Manual on wave overtopping of sea defences and related structures

This Overtopping Manual gives guidance on analysis andor prediction of wave overtopping for flood defences attacked by wave action. It is primarily, but not exclusively, intended to assist government, agencies, businesses and specialist advisors consultants concerned with reducing flood risk. Methods and guidance described in the manual may also be helpful to designers or operators of breakwaters, reclamations, or inland lakes or reservoirs

Background

Previous and related manuals

The first edition of EurOtop (2007) was widely praised within the coastal engineering community and has become the industry standard This influential manual was developed, at least in part, by consolidating existing industry resources, including the UK Environment Agency Manual on Overtopping edited by Besley (1999).

(Netherlands) TAW Technical Report on Wave run-up and wave overtopping at dikes, edited by Van der

The EurOtop (2007) manual was developed to revise, expand, and improve upon earlier safety guidelines outlined in Meer (TAW, 2002) and the German publication Die Küste (EAK, 2002) edited by Erchinger, specifically focusing on critical coastal phenomena such as wave run-up and overtopping.

Since EurOtop (2007), significant advancements have been made in understanding wave overtopping, including new techniques for steep slopes up to vertical, improved formulae for zero relative freeboard, and enhanced insights into wave behavior over vertical structures such as foreshores and storm walls Systematic videos now visually demonstrate how specific overtopping discharge appears in real-life scenarios, available on the website This manual incorporates these recent developments, updating and expanding guidance on wave overtopping predictions, and revises previous recommendations from the Rock Manual (2007) and the Revetment standards.

Manual by McConnell (1998), British Standard BS6349, the US Coastal Engineering Manual (2006), and

Sources of material and contributing projects

Recent advancements in hydrodynamic modeling have been achieved through new methods and data obtained from various European and national research programs, complementing the earlier manuals Notably, the EurOtop (2007) manual was significantly contributed to by projects such as OPTICREST, PROVERBS, CLASH, VOWS, and Big-VOWS, which have enriched the understanding and development of coastline protection techniques These contributions ensure that the latest research findings are integrated into best practices for coastal engineering and erosion management.

ComCoast's second version incorporates new insights gained from extensive testing with wave run-up and overtopping simulators in the Netherlands, alongside collaborative efforts between researchers Notable collaborations include Bruce and Van der Meer developing new wave overtopping formulae, and Zanuttigh and Van der Meer expanding the CLASH database to enhance artificial neural networks for predicting wave overtopping, transmission, and reflection The project also acknowledges Infram in the Netherlands for providing systematic videos of wave overtopping discharges, which are accessible on their website All information in this manual is backed by relevant research papers and authoritative manuals listed in the bibliography, ensuring accuracy and credibility.

Use of this manual

This manual is designed to assist engineers in analyzing the overtopping performance of various sea defenses and shoreline structures worldwide It leverages research from Europe and beyond to predict wave overtopping discharges, the frequency of overtopping waves, and the distribution of overtopping wave volumes Suitable for both current performance assessments and long-term design calculations, the methods outlined can be applied to evaluating existing defenses or planning rehabilitation and new construction projects.

The analysis methods described in this manual are primarily based upon a deterministic approach in which overtopping discharges (or other responses) are calculated for wave and water level conditions

Design equations for coastal defenses rely on accurate data of water levels and wave conditions at the structure's toe, where the water level input must include tidal and surge components such as wind set-up and barometric pressure effects It is essential to consider nearshore wave transformations, including shoaling and breaking, when determining wave conditions Methods for calculating depth-limited wave conditions are detailed in Chapter 2, ensuring comprehensive assessment of the design parameters for effective coastal defense planning.

All prediction methods discussed in this report have inherent limitations in their accuracy due to factors such as inherent scatter in empirical equations derived from physical data While some equations incorporate safety margins and are presented for design and assessment purposes, their reliability should be treated with caution Specifically, overtopping rate predictions based on empirical equations can only be considered accurate within a factor of 1 to 3, meaning the actual rate could be three times smaller or larger than the predicted mean The largest discrepancies are typically observed with small overtopping discharges, and confidence intervals, often depicted as a 90% confidence band on graphs, help illustrate the uncertainty in these predictions.

Many practical hydraulic structures often deviate from idealized models tested in laboratories, making overtopping rates highly sensitive to small variations in structure geometry, local bathymetry, and wave climate Empirical methods based on generic structural tests, such as vertical walls and armored slopes, can result in significant differences in overtopping predictions due to these variations While the methods discussed may not match the accuracy of structure-specific model tests, Artificial Neural Network tools can provide reasonably accurate overtopping estimates for very specific structures, comparable to traditional formulae.

This manual provides a focused overview of sea defenses, complementing existing detailed resources such as the Rock Manual (2007), British Standards BSI (2000), and guidelines from Simm et al (1996), Brampton et al (2002), and Dutch TAW or ENW standards It is intentionally concise to ensure clarity and ease of understanding, guiding professionals in relevant aspects of sea defense analysis, design, construction, and management For comprehensive technical details and the rationale behind recommended methods, a complete list of references is provided for further study.

Principal types of structures

Wave overtopping is a major concern for flood defense structures, commonly known as sea walls or sea defenses, designed to protect coastal areas from flooding Similar structures also serve to prevent coastal erosion, often referred to as coast protection measures Additionally, structures like breakwaters and moles are built to safeguard ports, harbors, and marinas, ensuring safe navigation and mooring conditions While some sea defense structures are located offshore or detached from the shoreline, most are integrated parts of the shoreline infrastructure, providing essential protection against wave action and erosion.

This manual focuses on the three main types of sea defense structures: sloping sea dikes and embankment seawalls, armored rubble slopes and mounds, and vertical or steep battered walls These structures are essential for protecting coastlines from erosion and wave action, offering a range of solutions tailored to different shoreline conditions Understanding the design and construction of each type is crucial for effective coastal protection and long-term durability.

Historically, sloping dikes have been the most widely used sea defense along coasts in the Netherlands, Denmark, Germany, and the UK, providing effective flood protection and added amenity value These embankments, built from materials such as clay or vegetated shingle ridges, often feature steeper side slopes, especially in the UK To guard against wave erosion, seawalls typically incorporate revetment facings on the seaward side, which can include concrete blockwork, cast-in-place concrete slabs, or asphaltic materials Dike structures are most commonly found along rural coastlines, offering reliable protection against flooding while blending into the natural landscape.

A second type of coastal structure involves a mound or layered quarried rock fill protected by rock or concrete armor units, designed to withstand wave action with minimal displacement Underlying layers of quarry or crushed rock support the armor and help separate it from finer materials, dissipating wave energy through breaking and friction Simplified rubble mound designs are often used for seawalls, vertical wall protection, or revetments, particularly to safeguard embankments built from relic sand dunes or shingle ridges These rubble mound structures are more commonly employed in regions where hard rock is readily available, offering effective coastal protection.

Urban coastal frontages near ports often utilize vertical erosion and flooding defense structures, such as stone, concrete block, mass concrete, or sheet steel pile seawalls, which may also function as retaining walls Caissons are commonly used as breakwaters to protect harbor areas, while shaped or recurved wave return walls can serve as standalone structures or be integrated into sloping defenses Some seawalls, constructed from blockwork or mass concrete with vertical or steeply sloping faces, are highly impermeable to wave action but are susceptible to intense local wave pressures, overtopping, and reflection of wave energy These reflected waves can increase local disturbance and accelerate bed scour, highlighting the need for careful design of coastal defense structures.

