The Analysis of Exponential Beach Profiles

Abstract

The overall form of many beach profiles has an exponential shape where the depth h is given by h = S₀ / k) (1-e^-kx) with S₀ the beach-face slope at x = 0 and k is an adjustable coefficient that determines the degree of concavity. The cross-shore variation in beach slope is then S = S₀e^–kx or S = S₀ - kh. Beach profiles can be analyzed in terms of both the cross-shore variations in hand S. Based on previous studies, the beach-face slope S₀ is predictable as a function of the sediment grain size and wave parameters. The evaluation of k can be based on best-fit comparisons with the measured profile depths or from bottom slope variations across the profile. Equations are derived for the evaluation of k from the offshore closure depth of the envelope of profile changes, or from some arbitrarily selected coordinate of the profile. An example of the analysis approach is provided by a beach profile from the Nile Delta coast of Egypt. This measured profile shows good agreement with the exponential form for cross-shore variations in both the absolute depth h and local bottom slope S. There is poor agreement with the h = Ax^2/3 profile relationship, in part because this Nile Delta profile is more reflective and has a greater concavity than allowed by the x^2/3 dependence. The failure of the x^2/3 profile form is still more evident in analyzing the beach-slope variations since it predicts an infinite slope at the shoreline. The exponential beach profile is a convenient mathematical relationship that should be useful in many applications.