An Approximate Analytical Method for Describing the Kinematics of a Bore Over a Sloping Beach

This article describes an approximate analytical method for determining the evolutionary behavior of a bore of arbitrary strength as it approaches the shoreline on a sloping beach. An infinite system of transport equations for the bore strength, that holds on the bore front, allows investigation of the dynamical coupling which exists between an arbitrary bore strength and the rearward flow. The evolutionary behavior of weak and strong bores described by the lowest order truncation of the system is in excellent agreement with the result predicted by the characteristic rule. However, for bores of intermediate strength 1< M <5, the results obtained from higher order truncation approximations differ qualitatively from those obtained from the characteristic rule. Strangely, on using the characteristic rule, the critical values of the bore strength for which the bore height and bore speed attain respectively a maximum and a minimum, remain uninfluenced by the initial bore strength and the undisturbed water depth. The present method takes care of this situation in a natural manner, and the first few approximations yield results with reasonably good accuracy. It should be remarked that even the lowest order of approximation describes the evolutionary behavior of a bore with good accuracy, though when the initial bore is of moderate strength, i.e., 1< M 0 <2, even the first order approximation provides an excellent description of the evolutionary behavior; indeed, the improvement takes place only on a subinterval, whose length diminishes as M 0 moves closer to either 1 or 2.

Keywords:

• Bore
• Hyperbolic Equations
• Whitham's Characteristic Rule