Definitions of key parameters and principal responses

Wave height

The wave height used in the wave run-up and overtopping formulae is the incident significant wave height

Spectral wave height, Hm0, is measured at the toe of the structure and is calculated as four times the square root of the zero-moment of the wave spectrum (Hm0 = 4 * √m0) An alternative definition of significant wave height is H1/3, which represents the average height of the highest third of the waves; however, this measurement is not typically used in this manual unless specific formulae are derived based on it In deep water conditions, both spectral wave height and significant wave height provide essential insights into wave behavior and structural impact assessment.

Although four different definitions produce similar values, shallow water conditions can cause variations of 10-15% Currently, there is limited testing on overtopping where significant differences in wave heights are observed using these definitions The selection of Hm0 as the preferred measure is primarily because design wave heights are frequently estimated through numerical models, which typically predict this particular wave height.

The significant wave height Hs is often used for Hm0 as well as H1/3 In this manual Hm0 has consequently been used

A foreshore is often present, providing a zone where waves shoal, break, and reduce their significant wave height Simplified models can predict the energy reduction caused by wave breaking, enabling estimation of wave height at the structure's toe Accurate wave height calculation should encompass the entire wave spectrum, including long-wave energy Utilizing the spectral significant wave height, the wave height distribution and the significant wave height H1/3 can be effectively determined using the Battjes and Groenendijk (2000) method.

Recent research highlights the critical role of long waves generated by wave breaking in predicting wave overtopping, particularly on steep foreshore slopes steeper than 1:50 When water depths at the structure decrease to a few decimetres, the short wave spectrum may diminish entirely, giving way to a long wave spectrum with peak periods of one minute or more These complex conditions are not yet fully understood through numerical modeling or wave flume experiments, emphasizing the need to supplement manual guidance—deseribed for very shallow water with long waves—with physical model tests for accurate prediction.

Wave period

Various wave periods, such as the peak period (Tp), average period (Tm), and significant period (T1/3), are used to analyze wave spectra and records The peak period (Tp) corresponds to the highest point in the spectrum, while the average period (Tm) is calculated from spectral data or wave records and typically falls within a ratio of 1.1 to 1.25 times Tp The significant period (T1/3), representing the average of the highest one-third of waves, is nearly identical to Tp, providing valuable insights into wave energy and behavior for engineering and oceanographic applications.

The wave period used for some wave run-up and overtopping formulae is the spectral period

Tm-1,0 = (m-1) / m0 is a spectral period that emphasizes longer wave periods compared to the average, providing more accurate assessments of wave behavior This parameter remains consistent across different spectrum types, resulting in similar wave run-up and overtopping for identical Tm-1,0 values and wave heights Utilizing Tm-1,0 allows for straightforward determination of wave run-up and overtopping even in bimodal and flattened spectra, eliminating the need for complex calculations.

In cases of a uniform (single-peaked) spectrum, there is a consistent relationship between the spectral period Tm-1,0 and the peak period When the peak period is known or has been determined but the spectral period has not, a conversion factor of Tp = 1.1 Tm-1,0 can be used to accurately estimate the spectral period This relationship facilitates effective seismic analysis and site response assessments by linking spectral and peak periods through a simple conversion.

In very shallow foreshores with extensive wave breaking, the wave period Tm-1,0 may be predominantly influenced by long waves, resulting in significantly longer wave periods—sometimes lasting minutes or more—compared to areas with minimal or no wave breaking.

Wave steepness and breaker parameter

Wave steepness, defined as the ratio of wave height to wavelength (s0 = Hm0 / L0), provides valuable insights into a wave’s history and characteristics A typical swell sea is characterized by a steepness of around 0.01, while wind seas tend to have steeper waves with values between 0.04 and 0.06 Additionally, swell seas are often associated with long-period waves, where the wave period becomes the primary factor influencing overtopping. -**Sponsor**Looking to optimize your article with SEO-friendly, coherent paragraphs? Let [Soku AI](https://pollinations.ai/redirect-nexad/mi1Hez1F?user_id=983577) rewrite it for you! Soku AI, the first fully autonomous Facebook/Meta advertising AI agent, uses expert-level automation to refine content, ensuring it resonates with your target audience while adhering to SEO best practices, just like understanding wave steepness (the ratio of wave height to wavelength) helps define a wave's characteristics Generally, a steepness of s0 = 0.01 indicates a typical swell sea, while s0 = 0.04 to 0.06 suggests a typical wind sea, linking wave characteristics to SEO keywords With Soku AI, transform your article into a high-performing asset, just as wave steepness transforms understanding wave dynamics.

But also wind seas may become seas with low wave steepness if the waves break on a gentle foreshore

Wave breaking initially causes a decrease in wave height while the wave period remains relatively unchanged, resulting in lower wave steepness This reduced steepness on deep water typically indicates the presence of swell waves, whereas in shallower, depth-limited areas, it often leads to broken waves on gentle beaches Understanding these wave dynamics is essential for predicting surf conditions and ensuring safety at coastal locations.

The breaker parameter, also known as the surf similarity or Iribarren number, is defined as ξm-1,0 = tan(α) / (Hm0 / Lm-1,0), where α represents the slope of the front face of the structure The term Lm-1,0 is the deep water wavelength, calculated as gT² / (2π), with g being gravitational acceleration, T the wave period, and π the constant pi This parameter is essential for assessing the breaking behavior of waves approaching coastal structures.

The analysis uses the deep water wavelength based on the wave period at the structure's toe, rather than the actual wavelength near the toe This approach results in a notional wave steepness, which is utilized to calculate a dimensionless wave period Consequently, the calculated wave steepness serves as an approximate value for assessing wave behavior, ensuring more accurate modeling of wave-structure interactions in deep water conditions.

The combination of structure slope and wave steepness gives a certain type of wave breaking, see

In wave dynamics, the classification of breaking waves depends on their wave steepness parameter, denoted as ξ_m-1,0 When ξ_m-1,0 exceeds approximately 2, waves are considered non-breaking surging waves, though some may still exhibit partial breaking Conversely, when ξ_m-1,0 is less than about 2, waves are classified as breaking The transition from plunging to surging waves on slopes occurs near ξ_m-1,0 = 1.8, closely aligned with the value of 2 This distinction is crucial for understanding wave behavior on different foreshore types; gentle beaches often produce spilling waves with multiple breaker lines, while plunging waves feature steep, overhanging fronts capable of striking structures or causing backwash, as illustrated in relevant figures The transition known as collapsing occurs when wave fronts become almost vertical, resulting in larger water movements on the slope during run-up and run-down Generally, larger waves in a sea state are characterized by specific values, but individual waves can vary, occasionally surging under typical plunging conditions or plunging under usually surging conditions.

