# 6.5.5: Summary and concluding remarks

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Some theoretical and many (semi-) empirical bed load formulations have been proposed in the literature. Many formulas were originally developed for rivers, and later applied to coastal environments by using a bed shear stress due to waves and currents in the original formulations (Meyer-Peter Müller, Kalinske et cetera). These formulations can normally be written in the form Eq. 6.5.2.5.

Unfortunately there is a massive difference in the results of the different (bed load) formulas, making the uncertainty in sediment transport computations rather large. The transport formulations can only be used with enough confidence if they have been properly calibrated, preferably with data for the considered site and for representative hydrodynamic conditions.

A common attribute in the bed load formulations is that the shear stress is raised to a certain power, say 1.5 to 2. In Eq. 6.5.2.6, for instance, at any moment in time, the transport is in the direction of the intra-wave velocity and has a magnitude proportional to the shear stress magnitude to the power 1.8. Since the shear stress itself is related quadratically to the velocity signal, we may simply state that the magnitude of the instantaneous bed load transport depends on the modulus of the near-bed velocity raised to a power \(n\) = 3 to 4. As expected, the velocity direction does not influence the sediment load. Since the transport is in the *direction* of the instantaneous velocity, we have \(S(t) \propto \text{sign} (u) |u|^n\) or equivalently \(S(t) \propto u|u|^{n - 1}\). The latter function may be interpreted as the product of a transporting velocity \(u\) and the sediment load stirred by waves and currents proportional to \(|u|^{n - 1}\). This concept we had already encountered in Sect. 6.5.4 for the more specific case of transport dominated by a mean current. Averaged over time, for instance over the short-wave period or the tidal period, we have \(\langle \propto \langle u |u|^{n - 1} \rangle\), with brackets denoting time-averaging. Intermezzo 6.4 demonstrates that even a purely oscillatory motion may give rise to a net (time-averaged) sediment transport.

An oscillatory velocity signal may, under certain circumstances, result in a wave-averaged sediment transport. First imagine a velocity signal \(u(t)\) that is purely symmetric about the horizontal axis (a sine or saw-tooth wave). In that case \(u |u|^{n - 1}\) is also symmetric and it follows that the wave-averaged transport \(\langle S \rangle = 0\) (check this by sketching \(u\), \(|u|^2\) and \(u|u|^2\) as a function of time). The symmetrical orbital motion simply moves an amount of sediment back and forth without a net wave-averaged transport. Now consider a positively skewed velocity signal, characteristic for shoaling waves (Sect. 5.3) or a flood-dominant tide (Sect. 5.7.4), with larger peak velocities in the wave propagation direction than in the opposite direction. Even though \(\langle u \rangle = 0\), we now find \(\langle S \rangle \ne 0\). This is because the sediment load responds non-linearly to the velocity, such that more sediment is stirred up during the part of the wave cycle with velocities in the propagation direction (again draw \(u|u|^2\)). The result is a net (bed load) transport in the propagation direction. Similarly, a net current superimposed on a sinusoidal velocity signal introduces asymmetry about the horizontal axis, leading to a net sediment transport (in the current direction) that is larger than the transport for the current alone situation. Verify this!