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Wave transmission at coastal structures has been studied extensively. However, most of the studies have focused on wave transmission at the low-crested coastal structure without considering another structure behind it (see Figure 1).

Figure 1: Submerged low-crested structure (LCS) in front of an (emerged) coastal structure (RMB).

A low-crested structure (LCS) in front of another (emerged) coastal structure can cause the mean water level in between them to increase (see Figure 1). This structure-induced wave set-up is expected to be at its maximum if the water between the two structures cannot flow sideways, for instance for a continuous LCS along the (emerged) coastal structure, or for a coastal system with segmented low-crested structures for which the lateral flow is blocked or reduced by another LCS (see left part of Figure 2). Blocking or reducing the lateral flow can reduce seabed scouring compared to a coastal system without structures to block or reduce the lateral flow (middle and right part of Figure 2).

Figure 2: Top view of a segmented system of submerged low-crested structures (LCS) in front of an emerged coastal structure (dike or breakwater), with a blocked or reduced lateral flow on the left side.

The present 2D study is focused on a system of coastal structures where the lateral flow is blocked, leading to an upper limit of the potential effects of the structure-induced wave set-up between the submerged LCS and the emerged coastal structure. The following research questions are dealt with:

What is the magnitude of the structure-induced wave set-up between the submerged LCS and the (emerged) coastal structure, and is this mean water level set-up affected by the permeability of the structures and by the distance between the structures?

Is the structure-induced wave set-up affecting the amount of wave transmission at the submerged low-crested structure?

How does the structure-induced wave set-up affect the wave overtopping discharge at the (emerged) coastal structure?

To answer above research questions, physical model tests have been performed in a wave flume at Deltares, Delft. While wave transmission tests by Van Gent et al. (2023) serve as reference tests without an emerged coastal structure behind the submerged LCS, the following configurations with a combination of structures were tested:

Impermeable submerged LCS in front of an impermeable slope with a long distance in between them.

Impermeable submerged LCS in front of an impermeable slope with a short distance in between them.

Impermeable submerged LCS in front of a permeable rubble mound breakwater.

Permeable submerged LCS in front of a permeable rubble mound breakwater.

The physical model tests are described in Section 2. In Section 3 the test results are analyzed. Section 4 provides a discussion of results. The conclusions and recommendations are described in Section 5.

Physical model tests

The physical model tests were performed in a 55 m long section of the Scheldt Flume (110 m long, 1 m wide, and
1.2 m high) at Deltares, Delft. The wave generator is equipped with active reflection compensation, accounting for short-waves and long-waves effects. This means that the motion of the wave paddle compensates for the waves reflected by the structure, preventing them from being re-reflected at the wave paddle and propagating towards the model. Second-order wave generation was applied.

A horizontal foreshore was constructed on which the structures were placed. A transition slope was made between the bottom of the flume and the horizontal foreshore. At deep water, seaward of the submerged low-crested structure, and between the LCS and the (emerged) structure, wave gauges were positioned to measure the structure-induced wave set-up and to separate incident and reflected waves from the measured surface elevations (based on Zelt and Skjelbreia, 1992, and De Ridder et al., 2023).

Figure 3: Tested impermeable and permeable submerged low-crested structures (pictures from Van Gent et al, 2023).

Impermeable low-crested structure in front of impermeable structure

Two test set-ups were applied with an impermeable (trapezoidal) LCS (left panel of Figure 3) in front of an inshore impermeable slope (upper left panel of Figure 5). These tests were focused on determining the structure-induced wave set-up between the two structures (see Figure 1). To analyze the potential influence of the distance between the two structures, this distance was varied; in one set-up the distance between the two structures was 2 m in the model (measured at the seabed) and in another set-up the distances was 10 m, see Figure 4.

Figure 4: Test set-ups with impermeable LCS in front of impermeable slopes (resp. with 2 m and 10 m in between).

Both structures were made of wood, leading to smooth slopes. The impermeable slope of the emerged structure was non-overtopped. Wave gauges were positioned at deep water, on the horizontal part seaward of the LCS, and between the two structures. At these locations a set of wave gauges was used to measure the mean water level set-up (average of the corresponding wave gauges) and to separate incident from reflected waves (except for the set-up, where only 2 m existed between the two structures, which is not sufficient to separate incident from reflected waves). At deep water and seaward of the LCS a set of 4 wave gauges was positioned, while between the two structures 3 to 5 wave gauges were positioned.