Figure 1.1: Type of breaking on a slope

Figure 1.2: Spilling waves on a beach; m-1,0 < 0.2

Parameter h * , d * and EurOtop (2007)

EurOtop (2007) initially used two combination parameters, which have been updated in this manual to focus on explicit parameters such as water depth, wave height, wave length, and wave steepness These are represented by the h* and d* parameters, which help distinguish between non-impulsive and impulsive wave structures The h* parameter is used to characterize non-impulsive waves on a vertical structure, while the d* parameter is applied to vertical walls with berms or toe mounds in front, providing clearer insights into wave behavior and structure interactions.

Wave parameters such as wave height and wavelength are described relative to the local water depth (h) at the structure's toe or above the berm or mound (d) Non-impulsive waves typically occur when the ratio (h* or d*) exceeds 0.3, while impulsive waves dominate at ratios of 0.3 or less Although initial formulas in EurOtop (2007) used these ratios raised to certain powers for predicting impulsive overtopping, they are no longer employed in current models However, these parameter groups—specifically h²/(Hm0 Lm-1,0) and hδ/(Hm0 Lm-1,0)—are still crucial for distinguishing between impulsive and non-impulsive wave conditions in modern wave overtopping predictions.

Toe of structure

The toe of a structure is typically identified at the point where the foreshore meets the front slope or the toe structure itself; for vertical walls, it is usually at the base of the main wall or the rubble mound toe Seasonal changes and severe wave conditions can cause the sandy foreshore and toe levels to vary, especially during storms and high tide cycles, leading to maximum erosion at these points It is important to account for increased wave heights as the toe depth increases, since wave overtopping calculations rely on the incident wave height at the toe If the structure's toe is above the still water level, defining a wave height becomes challenging, which impacts the assessment of overtopping risks, as illustrated in Section 7.3.2 with a vertical structure example.

Foreshore

The foreshore is the area in front of a breakwater, coastal structure, or seawall, and can be horizontal or gently sloped up to a maximum of 1:10 Its depth can vary from deep to very shallow, requiring consideration of shoaling and depth limiting effects in shallow zones to accurately assess wave height and wave period at the toe of the foreshore Additionally, a foreshore is defined as having a minimum length of one wavelength (Lm-1,0), ensuring adequate space for wave dynamics.

Transitioning from a shallow to a very shallow foreshore involves complex wave spectrum changes, making a precise delineation challenging At shallow foreshores, waves break and decrease in height while largely retaining their original spectral shape, whereas at very shallow foreshores, the spectrum changes drastically, often becoming flat with no distinct peaks As waves continue to break and diminish, a variety of wave periods emerge, including long waves with periods of one minute or more caused by breaking The impact of these conditions on wave overtopping remains poorly understood, especially in steep foreshore slopes with minimal water depths at the structure's base, highlighting the need for further research and guidance.

The transition from shallow to very shallow foreshores occurs where the original incident wave height, reduced by at least 50% due to breaking, marks this change At structures situated on very shallow foreshores, wave heights are significantly smaller compared to deep water conditions, leading to reduced wave steepness As a result, the breaker parameter—used in calculating wave run-up and overtopping—increases substantially in these zones.

Breaker parameters typically range from approximately 4 to 10, with maximum values corresponding to gentle slopes of 1:3 or 1:4 These higher values are usually smaller than those around 2 or 3, indicating a more gradual slope When wave steepness (sm-1,0) is below 0.01, it suggests the presence of a shallow or very shallow foreshore, providing key insights into coastal bathymetry.

In Chapter 7 on vertical structures, a distinction is made between structures without an influencing foreshore and those with a sloping foreshore, which requires further clarification since almost all coastal structures interact with the foreshore A vertical seawall at the end of a sloping foreshore can help limit wave energy, while vertical walls without an influencing foreshore are typically characterized by a nearly horizontal foreshore and deeper water relative to wave height In physical models, the foreshore is often represented by the bottom of the wave flume or basin, illustrating different interaction scenarios between coastal structures and the shoreline.

This article presents three examples of vertical walls that do not influence the foreshore First, storm flood walls in harbors are designed where wave heights are relatively small compared to water depth, minimizing their impact during storm events Second, caisson breakwaters founded on berms often feature berms located well below water level, which are too small to significantly affect incoming waves Third, structures such as lock gates during high water levels act as vertical walls with minimal wave impact, as wave heights are negligible relative to the deep water conditions.

Slope

A structure's profile includes slopes that are defined as those with a gradient between 1:1 and 1:8, which applies to both individual sections and the average slope The average slope is determined by drawing a line between -1.5 Hm0 and +Ru2% relative to the still water line, excluding berms, with Ru2% representing the run-up level exceeded by only 2% of incident waves While calculating continuous slopes between 1:8 and 1:10 can be initially estimated using simple slope formulas, these calculations are less reliable for gentler slopes Additionally, interpolation between a slope of 1:8 and a berm of 1:15 isn't feasible because berms serve as transitional gentle slopes rather than continuous slopes.

A structure with a slope steeper than 1:1 but not vertical is classified as a battered wall and is addressed as a complete structure in Chapter 7 In contrast, if the structure is merely a wave wall placed on a gently sloping dike, it is covered under the guidelines in Chapter 5.

Berm and promenade

A berm is a component of a structure profile where the slope transitions between horizontal and a 1:15 incline Its position relative to the still water line is defined by the depth, db, which is the vertical distance from the berm's center to the still water line The width of a berm, B, should not exceed one-quarter of the wavelength (B < 0.25 Lm-1); exceeding this limit means the structure is treated as a combined berm and foreshore, allowing wave run-up and overtopping calculations via interpolation For more detailed information, refer to Section 5.4.6.

A berm is typically positioned on a sloping structure such as a dike or levee, near the designated water level to maximize its effectiveness It helps create a gentler "equivalent slope," which can result in a lower crest level compared to similar structures without a berm Placement close to the design water level ensures optimal performance and enhanced safety of the containment structure.

Horizontal slopes are commonly found at promenades, such as along Belgium's North Sea coast, often positioned at a higher level than the berm in sloping structures The promenade can serve as the crest level, but if a storm wall is installed on top of it, the storm wall's crest becomes the critical water defence element In such cases, the promenade becomes an integral part of the coastal protection structure, characterized by its width, Gc Section 5.4.7 provides examples illustrating promenades equipped with and without storm walls, highlighting their role in coastal defence.

Crest freeboard, armour freeboard and width

The crest height of a structure, relative to water level, is known as the crest freeboard (Rc), which marks the point beyond which overtopping water cannot flow back to the sea In rubble mound structures, the crest freeboard is typically measured at the top of a crest element rather than the overall height of the rubble mound armor This critical measurement ensures the structure's effectiveness in preventing overtopping and protecting coastal areas Proper understanding of crest freeboard is essential for designing durable and resilient coastal defenses.

The armour freeboard (Ac) is the height of the horizontal part of the crest measured relative to SWL, with the horizontal crest part called Gc For rubble mound slopes, Ac may be higher, equal, or sometimes lower than the crest freeboard (Rc) When calculating wave overtopping, it is advisable to use the maximum of Rc and Ac, although doing so might slightly underestimate overtopping if Ac exceeds Rc, since some water can pass through the upper part of the rock and contribute to overtopping However, using the maximum of the two provides a more accurate assessment than relying on the smaller Rc, which could lead to an overestimation of wave overtopping.

