Revetments are used to protect banks and shorelines from erosion caused by waves and currents. This paper briefly addresses the application of revetments in wave environments using riprap and articulated concrete blocks. The discussion is limited to low-energy wave conditions where wave heights are less than 5 feet. These conditions occur in sheltered waters such as lakes, reservoirs, rivers, channels, canals, estuaries, and bays. High-energy wave conditions that are encountered on open ocean coastlines are more appropriately addressed using armor stone or concrete armor units.

In many coastal engineering projects, determination of the design condition is a major component of the design effort. In the case of wave-induced bank erosion, it is first necessary to determine the cause of the erosion. Then, the wave and water level conditions must be determined. Wave data are usually not available in sheltered waters. For these cases, the waves must be estimated from historical wind conditions using hindcast methods such as those described in the U.S. Army Corps of Engineers Shore Protection Manual (SPM) and Coastal Engineering Manual (CEM) (COE, 1973, 1984; COE, 2001). High water level data may be available from gauging stations.

In the United States, it is common to design revetments based on an event of a specified occurrence (1 percent annual chance occurrence or 100 year event). The choice of the design event is a key consideration in the design. Once the occurrence level has been selected, the joint probability of waves and water levels must be determined. A conservative approach is to design the high water condition for the 1 percent wave occurring at the 1 percent water level. If the revetment does not extend to the bed or channel bottom, then a low water level design condition must also be determined.

Waves are specified by the wave height, H (vertical distance from crest to trough), the wave period, T (the time between the passage of successive waves), wave direction, ? (angle between wave crest and shoreline), and the still water depth, h (the water depth in the absence of waves). The wave length, L (horizontal distance from wave crest to wave crest), is determined from the wave period and water depth by the dispersion equation. If the water depth is greater than half the wave length (h > L/2), then conditions are considered deep water and the deep water wave length, L_{0}, may be written as

Equation 1:

Where:

g is the acceleration due to gravity.

When wave conditions are generated by winds, there are always multiple waves. There are many wave heights, periods, and directions existing simultaneously. It is convenient to represent this collection of waves by a single representative wave height, period, and direction. The significant wave height, H, is related to the average of the highest 1/3 of the waves. The spectral wave height, H_{m0}, is equal to four times the square root of the total energy in the waves. In deep water, these two are taken to be equivalent. The peak wave period, T_{p}, is the wave period corresponding to the most energetic wave. The mean period, T_{m}, is determined from the distribution of wave heights and periods. A rule of thumb is

Equation 2:

The wave direction is usually specified by the peak direction, ?_{p}, corresponding to the most common direction or the spectral direction, ?_{m}, corresponding to the energy propagation direction.

When waves propagate into shallow water, the depth eventually limits the maximum wave height and the waves break. The breaker index, ? , relates the depth-limited breaking wave height, H_{B}, to the breaking depth, h_{B}.

Equation 3:

Values for ? range from about 0.6 to 1.2, depending on the wave height and period and the bottom slope. For a flat bottom, ? =0.78, and this value is commonly used. The type of breaking wave can be estimated from the surf similarity parameter,

Equation 4:

in which a is the slope. The subscript m is used to denote the use of the mean period.

A riprap revetment is distinguished from an armor stone revetment in that riprap is more widely graded. Stone armor is usually specified with a very narrow range of sizes. A typical weight range is 0.75W_{50} to 1.25W_{50}, where W_{50} is the median weight of the stone. The nominal diameter, D_{n50}, is defined as

Equation 5:

in which ?is the weight density of the stone. This gives an r allowable diameter range of 0.91D_{n50} to 1.08D_{n50}. The corresponding range for riprap is 0.125W_{50} to 4.0W_{50} (0.50D _{n50} to 1.59D_{n50}). Riprap is less stable than armor stone because the smaller sizes are susceptible to removal and it is difficult to obtain uniform placement of the size distribution on the structure. Without special considerations, riprap revetments are not recommended for large wave conditions.

