. As with the previous example, this heach has all the requisites to satisfy the static-equilihrium concept. The angle 13 hetween the dominant incident waves and the control line is 50° and the same as before. In this case, the The authors propose a method to estimate the equilibrium planform of pocket beaches similar to the previous work of Hsu and Evans (1989) but depending on only one calibration parameter. The standpoint is that in the neighborhood of the control point the tangent to the beach is parallel to the wave crest of the incident waves. From this, the authors derive a secondorder polynomial equation for R/Ril in terms of 13/8 to describe the planform of bayed beaches [(4)]. After imposing the conditions at the control point, the authors found that their expression depends on one parameter, 0'. This expression must fit a bayed beach in a way similar to the original parabolic method, although improving the fit at the control point, because now they take into account the local properties at this location. The adjusted coefficients obtained by Hsu and Evans (1989) do not satisfy the condition C, + C 1 + C2 = I at the downcoast control point. However, analyzing the example presented by the authors in Fig. 3, the fitted curve departs from Hsu and Evans' bay planform in the extreme opposite of the control point. Theoretically, this curve should improve the fit at the control point, but far away from this point the fit should be similar. To see if this behavior is due to the selected example, in which the authors reproduced the bay planform using Hsu and Evans' coefficients, the proposed method has been applied to several bayed beaches of the Catalonian coast (northwest Mediterranean Sea, Spain). For illustration, Fig. 5 shows the relation between R/Ril and 13/8 for one of the beaches considered (called Platja del Castell) as predicted by the authors' method and Hsu and Evans' method. Also shown is the empirical fit of the bay planform to a parabolic beach (second-order polynomial) by the least-squares method. This beach is located in a small hay, in such a way that at the extremes the condition of zero transport can be reasonahly assumed, and there is no river inside the hay. Under these conditions, it can he assumed that the planform of the beach is in static equilihrium and the method can he applied. The empirical fit of the heach gave an r 2 value of 1.00, and the coefficients ohtained were Cll = ~0.1O, C = 1.61, and C2 = ~().51.