Figure 1.4: Crest freeboard different from armour freeboard Rc can also be equal or larger than Ac

Estimating the effect of a permeable crest on wave overtopping is complex due to the permeability of the quarry stone armour layer The crest height for wave overtopping calculations should be based on the average of Rc and Ac for slopes without a wave wall, as using only the highest or lowest levels can lead to inaccurate results When a wave wall is present, the maximum of Rc and Ac should be used to determine the crest height, ensuring more accurate predictions of wave overtopping.

Figure 1.5: Crest with a permeable layer and no crest element present: take the average of Rc and Ac

The crest of a smooth dike or embankment without a wave wall is generally considered to be horizontal and of limited width, which means the crest width does not influence overtopping discharge However, in reality, the crest is often slightly rounded and of a certain width, a factor not typically accounted for in models of smooth impermeable crests The crest height (Rc) is defined as the height of the seaward crest line at the transition from the seaward slope to the crest, a key parameter for wave run-up and overtopping calculations In theory, neither the width nor the middle height of the crest significantly impact wave overtopping estimates, assuming no wave walls (Rc = Ac) and no crest width effect (Gc = 0) Nevertheless, if the crest is very wide, it can influence the actual wave overtopping, so the current approach is somewhat conservative to ensure safety.

The height of the wave wall located at the rear of an impermeable slope or vertical wall with a horizontal crest directly influences the value of Rc, while the height of the horizontal section determines Ac, as illustrated in Figure 1.6.

For promenades as well as crests at vertical or sloping structures, the horizontal part is given by Gc

Figure 1.6: Crest configuration for a vertical wall

Bullnose or wave return wall

Waves hitting vertical walls can cause upward rushing water that may overtop the crest and subsequently fall back into the water, leading to potential erosion and safety concerns To mitigate this overtopping, structures such as bullnose parapets or wave return walls are often positioned at the top of the vertical wall These structures are designed to deflect or return the up-rushing waves seawards, thereby reducing overtopping levels Although there are no specific guidelines for the geometric design of these structures, their size significantly influences their effectiveness in minimizing wave overtopping Proper design and sizing are essential for enhancing the durability and safety of coastal defenses.

A bullnose is a relatively small structure compared to the height of the vertical wall and the intensity of governing waves For example, Figure 1.7 illustrates a bullnose at a high crest wall on a caisson, depicted during its under-construction phase for the deck and crest wall In this scenario, the absence of impulsive waves means that the upward rushing water along the vertical wall reaching the bullnose is minimal and can be easily diverted seawards, enhancing structural safety and stability.

A bullnose may have significant effect on wave overtopping if it is situated fairly high above the water level

If not, a large overtopping wave will easily overtop and will not “feel” the small structure This manual gives guidance for this type of relatively small bullnose in Section 7.3.6

Figure 1.7: A relatively small bull nose on the crest wall of a large caisson The caisson under construction, Aỗu, Brasil, is 25 m wide and the crest level is 10 m above sea level

Figure 1.8: Effective fairly significant bullnose at Cascais, Portugal Waves are breaking on the foreshore and give impulsive wave conditions There was no wind Courtesy L Franco

Figure 1.8 illustrates a significant bullnose that causes impulsive swell waves to leap high into the air before returning seawards, highlighting its substantial impact on wave behavior In contrast, a smaller bullnose, such as the one shown in Figure 1.7, produces a much less pronounced effect For designing larger bullnoses, Section 7.3.6 provides essential guidance grounded in foundational research to optimize performance.

A high wave return wall can be designed to limit wave overtopping and maintain a minimal crest level, moving beyond traditional bullnose designs For example, a vertical seawall protecting a city center from flooding was enhanced with a large, nearly horizontal wave return wall integrated into a promenade, reducing the overall structure height Model testing demonstrated that even at elevated water levels, most waves could not overtop this multi-functional seawall design, effectively combining flood protection with aesthetic and recreational purposes.

Although the manual does not provide specific guidance on overtopping for large wave return walls, the Artificial Neural Network described in Section 4.5 offers accurate predictions, as it was trained on similar structures It is important to note that wave return walls experience increased wave forces, which should be considered in their design and analysis.

The wave return wall at Harlingen (NL) is a large and effective structure designed to protect the coastline from strong waves It forms an integral part of the promenade built atop the wall, combining functionality with aesthetic appeal The structure's effectiveness has been verified through model testing under simulated wave conditions, ensuring its durability and performance in real-world scenarios This innovative wave management system exemplifies advanced coastal protection measures, highlighting its role in enhancing shoreline resilience and safety.

A bullnose may also be applied at a storm wall on a promenade and reduces wave overtopping significantly Guidance on these kind of structures is given in Section 5.4.7.

Permeability, porosity and roughness

A smooth, impervious structure like a dike or embankment typically features a slope with minimal or no roughness, such as those covered with block revetments, asphalt, concrete, or grass on clay Incorporating roughness—through irregular block revetments or artificial ribs—helps dissipate wave energy during wave run-up, significantly reducing wave overtopping These design features enhance the durability and effectiveness of coastal defenses by managing wave impact efficiently.

A rubble mound slope with rock or concrete armour is generally rougher than impermeable dikes or embankments, which incorporate artificial roughness elements However, a key difference lies in their permeability and porosity, as rubble mound structures typically exhibit much higher porosity Porosity, defined as the percentage of voids between units or particles, is inherent in loose materials, which usually have some degree of porosity For rock and concrete armour, porosity generally ranges between 30% and 55%, while sand also shares comparable porosity levels Despite similar porosity, the wave behavior on a sand beach differs significantly from that on a rubble mound slope due to their distinct material properties.

The key difference in rubble mound breakwaters lies in permeability; their highly permeable armor slopes allow waves to penetrate between armor units and dissipate energy effectively However, penetration becomes more challenging for the under layers and the core of the structure This distinction is based on whether the under layers or core are impermeable or permeable, with both configurations sharing the same armor layer but differing in structural composition.

A rubble mound breakwater is typically constructed with an underlying layer of large rocks, which constitute approximately one-tenth of the armor's weight, providing stability and support Sometimes, a second layer of smaller rocks is added beneath the large rocks to enhance durability and reduce settlement The core of the breakwater is made up of even smaller rocks, ensuring a stable foundation and facilitating effective wave dissipation This layered design guarantees the structure’s robustness and longevity in marine environments.

Up-rushing waves can penetrate into the armour layer and will then sink into the under layers and core This is a structure with a “permeable core”

An embankment can be reinforced with an armor layer of rock, providing effective wave protection Beneath the armor layer, a thin, small-sized inner layer is typically installed on a geotextile fabric that offers additional stability Underneath the geotextile, impermeable materials such as sand or clay may be present to prevent up-rushing waves from penetrating the core An embankment with a rock armor layer features an "impermeable core," which plays a crucial role in controlling wave run-up and overtopping, as these processes depend significantly on the core’s permeability Proper design of the impermeable core enhances the embankment's overall durability and effectiveness in coastal protection.