Impermeable low-crested structures in front of permeable structure

To analyze the potential influence of the structure-induced wave set-up on wave overtopping discharges, tests have been performed with an impermeable LCS in front of an inshore overtopped rubble mound breakwater (see Figure 6). The impermeable LCS is again the same structure as before (see left panels of Figure 3), while the rubble mound breakwater has a configuration based on tests by Van Gent et al (2022) where no LCS was present in front of the rubble mound breakwater (see right and lower-left panel of Figure 5). Since the rubble mound breakwater is now placed on a shallower foreshore while the wave loading is reduced due to the presence of the LCS, the total height and freeboard of the rubble mound breakwater were reduced compared to the structure tested by Van Gent et al (2022). The rubble mound breakwater consisted of a 1:2 armour layer with stones Dn50 = 0.0317 m (layer thickness 0.060 m), a filter layer with
Dn50 = 0.0168 m (layer thickness 0.030 m), and a core with Dn50 = 0.0065 m. At the crest, a protruding crest wall with a total height of 0.14 m was present with a protruding part of 0.05 m. Overtopping water was collected in an overtopping box using an overtopping chute starting at the crest wall. The tested cross-section is shown in Figure 7.

Figure 6: Test set-up with impermeable LCS in front of (permeable) rubble mound breakwater (RMB).

Analysis of test results

Structure-induced wave set-up

The mean water level set-up induced by structures has been studied by Longuet-Higgins (1967) who provided an analytical solution for small-amplitude non-breaking monochromatic waves and frictionless structures. Obviously, the solution is not valid for structures under severe irregular wave loading, as tested in the present study. Diskin et al (1970) carried out physical model tests on permeable structures with monochromatic waves. Diskin et al (1970) derived an empirical expression in which the non-dimensional structure-induced wave set-up (/H) depends only on the non-dimensional freeboard (Rc / H), although they observed also an influence of the wave period.

(1)

The main parameter affecting the structure-induced wave set-up () is the freeboard (Rc). In Figure 9 the measured non-dimensional set-up (/Hm0) is shown versus the non-dimensional freeboard (Rc /Hm0). The left panel shows all tests with an impermeable LCS and the right panel all tests with a permeable LCS.

Figure 9: Measured non-dimensional wave set-up (/Hm0) versus the non-dimensional freeboard (Rc /Hm0); Left panel: All tests with an impermeable LCS; Right panel: All tests with a permeable LCS.

Figure 9 shows that for both types of structures the wave set-up clearly depends on the non-dimensional freeboard. Figure 9 also shows that for the impermeable LCS there is also a clear dependency on the wave steepness (i.e. larger wave set-up for larger wave steepnesses), while for the permeable LCS this dependency is less clear. Comparing the left and right panels shows that the set-up is larger for the impermeable LCS. This is expected to be due to the seaward flow through the permeable LCS that reduces the wave set-up. If the seaward flow through the permeable LCS is affecting the wave set-up, the height of the permeable LCS may play a role. Besides the height of the structure, a lower permeability resulting from a smaller stone size, smaller porosity, or wider structure may increase the wave set-up, but this was not examined in the present test programme.

For an impermeable submerged LCS in front of another structure an empirical expression for the wave set-up has been derived using the non-dimensional freeboard (Rc / H) and the wave steepness (for 0.01 ≤ sm-1,0 ≤ 0.04 and -1 ≤ Rc / Hm0 ≤ 0):

(2)

The left panel of Figure 11 shows all data with impermeable low-crested structures in front of three types of emerged structures. The right panel shows the same data but with the symbols denoting the tests with different wave steepnesses. Figure 11 shows that Equation 2 provides an accurate description of the measured structure-induced wave set-ups, for each of the configurations with an impermeable LCS and for each wave steepness. The RMSE value based on the non-dimensional set-up is 0.0082.

Wave transmission

Wave transmission at coastal structures has been studied by a large number of researchers (e.g. Sollitt and Cross, 1972; Daemen, 1991; d’Angremond et al, 1996; Seabrook and Hall, 1998; Bleck and Oumeraci, 2001; Calabrese et al, 2002; Briganti et al, 2003; Van der Meer et al, 2005; Van Oosten et al, 2006; Koutandos et al, 2006; Buccino and Calabrese, 2007; Makris and Memos, 2007; Panizzo and Briganti, 2007; Goda and Ahrens, 2008; Tomasicchio et al, 2011; Metallinos et al, 2016; Mahmoudi et al, 2017; Lokesha et al, 2019; Brancasi et al, 2022; Le Xuan et al, 2022; Kurdistani et al, 2022; Van Gent et al, 2023). For a discussion of the various wave transmission studies see the latter two studies. Many of those studies resulted in methods to estimate wave transmission at low-crested structures. However, most of the research was focused on wave transmission at coastal structures without another structure behind these low-crested structures. In the present study the tested LCS configurations correspond to those tested by Van Gent et al (2023) and therefore the derived empirical expression based on those tests is applied here:

(4)

where the freeboard Rc is made non-dimensional using the wave height Hm0 and the crest width B is made non-dimensional using the wavelength based on the spectral wave period: Lm-1,0 = (g/2) Tm-1,02. The coefficients in Equation 4 depend on the type of structure. The coefficients for smooth impermeable and permeable homogeneous trapezoidal low-crested structures are shown in Table 1.

