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Formulation of the undertow using linear wave theory
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The undertow is one of the most important mechanisms for sediment transport in nearshore regions. As such, its formulation has been an active subject of research for at least the past 40 years. Still, much debate persists on the exact nature of the forcing and theoretical expression of this current. Here, assuming linear wave theory and keeping most terms in the wave momentum equations, a solution to the undertow in the surf zone is derived, and it is shown that it is unique. It is also shown that, unless they are erroneous, most solutions presented in the literature are identical, albeit simplified versions of the solution presented herein. Finally, it is demonstrated that errors in past derivations of the undertow profile stem from inconsistencies between (1) the treatment of advective terms in the momentum equations and the wave action equation, (2) the expression of the mean current equation and the surface shear stress, and (3) the omission of bottom shear stress in the momentum equation.
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