In summary, the following types of structures can be described:

Smooth dikes and embankments: smooth and impermeable

Dikes and embankments with rough slopes: some roughness and mostly impermeable Rock cover on an embankment: rough with impermeable core

Rubble mound breakwater: rough with permeable core

Wave run-up height

The wave run-up height, indicated by Ru2%, refers to the vertical measurement above the still water line that is exceeded by only 2% of incident waves This metric directly relates the frequency of waves surpassing this level to the total number of incoming waves, rather than the number of waves that actually run up the slope Understanding Ru2% is essential for accurately assessing coastal design and erosion risks, making it a crucial parameter in marine and shoreline engineering.

Measuring extremely thin water layers on run-up tongues is challenging, as layers thinner than 2 mm in models correspond to about 2 cm in real-world waves, depending on scale Strong winds can blow these thin water layers far up smooth slopes, a phenomenon that small-scale models cannot accurately simulate Water tongues less than 2 cm thick contain minimal water and should not be considered true wave run-up Therefore, the wave run-up level on smooth slopes is best identified when the water layer diminishes below 2 cm, excluding thin layers blown onto the slope from the definition of wave run-up.

Run-up is a critical factor for ensuring stability on smooth slopes and embankments, as well as on rough slopes armored with rock or concrete While wave run-up is essential for these structures, there is no equivalent parameter for vertical structures Instead, the percentage or frequency of overtopping waves serves as the key indicator of their performance against wave impacts.

Wave overtopping discharge

Wave overtopping refers to the average volume of water passing per meter width, expressed as discharge rate per meter, such as m³/s per meter or liters per second per meter The methods outlined in this manual primarily calculate overtopping discharges in cubic meters per second per meter, unless specified otherwise For convenience, these results are often multiplied by 1000 to express the discharge in liters per second per meter, facilitating easier interpretation and comparison.

Wave overtopping is a highly unpredictable process, with no constant discharge over the crest of a structure The occurrence, timing, and volume of overtopping are random, with high waves causing significant water volumes to spill over quickly, often within less than a wave period, while smaller waves may not cause any overtopping at all Measurements, such as those shown in Figure 1.10 over a 30-second period, illustrate this irregularity, with the lower graph highlighting the fluctuating flow depths and the upper graph displaying the cumulative overtopping volume recorded by a load cell in the tank Multiple waves contribute to overtopping, with at least nine waves responsible for approximately 15 liters of total overtopping volume; individual wave contributions are difficult to distinguish due to their grouping Accurate calculation of average overtopping discharge requires considering measurement duration and the width of the chute channeling water into the tank.

Figure 1.10: Example of wave overtopping measurements, showing the random behaviour

C u m u lat iv e o v erto pp in g (l)

A mean overtopping discharge is widely used as it can easily be measured and also classified:

 q < 0.1 l/s per m: Insignificant with respect to strength of crest and rear of a structure

 q = 1 l/s per m: On crest and landward slopes bad grass covers or clay may start to erode It will not give erosion to rubble mound structures

 q = 10 l/s per m: Significant overtopping for dikes, embankments For large wave heights it may lead to severe erosion on the harbour side of rubble mound breakwaters

For effective protection, the crest and inner slopes of dikes must be reinforced with asphalt or concrete When constructing rubble mound breakwaters, wave transmission can occur, necessitating an armor layer that covers the crest and landward slope to ensure durability and stability.

The severity of overtopping is determined not only by the average overtopping discharge but also by the wave height responsible for the overtopping Larger wave heights result in more severe overtopping, even when the overtopping discharge remains the same For detailed guidelines on allowable wave overtopping, please refer to Chapter 3.

Wave overtopping volumes

Overtopping discharge alone does not specify the number of waves or the volume of water overtopped per wave The overtopping wave volume (V) is measured in cubic meters per wave per meter of structure width While most overtopping waves are relatively small, a few waves can result in significantly larger water volumes overtopping the structure.

The maximum volume of overtopped water in a sea state is influenced by the mean discharge (q), storm duration, and the percentage of overtopping waves A longer storm duration results in more overtopping waves and a statistically higher maximum overtopping volume While numerous small overtopping waves can produce the same mean discharge as fewer large waves in calmer conditions, rough sea conditions with large waves lead to significantly greater maximum overtopping volumes This manual provides a method to calculate the distribution of overtopping wave volumes based on specific wave conditions and average overtopping discharge, aiding in accurate risk assessment and infrastructure design.

Description and use of reliability in this manual

Definitions

Uncertainty in coastal engineering refers to the relative variation or error in model parameters, indicating that instead of a single value, parameters are best represented as a range of possible outcomes Given the inherently random nature of many variables in coastal engineering, parameters should be modeled stochastically rather than deterministically, acknowledging their multiple realizations within a range This approach treats uncertainty as a statistical distribution of the parameter, often assuming a normal distribution for simplicity When a normal distribution is assumed, uncertainty can be expressed as a relative error, specifically through the coefficient of variation (σ’(x)), which quantifies the variability of a parameter’s possible values.

The standard deviation, denoted as σ₁.₂, measures the variability of a parameter around its mean value, μ(x) While this definition may be considered imperfect, it offers practical utility and is straightforward to apply in real-world scenarios Understanding this relationship helps in assessing the dispersion and reliability of data, making it a valuable concept in statistical analysis and data interpretation.

A normal distribution is often assumed for parameter x, but in cases where x cannot be negative—such as thickness—the distribution may need to be modeled as a log-normal distribution For example, with a mean μ(x) = 2.0 and a standard deviation σ(x) = 0.75, the normal distribution can sometimes produce negative values, which are physically impossible In such scenarios, the log-normal distribution provides a more accurate representation by being slightly skewed and always positive When the mean is significantly distant from zero and the standard deviation is relatively small, the differences between normal and log-normal distributions become minimal, reducing concerns about skewness.

Figure 1.11: Normal and associated log-normal distribution For the normal distribution μ(x) = 2.0 and σ(x) = 0.75.

Background on uncertainties

Many parameters used in engineering models are uncertain, and so are the models themselves The uncertainties of input parameters and models generally fall into certain categories; as summarised in

 Fundamental or statistical uncertainties: elemental, inherent uncertainties, which are conditioned by random processes of nature and which cannot be diminished (always comprised in measured data)

 Data uncertainty: measurement errors, inhomogeneity of data, errors during data handling, non- representative reproduction of measurement due to inadequate temporal and spatial resolution

 Model uncertainty: coverage of inadequate reproduction of physical processes in nature

Human errors during production, abrasion, maintenance, and other operational activities are critical factors that can impact system performance However, these mistakes are typically not included in standardized models due to their problem-specific nature and the lack of universal approaches for their mitigation.

If normal or Gaussian distributions for x are used 68.3% of all values of x are within the range of

Approximately 95.5% of all values fall within the range of μ(x) ± 2σ(x), and nearly 97.7% of values lie within μ(x) ± 3σ(x), as illustrated in Figure 1.13 Considering uncertainties in a design involves treating input parameters as variables that can vary within certain ranges rather than fixed deterministic values This approach has two key implications: first, it requires verifying whether all possible realisations of the input parameters comply with the design criteria, ensuring robust and reliable performance under variability.

Normal distributionLog-normal distribution

When modeling wave heights, it's essential to ensure the parameters are physically realistic, as some effects, like a normally distributed wave height, can mathematically produce negative values, which are impossible in reality To address this, a log-normal distribution can be used, as it naturally excludes negative values Additionally, parameters must be validated against real-world data, considering physical constraints such as water depth, since certain wave heights can only occur under specific conditions and not in all combinations with wave periods.