The design of riprap revetments in the United States follows recommendations provided by the U.S. Army Corps of Engineers (COE 2001, 1984, 1973). The required riprap size can be determined using the Hudson equation or the van der Meer equation. The Hudson equation is

Equation 6:

in which K_{D} is the empirical Hudson stability coefficient and ? is the immersed relative density defined as ? = ?_{r}/?_{w}-1, where ?_{r} is the specific weight of the rock and ?_{w}is the specific weight of the water (62.4 pounds per cubic foot (pcf) for fresh water; 64.0 pcf for sea water). The stability coefficient for angular riprap in breaking waves is K_{D} = 2.2 (COE 1984, 1973). The wave height is computed as the wave height at the toe of the structure. The 1973 and 1984 versions of the SPM are inconsistent in the use of the Hudson equation. The 1984 SPM recommends using a wave height that is 1.27H_{s}. If this wave height is used in the Hudson equation, the required stone weight doubles.

The van de Meer equation is

Equation 7:

in which N is the number of waves (maximum value of 7,500), P is the notational permeability, and S is the damage level. The two equations account for different breaking wave types based on the value of ?_{m}. For slopes of 1V:1.5H to 1V:3H, S = 2; and for slopes 1V:4H to 1V:6H, S = 3. These equations are for deep water, but can also be used in shallow water wave conditions (CUR, 1991). Guidance for selecting the notational permeability is given in CUR (1991) and COE (2001). Since riprap is well graded, the permeability is lower and a value of 0.2 is suggested in the absence of additional information.

The thickness of the riprap layer should be 2D_{n50}, but not less than 1 foot. Care must be taken when placing the riprap to ensure that the stone sizes are uniformly distributed over the full slope. End dump construction often results in the larger stones at the toe with smaller material on the upper slope.

An underlayer beneath the riprap provides pressure dissipation, drainage, and containment of the fines in the subgrade. Because a riprap revetment is widely graded, the underlayer size requirements are more restrictive than a stone armor revetment. The size of the underlayer is given by

Equation 8:

in which D_{85 underlayer }is the diameter at which 85 percent of the underlayer sizes are finer and D_{15cover} is the diameter at which 15 percent of the riprap stone sizes are finer. An approximate underlayer size is given by

Equation 9:

The underlayer beneath the riprap should have a thickness of 3D_{n50underlayer}, but not less than 1 foot. A typical riprap revetment cross section is shown in Figure 1.

Articulating concrete blocks (ACBs) are designed to provide stability and erosion control in a wide variety of hydraulic applications. Made on dry cast block machines, the individual units are engineered to capitalize on the weight of concrete, friction between units, and the interconnection of units into flexible mattresses. Flexibility between units is provided to allow the mat to conform to minor deformations in the subgrade. Classes of individual units can be produced at varying thicknesses, providing the designer flexibility in selecting appropriate levels of protec- tion. The range of block classes allows selection of the proper combination of unit weight, surface roughness, and open area for hydraulic stability.

For example, an ArmorFlex armor unit, shown in Figure 2, is substan- tially rectangular, having a flat bottom to distribute the weight evenly over the subgrade. The upper sides of the unit are sloped to permit articulation of the armor layer and to accommodate underlayer irregularities when the armor units are connected into mats. The units have two vertical openings providing for permeability of the armor layer. This reduces uplift forces on the armor by allowing release of dynamic pressures that occur during wave break- ing. The vertical cells also increase surface roughness and allow a flux of water into the underlayer, reducing waving runup.

Current industry standards utilize the Pilarczyk (1990) equation to select an appropriate thickness of articulating concrete block. The Pilarczyk equation was developed in the Netherlands based on analysis of numerous large-scale tests at the Delft Hydraulics Laboratory. Interestingly, full-scale testing on the ArmorFlex unit as depicted above was used in development of the design methodology. The Pilarczyk equation is as follows:

Equation 10:

in which D is the block thickness, ?_{u} is an empirical stability upgrading factor ( ?_{u} = 2.50 for cabled blocks on a granular sublayer), F is a stability factor for incipient motion ( F = 2.25 for blocks placed on a permeable core), and b is a coefficient related to the interaction process between waves and the revetment (b = 2/3 is acceptable for ArmorFlex open-block system). The Hudson equation has also been used to estimate block stability in waves. For this case, ? corresponds to r the specific weight of the block and an appropriate K_{D} value is used. However, the Pilarczyk equation considers additional variables associated with revetment stability in wave environments and is the recommended approach. A typical ACB revetment cross section is shown in Figure 3.

