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Free Long Wave Transformation in the Nearshore Zone Through Partial Reflections

Stephanie ContardoaCSIRO Oceans and Atmosphere, Crawley, Australia
bSchool of Earth Sciences, The University of Western Australia, Crawley, Australia

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Ryan J LowecOceans Graduate School, The University of Western Australia, Crawley, Australia

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Francois DufoisdPacific Community Center for Ocean Science (PCCOS), Pacific Community (SPC), Nouméa, New Caledonia

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Jeff E HansenbSchool of Earth Sciences, The University of Western Australia, Crawley, Australia

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Mark BuckleyeSt. Petersburg Coastal and Marine Science Center, U.S. Geological Survey, St. Petersburg, FL 33701 U.S.A.

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Graham SymondsbSchool of Earth Sciences, The University of Western Australia, Crawley, Australia

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Abstract

Long waves play an important role in coastal inundation and shoreline and dune erosion, requiring a detailed understanding of their evolution in nearshore regions and interaction with shorelines. While their generation and dissipation mechanisms are relatively well understood, there are fewer studies describing how reflection processes govern their propagation in the nearshore. We propose a new approach, accounting for partial reflections, which leads to an analytical solution to the free wave linear shallow-water equations at the wave-group scale over general varying bathymetry. The approach, supported by numerical modelling, agrees with the classic Bessel standing solution for plane sloping beach but extends the solution to arbitrary alongshore uniform bathymetry profiles and decomposes it into incoming and outgoing wave components, which are a combination of successively partially reflected waves lagging each other. The phase lags introduced by partial reflections modify the wave amplitude and explain why Green’s law, which describes the wave growth of free waves with decreasing depth, breaks down in very shallow water. This reveals that the wave amplitude at the shoreline is highly dependent on partial reflections. Consistent with laboratory and field observations, our analytical model predicts a reflection coefficient that increases and is highly correlated with the normalised bed slope (bed slope relative to wave frequency). Our approach shows that partial reflections occurring due to depth variations in the nearshore is responsible for the relationship between the normalised bed slope and the amplitude of long waves in the nearshore, with direct implications for determining long-wave amplitudes at the shoreline and wave runup.

Corresponding author: Stephanie Contardo, stephanie.contardo@csiro.au

Abstract

Long waves play an important role in coastal inundation and shoreline and dune erosion, requiring a detailed understanding of their evolution in nearshore regions and interaction with shorelines. While their generation and dissipation mechanisms are relatively well understood, there are fewer studies describing how reflection processes govern their propagation in the nearshore. We propose a new approach, accounting for partial reflections, which leads to an analytical solution to the free wave linear shallow-water equations at the wave-group scale over general varying bathymetry. The approach, supported by numerical modelling, agrees with the classic Bessel standing solution for plane sloping beach but extends the solution to arbitrary alongshore uniform bathymetry profiles and decomposes it into incoming and outgoing wave components, which are a combination of successively partially reflected waves lagging each other. The phase lags introduced by partial reflections modify the wave amplitude and explain why Green’s law, which describes the wave growth of free waves with decreasing depth, breaks down in very shallow water. This reveals that the wave amplitude at the shoreline is highly dependent on partial reflections. Consistent with laboratory and field observations, our analytical model predicts a reflection coefficient that increases and is highly correlated with the normalised bed slope (bed slope relative to wave frequency). Our approach shows that partial reflections occurring due to depth variations in the nearshore is responsible for the relationship between the normalised bed slope and the amplitude of long waves in the nearshore, with direct implications for determining long-wave amplitudes at the shoreline and wave runup.

Corresponding author: Stephanie Contardo, stephanie.contardo@csiro.au
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