Figure 1.13: Gaussian distribution function, variation of parameters and 90%-confidence interval

A common approach to addressing uncertainties involves presenting the mean along with a 90% confidence interval, which is widely used in this manual This interval is calculated by determining the 5% exceedance on both sides of the mean, using the formula μ(x) ± 1.64σ(x) based on a normal distribution These confidence bands provide a clear representation of uncertainty, as illustrated in Figure 1.13.

When designing with uncertainties, it is crucial to carefully select statistical distributions for most parameters to ensure accurate modeling Additionally, maintaining physical relationships between parameters is essential to preserve realism and consistency These considerations will be thoroughly discussed in the following sections to guide effective and reliable design practices.

Parameter uncertainty

Input parameter uncertainty refers to the inaccuracy arising from measurement errors or their inherent variability This uncertainty is typically characterized using statistical distributions or the relative variation of parameters Sources for these variations include observed measurement errors, expert opinions gathered through questionnaires, and error reports from scientific literature, ensuring comprehensive and reliable uncertainty assessment.

This article explores the uncertainties associated with parameters used to predict wave overtopping of coastal structures, with each chapter dedicated to different predictive methods It examines the physical relationships between parameters and discusses how these interactions influence the prediction accuracy Additionally, the study proposes restrictions and guidelines for assessing uncertainties effectively, ensuring more reliable and robust wave overtopping forecasts for coastal defense designs.

Model uncertainty

Model uncertainty refers to the accuracy with which a model or method can accurately describe a physical process or limit state function It captures the deviation between model predictions and measured data caused by the method itself This concept is complex due to the interplay between parameter uncertainty and model uncertainty, as discrepancies between predictions and observations may stem from either uncertain input parameters or inherent limitations within the model.

Model uncertainties may be described using the same approach than for parameter uncertainties using a multiplicative approach This means that:

The model for predicting wave overtopping is expressed as \(f(x_i) = m \cdot q\), where \(m\) is the model factor, and \(q\) is the mean overtopping rate The model factor \(m\) is assumed to follow a normal distribution with a mean of 1.0 and a coefficient of variation specific to the model, accounting for uncertainty Alternatively, \(m\) can be a coefficient within a formula treated as a stochastic variable, with its uncertainty represented by the standard deviation of the coefficient This approach often provides more accurate predictions, especially when the process is modeled logarithmically—for example, larger overtopping discharges (100–300 l/s per meter) are predicted more reliably than smaller ones (0.1–0.8 l/s per meter) When presenting this model graphically, including a 90% confidence band or interval, shown by two 5%-exceedance lines, enhances the clarity and reliability of the results for wave overtopping predictions.

Model factors in overtopping discharge predictions often exhibit coefficients of variation exceeding 30%, indicating significant variability A mean value (m) of 1.0 signifies that the model has no systematic bias, ensuring unbiased predictions Any consistent underestimation or overestimation must be corrected within the model; for instance, if a model consistently over-predicts by 20%, it should be adjusted to produce results 20% lower to improve accuracy and reliability in the predictions.

This concept is followed in all further chapters of this manual so that from here onwards, and the procedure to account for the model uncertainties is given in Section 4.10.1.

Methodology and application in this manual

All parameter and model uncertainties outlined previously are incorporated into the application of the formulas and the execution of the proposed models The results generated by these models are analyzed and presented using statistical methods to ensure accuracy and reliability.

18 distributions rather than being single deterministic values Hence, interpretation of these results is required and recommendations will be given on how to use formulae or outputs of the models

This manual describes the reliability of the formula often by taking one of the coefficients as a stochastic parameter and giving a standard deviation (assuming a normal distribution) The first EurOtop Manual

In 2007, a deterministic and probabilistic approach to prediction formulas was introduced, presenting two similar formulas with different coefficients The probabilistic method utilized the mean coefficient value, while the deterministic approach added one standard deviation to account for uncertainty The deterministic method thus incorporated a safety margin to address the substantial uncertainties associated with wave run-up and overtopping.

The "deterministic design or safety assessment" approach in the first EurOtop Manual (2007) should be considered a semi-probabilistic method, as it incorporates a partial safety factor of one standard deviation This approach provides a more nuanced safety evaluation by combining deterministic principles with probabilistic considerations The EurOtop Manual introduces enhanced methods that improve the accuracy and reliability of overtopping and wave impact assessments These advanced approaches help engineers better predict safety margins and ensure structural resilience against wave actions Incorporating semi-probabilistic methods aligns with modern best practices in coastal hazard management and risk assessment.

The mean value approach involves using the formula with the average of the stochastic parameters to predict or compare with test data Enhancing this method with 5% exceedance lines or a 90% confidence band in graphical representations offers a comprehensive comparison, a technique recognized as the probabilistic approach in EurOtop (2007).

The design or assessment approach discussed is a semi-probabilistic method that incorporates a partial safety factor, providing a more comprehensive evaluation by accounting for uncertainty in predictions This mean value approach includes stochastic parameters expressed as μ(m) + σ(m), where m is defined in Equation 1.3, representing an advancement over the purely deterministic method previously described in EurOtop (2007).

 Probabilistic approach Consider the stochastic parameter(s) with their given standard deviation and assuming a normal or log-normal distribution;

 The 5%-exceedance lines, or 90%-confidence band, can be calculated by using μ(m) ± 1.64σ(m) for the stochastic parameter(s)

This manual presents formulae based on a mean value approach, with key coefficients treated as stochastic variables to account for uncertainty through their standard deviation, σ(m) It provides graphical representations of the formulae and 5%-exceedance curves, offering clear insights into the probabilistic aspects The coefficients used for design or assessment purposes are specified to ensure accurate application An example illustrating this approach is demonstrated for wave run-up, which is consistently followed throughout the manual Note that a fully probabilistic method is not employed here, except for a specific example in Section 5.6.

The formula for the 2%-wave run-up level is given by (assuming breaking waves only):

The coefficient 1.65 is treated as a stochastic variable with a mean value of 1.65 and a standard deviation of 0.10, representing the mean value approach For more accurate design and assessment purposes, it is recommended to use the conservative value of 1.75 rather than 1.65, ensuring safety and reliability in engineering calculations.

2 Water levels and wave conditions

Introduction

This Overtopping Manual focuses exclusively on wave run-up and wave overtopping, not on the entire design process of coastal structures It does not serve as a comprehensive design guide but highlights key activities involved in deriving water levels and wave conditions, including depth-limited scenarios The manual outlines essential parameters, provides a check-list of critical processes, and offers comprehensive references to authoritative sources It includes brief method descriptions, summarizes relevant tools and models, and cross-references other manuals to support accurate assessment of wave overtopping risks.

The primary manuals and guidelines for the design and safety assessment of coastal and inland structures, including water levels and wave conditions, are outlined in The Rock Manual (2007) This authoritative resource provides comprehensive guidance to ensure structural resilience and safety in marine and freshwater environments.