In addition to selecting the appropri- ate riprap or block size in waves, there are other important components of revetment systems to consider in the design. These include the underlayer, filter fabric, articulation, toe, flanks, and runup/overtopping.

**Underlayer —** A permeable under- layer is placed beneath the armor layer. This layer provides drainage to avoid build-up of excess hydraulic pressures beneath the armor, prevents migration of fines out of the bank, and provides a suitable surface for place- ment of the armor. Build-up of excess pressure beneath the revetment is one of the most important failure modes for revetments. Permeable revetments and underlayers allow dissipation of this pressure as water can flow out of the armor layer. Also, the subgrade must be geotechnically stable for the static and dynamics conditions associated with the design. This may require compaction or other improvements of the subgrade prior to placing filter and armor layers.

**Filter fabric —** If filter fabrics are used, ensure that the porosity and permeability requirements are satisfied. The fabric provides separation between the underlayer and the subgrade, preventing loss of fines but allowing the flow of water. In general, the equivalent opening size (EOS) of the fabric (EOS = 95 percent smaller opening size) should be EOS = D_{50 subgrade.} The fabric permeability should be at least 10 times the subgrade permeability. The fabric must have suitable strength capabilities in elongation, puncture, and shear. If the fabric will be exposed to sun light, it must also be UV stabilized. Additional information on using filter fabrics in wave environments is given in Pilarczyk (2000).

**Articulation —** Flexibility of the armor layer is a consideration with using articulating concrete block systems. The revetment system should allow for individual units to adjust to differential settling of the underlying material. Any settlement beneath a rigid revetment system may result in voids beneath the armor layer, causing points of weakness which will lead to failure. Toe protection — Toe protection may be necessary to prevent failure of the structure caused by scour and undermining. Common alternatives are: 1) place a scour blanket of hydraulically stable material, 2) place larger stones or blocks on the toe, 3) trench the toe beneath the depth of maximum anticipated scour, 4) use filter fabric to contain the armor at the toe (Dutch toe), or 5) use ground anchors or screws for restricting block motion.

**Flank protection — **The lateral ends of the revetment may be susceptible to damage. The flanks may be stabilized using techniques similar to the toe. If the revetment is placed on a shoreline experiencing chronic erosion, then it may be necessary to tie the revetment back into the slope using wing walls. This will reduce the tendency for the revetment to be flanked.

**Runup/overtopping —** If the revetment is intended to prevent backshore flooding caused by waves, then the height of the revetment must be sufficient to prevent wave overtopping. Guidelines for estimating wave runup and overtopping are given in the SPM and CEM. If wave overtopping is expected, but is allowable, then the berm of the revetment may require additional stabilization. The techniques used on the toe are applicable. Along the berm, it may also be possible to use biostabilization methods. An overtopping rate of 0.02 cubic feet per minute per foot is sufficient to cause structural damage to buildings behind the revetment.

Setting — A 200-foot reach of shoreline along the south bank on a 3-mile-long, 30-foot-deep lake is experiencing wave-induced erosion during large storms. The clay-silt bank to be protected has an average slope of 1V:3H slope (18.4 degrees). The backshore is a campground with limited infrastructure.

Design variables — Given that there is no critical development in the backshore, a 10-year design condition is selected. A gauging station in the lake indicates that the water depth at the toe of the bank is 4 feet during the 10-year design event. At this water level, the bank has 5 feet of freeboard. Analysis of historical wind measurements indicates that the 10-year, 2-minute wind speed from the north is 50 mph. Using hindcast methods given in the SPM, the wave height and period are estimated to be H_{s}= 4.4 feet, and T_{p}= 3.7 seconds. It is assumed that the storm lasts more than 40 minutes, allowing this wave condition to develop. The same storms that cause high waves also cause the high water levels, so the joint 10-year conditions correspond to the water level and wave conditions as determined.

There are no offshore islands, sand bars, vegetation, or other mechanisms that would modify the waves before they reach the shoreline, and the wave approach is normally incident to the shoreline. The breaking wave height at the shoreline can be estimated based on shoaling and the breaker index as

Using Equation 3, this wave height breaks in a water depth of approximately

Note that this depth is greater than the design depth of 4 feet. Therefore, the wave height at the toe is depth limited and, using Equation 3, is approximately

It will often be the case for revetments in shallow water that the wave heights are depth limited. In these cases, the wave height can simply be estimated using Equation 3 without the need for hindcast computations.

