Wave setup is the superelevation of mean water level caused by wave action (additional changes in water level may include wind setup or tides). Total water depth is a sum of still-water depth and setup:
where h = still-water depth and = mean water surface elevation about still-water level.
Wave setup balances the gradient in the cross-shore directed radiationstress, i.e., the pressure gradient of the mean sloping water surface balances the gradient of the incoming momentum.
Mean water level is governed by the cross-shore balance of momentum:
where Sxx is the cross-shore component of the cross-shore directed radiation stress, for longshore homogeneous waves and bathymetry.
Radiation stress both raises and lowers (setdown) the mean waterlevel across shore in the nearshore region (Figure 1).
Figure 1. Definition sketch for wave setup
Seaward of the breaker zone, Longuet-Higgins and Stewart (1963) obtained setdown for regular waves from the integration of Equation , assuming linear wave theory, normally incident waves, and in deep water:
The maximum lowering of the water level, setdown, occurs near the breakpoint .
In the surf zone, increases between the break point and the shoreline (Figure 1). The MSL gradient, assuming linear theory and substituting () in Equation , is given by:
The value of depends on wave decay through the surf zone. Applying the saturated breaker assumption of linear wave height decay on a plane beach, Equation reduces to:
CombiningEquations and , setup at the still-water shoreline is given by:
The first term in Equation is setdown at the break point and the second term is setup across the surf zone. The setup increases linearly through the surf zone for a plane beach. For a breaker depth index b of 0.8, ≈ 0.15 hb. Note that, for higher breaking waves, hb will be greater and thus setup will be greater.Equation gives setup at the still-water shoreline; to calculate maximum setup and position of the mean shoreline, the point of intersection between the setup and beach slope must be found. This can be done by trial and error, or, for a plane beach, estimated as:
where Δx is the shoreward displacement of the shoreline and is the setup at the mean shoreline.
Wave setup and thevariation of setup with distance on irregular (non-planar) beach profiles can be calculated based on Equations and (e.g., McDougal and Hudspeth 1983, Larson and Kraus 1991).
Setup for irregular waves should be calculated from decay of the wave height parameter Hrms. Wave setup produced by irregular waves is somewhat different than that produced by regular waves (Equation ) because long waveswith periods of 30 sec to several minutes, called infragravity waves, may produce a slowly varying mean water level. See Part 2 for discussion of magnitude and generation of infragravity waves. Figures 2 and 3 show irregular wave setup, nondimensionalized by Hrmso, for plane slopes of 1/100 and 1/30, respectively. Setup in these figures is calculated from the decay of Hrms given by the irregularwave application of the Dally, Dean, and Dalrymple (1985) wave decay model (see Figure II-4-4). Nondimensional wave setup increases with decreasing deepwater wave steepness. Note that beach slope is predicted to have a relatively small influence on setup for irregular waves.
Figure 2. Irregular wave setup for plane slope of 1/100
Figure 3. Irregular wave setup for plane slope of 1/30Problem
A plane beach having a 1 on 100 slope, and normally incident waves with deepwater height of 2 m and period of 10 sec (where the incipient breaker height and depth were determined as 2.7 m and 3.2 m, respectively; the breaker index is 0.84).
Setup across the surf zone.
1.- Setdown at the breaker point is determined from Equation . At breaking (assuming...
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