The Coastal Engineering Manual (2006); The British Standards (2000); The German “Die Küste”

(EAK, 2002); ), the Dutch ENW manuals (ENW: http://kennisbank-waterbouw.nl/dicea/TAW-ENW.htm); and the DELOS Design Guidelines (2007).

Water levels, tides, surges and sea level changes

Mean sea level

In open coastal waters connected to the sea, the mean water level can typically be considered a site-specific constant, linked to the global mean sea level For short-term safety assessments spanning up to five years, this mean water level can be treated as a fixed reference point However, due to anticipated global warming, sea level rise projections for the next century vary significantly, ranging from approximately 0.2 meters to over 1.0 meter.

When designing durable structures, it is essential to account for anticipated sea level rise to ensure long-term resilience Some countries specify a required sea level rise to be incorporated into flood defense designs, with the chosen period depending on potential future modifications Earthen dikes are relatively easy to heighten, so a 50-year sea level rise projection may suffice; however, permanent flood defenses through urban areas are more challenging to adapt, often necessitating considerations for sea level rise over 100 years or more.

Astronomical tide

Tidal movements are primarily driven by predictable astronomical forces, enabling accurate forecasting of tidal levels and currents The most significant fluctuations in water levels along the UK, North Sea coast, and worldwide are caused by these astronomical tides, resulting from the gravitational pull of the sun and moon as they orbit the Earth This differential gravitational effect creates well-defined tide periods, mainly semi-diurnal and diurnal, with semi-diurnal tides being dominant around the British Isles and North Sea coasts.

Tidal level fluctuations are influenced not only by Earth's rotation but also by various periodicities, with the most prominent being the fortnightly spring-neap cycle This cycle, occurring roughly every two weeks, aligns with the half period of the lunar cycle and significantly impacts tidal amplitudes Understanding these cyclical patterns is essential for accurate tidal predictions and coastal management.

For comprehensive information on the generation and dynamics of astronomic tides, refer to the Admiralty Manual of Tides, which provides daily predictions of high and low water times at key locations like ports It also includes methods for calculating water level differences between locations However, predicting extreme water levels in practice is more complex due to weather effects, as discussed further in the manual.

Surges related to extreme weather conditions

The difference between the highest astronomical tide and the largest predicted tide in a year is typically only a few centimeters, making it a relatively minor variance In practical terms, this small difference is overshadowed by the larger discrepancies between predicted and observed tidal levels caused by weather effects Understanding these subtle variations helps in accurate tide forecasting and coastal management.

Extreme high water levels result from a combination of high tidal elevations and positive surges, which typically involve three main components: a barometric effect caused by atmospheric pressure variations; a wind set-up, where strong winds in shallow seas like the English Channel or North Sea can rapidly raise sea levels within hours; and a dynamic effect, where the shape of the coastline amplifies surge-induced motions through phenomena such as seiching and funneling.

Wave set-up is a localized increase in water levels within the surf zone caused by waves breaking as they approach the shoreline, distinct from other surge components It is implicitly included in physical model tests used for overtopping calculations, but only over the foreshore length represented in the model Generally, there is no need to add a separate water level increase for wave set-up when estimating overtopping discharges, except in cases where the foreshore is very long and gentle In such scenarios, numerical models should account for wave set-up one or two wavelengths in front of the structure's toe to ensure accurate predictions.

Negative surges primarily consist of two components: the barometric effect from high atmospheric pressures and wind set-down resulting from offshore winds Large positive surges occur more frequently than negative ones because depressions that cause positive surges tend to be more intense and generate stronger wind conditions than anticyclones.

Surges in large, shallow regions like the southern North Sea are crucial for estimating extreme water levels, as they can reach several meters during events with long return periods The most straightforward approach to predicting these extreme water levels involves analyzing long-term water level data from the specific site However, in the absence of such data, predictive methods—either theoretical or empirical—are necessary to estimate surge levels Combining these surge predictions with tidal data enables accurate assessment of extreme water levels, essential for effective coastal risk management and infrastructure planning.

Figure 2.1 illustrates nearly a century of high water level measurements in the Netherlands, providing valuable data for understanding flood risks The chart also includes extrapolated results to extreme low exceedance probabilities, such as 10^-4, indicating a "once in 10,000 years" event This comprehensive analysis enhances the accuracy of flood risk assessments and supports robust flood management strategies in the Netherlands.

Extreme water levels are often assessed by analyzing long-term wind statistics, which are typically observed over extended periods longer than wave measurements Numerical modeling of water motion, driven by these identified extreme wind conditions, provides accurate predictions of potential water level extremes.

Hurricane surge levels differ significantly from general storm surge data, as they are highly dependent on the hurricane's landfall location The maximum onshore winds, which cause the highest surges, occur at varying distances from the eye and vary depending on whether winds are onshore or offshore Consequently, areas closer to the landfall point may experience very high localized surges, while regions farther away and on the opposite side of the storm typically see lower surge levels and wave impacts Accurate assessment of hurricane surge risks requires considering the probability of a hurricane making landfall at a specific location, although detailed procedures for this are beyond the scope of this manual.

Figure 2.1: An Example of measurements of maximum water levels for almost 100 years (from 1887 to

1985) and extrapolation to extreme return periods Hook of Holland, the Netherlands

High river discharges

Coastal flood defences face the sea or a (large) lake, but flood defences are also present along tidal rivers

Extreme river discharges lead to high water levels along river flood defenses, which can persist for a week or more During these events, storms may generate waves that cause overtopping of flood defenses, necessitating river dikes to be designed with sufficient height to prevent failure The required height of a river dike depends not only on the extreme water level but also on the potential for wave overtopping caused by storm-generated waves It is important to note that the occurrence of extreme river discharges and water levels is independent of storm events, with only average “normal” storms occurring roughly once every decade considered during high discharge events.

Where rivers meet the sea, both storm surges and high river discharges can cause extreme water levels, posing risks to coastal and inland areas While extreme storms and surges significantly impact water levels near the coast, their effects diminish upstream Joint probabilistic analysis of storm surges and river discharges is essential for accurately determining extreme water levels, ensuring effective design and safety assessments for coastal infrastructure.

Effect on crest levels

When designing or assessing the safety of a dike, the crest height must consider more than just wave run-up and wave overtopping; it also requires accounting for the reference level, local sudden gusts, and oscillations to ensure comprehensive protection.

(leading to a corrected water level), settlement and an increase of the water level due to sea level rise

The structure height of a dike in the Netherlands up to 2016 was determined by several key factors, including the reference level with a probability of being exceeded aligned with the legal standard, which ranges between a 1,250 and 10,000-year return period Additionally, sea level rise or lake level increases during the design period and anticipated local ground subsidence also significantly contributed to the final dike height These components are outlined in the Guidelines for Sea and Lake Dikes (TAW, 1999) and are illustrated in Figure 2.2.

High water level (m relative to reference level)

Average number of exceedances per year or storm season

The article highlights key factors influencing dike stability, including additional allowances for squalls, gusts, seiches, and other local wind conditions It also considers the expected decrease in crest height resulting from settlement of the dike body and foundation soils throughout the design period Furthermore, the discussion emphasizes the importance of assessing wave run-up height and wave overtopping height to ensure effective flood protection.

