A simple analytical expression to describe tidal damping or amplification

Can a single line describe tidal damping or amplification? Tidal damping and amplification in alluvial estuaries are constrained by an implicit feedback mechanism. The time lag ϵ between the occurrence of high water and high water slack is crucial in this regard. An analytical solution of the St. Venant equations appears to result in a surprisingly simple explicit relation for the tidal range, consisting of an exponential and a linear term. In alluvial estuaries, the linear term is dominant, particularly in the case of tidal amplification. In the case of tidal damping the exponential term only becomes important in the upper reaches of the estuary (preventing the expression for the tidal range from becoming negative). In tidal amplification, the exponential term is suppressed by the newly defined tidal Froude Number which (as it contains sin ϵ) tends to zero when the tidal wave gets a predominantly standing wave character. This negative feedback prevents the development of an exponentially increasing tidal range. Finally, the expression obtained is a very useful explicit equation to determine estuary parameters that are difficult to determine from direct observations, such as the roughness and the mean water depth.