# 8.3.3: Analytical solution for accretion near breakwater or jetty

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The first person to describe the single line theory was Pelnard-Considère (1956). He came up with an analytical solution that can be used for a quick assessment of the effects of a structure, for instance a breakwater, on a coastline.

In order to find analytical solutions, we need to simplify Eq. 8.3.2.5. If we restrict ourselves to small changes in the angle of wave attack, we can assume \(\partial S_x/\partial \varphi\) to be constant. Such an assumption implies that a segment of the (\(S, \varphi\))-curve has been replaced by a straight line, which is reasonable for relatively small changes in \(\varphi\). We can now write:

\[\dfrac{\partial S_x}{\partial \varphi} = s \label{eq8.3.3.1}\]

in which \(s\) is a coastal constant. For \(-20^{\circ} < \varphi < 20^{\circ}\), \(S\) is a linear function of the wave angle, such that \(s = S/\varphi '\) (see Sect. 8.2.5). Note that \(S\) is the transport for the original angle of incidence \(\varphi '\).

Substituting Eq. \(\ref{eq8.3.3.1}\) in Eq. 8.3.2.5 changes the parabolic differential equation to:

\[\dfrac{\partial Y}{\partial t} - \dfrac{s}{d} \dfrac{\partial^2 Y}{\partial x^2} = 0 \label{eq8.3.3.2}\]

This equation shows that the curvature in the coastline, scaled by the coastal constant, determines the coastal change. To solve Eq. \(\ref{eq8.3.3.2}\) we need one initial condition and two boundary conditions. Very often these conditions are the position of the coastline at \(t = 0\) (the initial situation) and the sediment transport on both borders of the coastal area as a function of time.

Consider the (sudden) construction of a breakwater or a jetty running perpendicular to an initially straight shoreline (Fig. 8.11). If the longshore transport is from left to right, then the coast to the left of the structure is called the updrift coast, whereas the coast to the right of the breakwater is called the downdrift coast. The breakwater is assumed to extend into the sea until at least the depth of closure.

The initial condition is the shape of the coastline at time \(t = 0\):

\[Y = 0, \ \text{ for } \ t = 0 \ \text{ and for } \ -\infty < x < 0\]

*One boundary condition* is that at a great distance from the breakwater, \(x = -\infty\), the sand transport remains constant and equal to its value before breakwater construction:

\[S_x = S, \ \text{ for } \ x = -\infty \ \text{ and for all } \ t\]

The *second boundary condition* is imposed at the breakwater; it is impermeable to sand (no sand particles slipping through the pores of the breakwater). This means that the local longshore transport at the breakwater must be zero (100% blocking):

\[S_x = 0, \ \text{ for } \ x = 0 \ \text{ and for all } \ t\]

This boundary condition is only valid as long as the breakwater still blocks the entire transport, viz. as long as the active zone has not moved past the tip of the breakwater. Note that a zero transport means that the wave angle with respect to the orientation of the coastline must be zero as well. Since the offshore wave direction does not change, this can only be accomplished by a shoreline rotation, such that the wave angle relative to this rotated coastline becomes zero (Fig. 8.11). The boundary condition at \(x = 0\) can therefore also be written as:

\[\partial Y/ \partial x = \varphi ', \ \text{ for } \ x = 0 \ \text{ and for all } \ t\]

In other words, the beach accretion progresses seaward always making angle \(\varphi '\) with respect to the \(x\)-axis at the breakwater. Hence, at the breakwater, the coastline and the depth contours tend to become parallel to the approaching waves.

With the help of Eq. \(\ref{eq8.3.3.2}\), applying the reference system of Fig. 8.11 and using the mentioned initial and boundary conditions, the shape of the coastline \(Y(x, t)\) on the updrift side of the breakwater can be solved as a function of time. Here, we only give some characteristics of the resulting solution (see also Fig. 8.12). First, the outward growth of the coastline at the breakwater \(L(t)\) at \(x = 0\) is found to be:

\[L(t) = 2 \sqrt{\dfrac{\varphi ' St}{\pi d}} = \varphi ' \sqrt{\dfrac{4at}{\pi}}\label{eq8.3.3.7}\]

where:

\(t\) | time | \(yr\) |

\(d\) | profile height | \(m\) |

\(a\) | \(=\dfrac{s}{d} = \dfrac{S}{\varphi ' d}\) | \((m^3/yr)/rad/m\) |

Secondly, it can be shown that the influence of the breakwater (in terms of accretion of the shoreline relative to the accretion at the breakwater) is negligible at a distance \(5\sqrt{at} = 2.5 \sqrt{\pi} L/\varphi '\) from the breakwater. So, the influenced shoreline is dependent on the steepness \(s\) of the considered segment of the (\(S,\varphi\))-curve and the height of the active zone \(d\) (both of which are strongly dependent on the wave height). Both the horizontal extent of the influenced zone and the seaward growth at the breakwater vary with \(\sqrt{t}\). This means that the progress of the shoreline gradually slows down; suppose that after a time \(t_1\) the accretion length at the breakwater is \(L_1\). At \(t_2 = 2t_1\) the accretion length has not doubled but only increased to \(L_2 = \sqrt{2} L_1\). Equation \(\ref{eq8.3.3.7}\) further shows that, of course, the accretion speed increases with increasing sediment transport \(S\). The surface area OCB in Fig. 8.12 is equal to:

\[\text{surface}\ \text{OCB} = \dfrac{1}{2} \dfrac{L^2}{\varphi '} = \dfrac{2}{\pi} \dfrac{St}{d}\]

Also, from continuity, the total surface area AOB is:

\[\text{surface}\ \text{OCB} = \dfrac{St}{d}\]

Therefore, 64% of the total accretion is stored in the shaded area OCB.

For the coastal changes on the lee side of the breakwater (downdrift) an analogous analytical solution can be derived. In principle, the coastline changes on the updrift side are mirrored on the downdrift side and the total volume of updrift accreted sediment equals the total volume of downdrift eroded material. This is illustrated in Fig. 8.13.

In reality however, there is a difference in the wave attack between the updrift and downdrift side. At the downstream side a part of the shore is sheltered from wave attack by the breakwater. Waves propagating past the end of the breakwater, diffract into the shadow zone (see Sect. 5.2.4). As a result the wave heights are significantly lower than in the undisturbed region (see Fig. 8.14). Also the wave angles are significantly altered, since the wave rays turn somewhat towards the breakwater. At the breakwater the rays run parallel to the breakwater. In principle, at the breakwater (zero transport), the coastline reorients itself normal to the local waves (hence parallel to the original coastline). There is however an additional shadow zone effect due to alongshore set-up differences. Due to the alongshore differences in wave height, the wave set-up in the surf zone changes along the shore (Sect. 5.5.4). The lower wave heights just behind the breakwater result in a lower set-up at the shoreline close to the breakwater than elsewhere along the coast. The alongshore water level gradients result in a secondary current pattern that – close to the coast – is directed towards the breakwater (Fig. 5.45) and induce a sediment transport towards the breakwater.

The resulting downdrift coastline development depends on the combination of all effects. The transport must increase from 0 at the breakwater to \(S\) in the undisturbed region. However, shadow zone effects will reduce the transport magnitude directed away from the breakwater or may even cause a net transport towards the breakwater. This leads to a coastline development as shown schematically in Fig. 8.14 (the solid line).

Analytical solutions also exist for different boundary conditions at the breakwater and for different breakwater layouts. For example, after some time a beach can build up on the updrift side to such an extent that sand will be transported around the break-water tip. This bypassing of sediment can be taken into account. Analytical solutions make hand computations of coastline changes possible and hence facilitate a quick assessment of the impact of certain coastal structures on the shoreline development. Neither tidal influences (see Fig. 5.60 for tidal currents passing a breakwater) nor variations along the coast of wave height, wave direction or sediment characteristics can be taken into account. Besides, shoreline evolution is calculated under the assumption of steady wave conditions. This means that, as in the above example, only a single wave condition is taken into account. This limits the applicability of these analytical solutions, especially in the vicinity of structures where sheltering effects are important and vary strongly with the wave direction. In Sect. 8.4.2 this is further explained.

Numerical methods based on Eq. 8.3.2.5 can often include the above-mentioned processes. These methods still suffer from limitations as a result of the basic assumption that coastline models are based on, namely the assumed equilibrium coastal profile. This assumption does not allow for profile changes in time or along the coast and assumes an instantaneous cross-shore redistribution of sediment when erosion of accretion occurs. This may be appropriate in the case of for instance a perturbation by a nourishment (Sect. 8.4.4). However, many situations are truly 2D which implies that cross-shore and alongshore processes are not physically independent and the assump- tion of an instantaneous equilibrium cross-shore profile is invalid. Two examples of two-dimensional effects are the lee-side circulation patterns as mentioned before and ‘outbreaking’ longshore currents as they approach the breakwater. The latter refers to the fact that the water flowing towards the breakwater on the updrift side has no other option than to go offshore, since the longshore current decreases to zero at the break-water (Fig. 8.15). This contributes to the redistribution of material in the cross-shore direction.