New Relationships for Equilibrium Shaped Bays

Several types of equations have been proposed in the past 30 years to describe the planform of crenulate-shaped bays in static equilibrium: a logarithmic spiral, a parabolic curve, a hyperbolic tangent curve, and a set of circle arcs. Although several authors pointed out that these models have some physical and mathematical weaknesses, they find much use in practical applications, particularly the parabolic form. Several expressions of the parabolic shape have been proposed. Hsu and Evans’ original ones fit best the experimental values of Ho’s model tests and of Australian bays in equilibrium used to determine these expressions. The formula proposed by Tan and Chiew respects the tangent condition of the incident wave crest line at the control point and the crossing of the beach line through this point. The expression proposed in Italy by Mita also respects the conditions on the critical angle of a bay in equilibrium, as experimentally determined by Hsu and Silvester; but it poorly fits the data of Ho’s model tests and of Australian bays in equilibrium. This article proposes a new cubic equation to describe crenulate-shaped bays in static equilibrium. The proposed relationship includes the advantages of the three parabolic expressions mentioned above. In fact this proposed third-order equation interpolates the experimental data used by Hsu and Evans to the same degree of approximation as in their original second-order relationship. It also rigorously respects the theoretical conditions at the control point on the downdrift beach line as well as the condition of the critical angle experimentally determined by Hsu and Silvester. Finally the indentation ratio values proposed by Silvester and Hsu for bays in equilibrium are well approximated by the cubic equation across the whole application range. In short, the current formalism is mathematically inconsistent and the proposed modification resolves this as a relatively straightforward amendment of the formulas which leads to moderate changes in the results of the shoreline, but reduces some of the errors in the process of applying this type of model in engineering practice.