Table 1: Coefficients in Equation 4 for impermeable and permeable low-crested structures.

Structure type	c1	c2	c3	c4	c5
A: Impermeable structure	0.47	3.1	0.75	0	0.5
B: Permeable structure	0.43	3.1	0.75	-0.25	0.5

The left panel of Figure 14 shows all data with an impermeable LCS in front of a structure (tests with an impermeable slope 10 m behind the LCS and tests with a rubble mound structure behind the LCS) together with the test conditions by Van Gent et al (2023) without a structure behind the LCS (red filled squares). The graph shows that the measured wave transmission from the tests with impermeable slopes behind the LCS (red open squares) are on average slightly higher than Equation 4 predicts, but still the match between the data and Equation 4 is rather good: RMSE=0.0322. The graph shows that the measured wave transmission from the tests with a permeable rubble mound structure behind the LCS (green filled squares) are on average equal to those obtained using Equation 4. The spreading around the trend is also small: RMSE=0.0153. Note that the data on which Equation 4 was based (red filled squares) lead to RMSE=0.0166.

Conclusions and recommendations

The influence of a submerged low-crested structure in front of an emerged coastal structure were studied by means of physical model testing in a wave flume. The tested trapezoidal low-crested structures were either impermeable or permeable, while the emerged structures were also impermeable or permeable. Between the low-crested structure and emerged structure wave set-up occurs. This structure-induced wave set-up was studied, as well as the effects on wave transmission at the low-crested structures and the effects on wave overtopping at the emerged coastal structures. The study provides the following insights:

Structure-induced wave set-up:

For the performed tests the structure-induced wave set-up reached levels up to 21% of the incident significant wave height. The wave set-up is dominated by the freeboard of the low-crested structure. The wave steepness also affects the wave set-up, while the effects of the wave steepness are more significant for an impermeable low-crested structure than for a permeable low-crested structure. For an impermeable low-crested structure the wave set-up is clearly larger than for a permeable low-crested structure. For a permeable homogeneous low-crested structure the height of the structure slightly affects the wave set-up.

For an impermeable low-crested structure in front of an emerged impermeable slope, the distance between the two structures does not affect the structure-induced wave set-up.

For a submerged low-crested structure in front of an emerged structure, the structure-induced wave set-up can be described by empirical expressions with reasonably good accuracy (Equations 2 and 3 for an impermeable and permeable low-crested structure, respectively).

Wave transmission:

For a low-crested structure in front of an emerged coastal structure, the influence of structure-induced wave set-up on wave transmission is negligibly small. For the performed tests with impermeable and permeable low-crested structures, the empirical expression by Van Gent et al (2023) (i.e. Equation 4) provides reasonably accurate estimates of the wave transmission coefficients.

Wave overtopping:

The structure-induced wave set-up between the submerged low-crested structure and the emerged structure affects the wave overtopping discharge. The effects can be taken into account by adding the structure-induced wave set-up to the water level in front of the emerged structure (rubble mound breakwater). With the modified freeboard, the existing expression to estimate wave overtopping discharges by Van Gent et al (2022) (i.e. Equation 10) can be applied with similar accuracy as for rubble mound breakwaters without a low-crested structure in front.

It is recommended to take structure-induced wave set-up into account when estimating wave overtopping discharges at coastal structures with a low-crested structure in front. For structures where a lateral flow can occur between the low-crested structure and the emerged structure, structure-induced wave set-up will be less than for situations in which this lateral flow cannot develop. For situations with a lateral flow, the presented method to estimate wave set-up and its effects on wave overtopping provides a conservative (safe) estimate of wave overtopping discharges. For situations with a lateral flow in reality, while such lateral flow obviously cannot occur in wave flume tests, special care must be taken in the analysis of wave flume tests.

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Notation

Name	Symbol	Unit

slope angle of structure	α

structure-induced wave set-up	m

influence factor for wave overtopping	-

crest level of armour at crest, relative to still water level	Ac	m

crest width of low-crested structure	B	m

stone diameter	Dn50	m

gravitational acceleration	g	m/s2

significant wave height of incident waves at toe of structures, based on wave energy spectrum	Hm0	m

significant wave height of transmitted waves at rear of structures, based on wave energy spectrum	Hm0-t	m

Abbreviations

LCS	Low-crested structure

RMB	Rubble mound breakwater

RMSE	Root mean square error

SWL	Still water level

References

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