**Riprap revetment — **The necessary riprap size is determined using the Hudson and van der Meer equations. The Hudson equation gives

in which it is assumed that ?_{r}= 165 pcf and K_{D} = 2.2. This stone weight has a nominal diameter of D_{n50} = 12.0 inches. The riprap size range is 21 to 668 pounds (6 to 19 inches) and the thickness of the riprap layer is 2D_{n50} = 24.0 inches. The underlayer size using Equation 9 is D=_{n50 underlayer}= D_{n50 cover}/3.7=3.2 inches. The underlayer thickness is 3D_{n50 underlayer,} but not less than 1 foot. In this case, the underlayer thickness is 1 foot. A fabric placed beneath the underlayer requires suitable strength, UV, porosity, and permeability properties as discussed above.

For the van der Meer equation, it is necessary to determine the mean period, the number of waves, the surf similarity parameter, and then which equation to use. The mean period is

The number of waves is the storm duration divided by the mean wave period or 7,500, whichever is less. The maximum number of waves times the mean period gives 7.0 hours. If the storm duration is longer than 7.0 hours, then N = 7,500. If the storm duration is less than 7.0 hours, a reduced value for N is used. For this example, it is assumed that storm conditions last for longer than 7.0 hours, so N = 7,500.

Next, the surf similarity parameter and critical value for the surf similarity parameter are determined.

The corresponding weight is W_{50} = 184 pounds. The riprap layer thickness and underlayer properties are similar to and follow the same steps as the Hudson equation results.

**ACB revetment — **The Pilarczyk equation is used to determine the required ACB thickness. The immersed relative density is

where the specific weight of the block has been taken as 140 pcf. The Pilarczyk equation is

in which coefficients appropriate for ArmorFlex have been used. For this case, ArmorFlex Class 60 block with a thickness of 7.5 inches would be selected. The individual blocks would be cabled together into appropriate mat sizes using either polyester, galvanized, or stainless steel cable depending on lake water salinity and project-specific logistics.

Prior to placement of the ACB cabled mat system, a site-specific filter fabric compatible with the subgrade soils would be placed on the graded formation, utilizing a minimum 1-foot overlap of successive rolls. Determination of the filter fabric follows design criteria mentioned above. A 4-to 6inch-thick, coarse, uniformly graded, granular material (#57 crushed granite typical) is then placed as a filter (bedding) layer over the site-dependent filter fabric. Blinding the ACB system with a gravel material can enhance the stability of the system by increasing the inter-block friction and providing a means of load transfer over a greater area of the revetment. Also, a site-specific soil dressing, followed by native grasses and plants with shallow root systems could be planted within the open cells of the blocks on the upper portions of the slope to provide vegetation and associated habitat. Note that stability equations have been developed for wind waves and not boat wakes. A rule of thumb is to multiply the wake height by 1.5 to estimate an equivalent H_{s}to use in the stability equations (CUR, 1991). Also, the number of boat wake waves is difficult to estimate. One boat passage only generates a few waves, but there are many boats. Hence, unless additional information is available, it is recommended to use a maximum value of N = 7,500 in boat wake designs.

**Christopher I. Thornton, Ph.D., P.E**., is director of the Hydraulics Laboratory and Engineering Research Center at Colorado State University.

**Richard Kane**, erosion control product manager, CONTECH Construction Products, Inc., has 10 years of experience in the geotechnical, environmental, and civil engineering industries.

- COE, 1973, Shore Protection Manual, U.S. Army Corps of Engineers, Fort Belvoir, Va.
- COE, 1984, Shore Protection Manual, U.S. Army Corps of Engineers, Vicksburg, Miss.
- COE, 2001, Coastal Engineering Manual, Part VI — Design of Coastal Project Elements, U.S. Army Corps of Engineers, Vicksburg, Miss.
- CUR, 1991, Manual on the Use of Rock in Coastal and Shoreline Engineering, AA Balkema Publishers, Brookfield, Vt., 607 pages.
- Pilarczyk, K.W., ed., 1990, Design of Seawalls and Dikes — Including Overview of Revetments, Chapter 7, Coastal Protection, Balkema, The Netherlands.
- Pilarczyk, K.W., ed., 2000, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering, Balkema, The Netherlands.