Figure 2.2: Important aspects during calculation or assessment of dike height

Contributions (a) to (d) typically cannot be influenced, whereas contribution (e) can be affected through management strategies The influence on contribution (f) depends significantly on the outer seaward slope, which may consist of various materials such as asphalt layers, cement-concrete dike covers (pitched block work), or grass on a clay base, with combinations of these materials also possible Slopes are often irregular, featuring differing upper and lower slopes, and may include berms, although this manual does not detail the design of cover layers concerning hydraulic and geotechnical stability Instead, it focuses on how berms, slopes, and roughness elements impact wave run-up and wave overtopping, which are critical factors in coastal protection assessments.

Since 2016, the Netherlands has shifted from designing flood defenses based on storm or river discharge return periods to a probabilistic approach that considers the overall likelihood of flooding This new system requires the assessment of all failure mechanisms across the entire flood protection system to ensure comprehensive risk management Longer dike sections inherently have a higher probability of failure compared to shorter segments under identical conditions, emphasizing the importance of considering length in design Despite this change, key factors a) to f) in Figure 2.2 remain influential in determining the appropriate height of flood defenses.

Wave conditions

Offshore wave conditions

Accurate wave climate assessment relies on long-term instrumental measurements at the specific site, though such data collection is rarely feasible An illustrative example includes nearly 25 years of wave height data from the Dutch North Sea, which, when extrapolated to extreme events, indicates a 10,000-year return wave height, aiding in comprehensive risk analysis and design considerations.

Deep water offshore data can be obtained through computational wave prediction models based on wind data or wave models, which can then be combined with wave transformation models to assess wave climate at coastal sites When instrumentally measured data is available for a short period, it can be used to calibrate or verify the wave transformation models, enhancing their accuracy and reliability Additionally, remote sensing via satellite databases offers a recent and valuable source of wave data for specific areas, improving wave climate assessments for coastal planning.

Wind-generated waves offshore of most coasts typically have wave periods ranging from 1 to 20 seconds, influenced by local wind conditions The height, period, and direction of these waves depend on factors such as wind speed, duration, and wind direction, which shape wave characteristics Understanding these parameters is essential for accurate wave prediction and coastal planning.

Fetch refers to the unobstructed distance of sea surface over which the wind has acted, influencing wave development In inland lakes or reservoirs, even short storms can generate large waves due to limited fetch, making waves "fetch limited," where the duration has minimal impact Conversely, on open coasts with large fetch but short wind duration, waves are "duration limited," with wave growth primarily constrained by the storm’s length rather than fetch distance.

Figure 2.3: Wave measurements at the North Sea (1979-2002) and extrapolation to very small probabilities of exceedance SON-platform north of the Wadden islands at a depth of 19 m

Oceanic shorelines often present more complex conditions, with both fetch and duration reaching extensive lengths Under these circumstances, waves become "fully developed," meaning their height is determined primarily by the wind speed, leading to larger and more powerful waves.

Long period waves, or swell, can travel great distances without significant loss, especially when wave periods are large Swell waves are generated by distant wind conditions and are not directly influenced by local winds, making their prediction more challenging They may arrive unexpectedly at coastlines, even without local wind activity, as they originate from storms far away and days prior Regions like the Brazilian coast and West Africa frequently experience such unpredictable swell conditions, which can become crucial for wave prediction models and coastal planning.

Wave conditions at depth-limited situations

Wave breaking remains a complex phenomenon that is difficult to describe mathematically due to an incomplete understanding of its underlying physics Despite this, wave breaking significantly influences wave behavior, sediment transport, forces on coastal structures, and overtopping responses, making it essential to include in computational models The most common approach to modeling wave breaking involves incorporating an energy dissipation term that activates when waves reach a critical depth relative to their height, effectively capturing the energy loss during breaking.

Empirical methods such as those by Goda (2000), Owen (1980), and the Rock Manual (2007, Figure 4.40) are commonly employed for initial estimates of incident wave conditions in the surf zone According to Goda (2000), inshore wave conditions are significantly affected by shoaling and wave transformation processes, making these methods valuable tools for surf zone analysis.

Wave breaking is influenced by parameters such as sea steepness and bathymetry slope Goda (2000) developed a series of graphs to accurately predict the largest and significant wave heights (Hmax and Hs) for various sloping bathymetries, including ratios of 1:10, 1:20, 1:30, and 1:100, ensuring comprehensive assessment of wave behavior in different marine environments.

Results from a basic 1D energy decay numerical model, based on Rock Manual (2007) Figure 4.40 and also described in the CIRIA/CUR Manual (1991), demonstrate accurate wave height predictions for slopes ranging from 1:10 to 1:100 For slopes flatter than 1:100, predictions should be based on the 1:100 slope data This model accounts for wave breaking effects but does not consider the influence of long waves generated by wave breaking or changes in wave period.

Figure 2.4: Depth-limited significant wave heights for uniform foreshore slopes

The method for using these graphs is:

1) Determine the deep-water wave steepness, sop = Hso/Lop (where Lop = gTp 2/(2π)) This value determines which graphs should be used Suppose here for convenience that sop = 0.043, then the graphs of Figure 2.4 for sop = 0.04 and 0.05 have to be used, interpolating between the results from each

2) Determine the local relative water depth, h/Lop The range of the curves in the graphs covers a decrease in wave height by 10 per cent to about 70 per cent Limited breaking occurs at the

When the ratio h/Lop exceeds the maximum value indicated on the graph, it signifies that wave breaking is either absent or minimal This allows us to assume that there is no significant wave breaking occurring, with deep-water wave height remaining consistent with shallow-water wave height The right-hand side of the graphs illustrates conditions with typical wave behavior, while the severe breaking points are depicted on the left-hand side, providing a comprehensive overview of wave dynamics under varying conditions.

3) Determine the slope of the foreshore (m = tan α) Curves are given for range m = 0.075 to 0.01

(1:13 to 1:100) For gentler slopes the 1:100 slope should be used

4) Enter the two selected graphs with calculated h/Lop and read the breaker index Hm0/h from the curve of the calculated foreshore slope

5) Interpolate linearly between the two values of Hm0/h to find Hm0/h for the correct wave steepness

Example: Suppose Hso = 6 m, Tp = 9.4 s, foreshore slope is 1:40 (m = 0.025) Calculate the maximum significant wave height Hm0 at a water depth of h = 7 m

1) The wave conditions in deep water give sop = 0.043 Graphs with sop = 0.04 and 0.05 should to be used

2) The local relative water depth h/Lop = 0.051

3) The slope of the foreshore (m = 0.025) is in between the curves for m = 0.02 and 0.033

4) From the graphs, Hm0/h = 0.64 is found for sop = 0.04 and 0.68 is found for sop = 0.05

5) Interpolation for sop = 0.043 gives Hm0/h = 0.65 and finally a depth-limited spectral significant wave height of Hm0 = 3.9 m

Wave breaking in shallow water influences not only the significant wave height (Hm0) but also alters the distribution of wave heights In deep water, wave heights follow a Rayleigh distribution, making the spectral wave height (Hm0) close to the statistical wave height (H1/3) However, in shallow water, wave heights diverge due to the breaking process, with the highest waves breaking first as they approach the bottom, while smaller waves remain unchanged This creates a non-homogeneous wave height distribution, comprising both broken and unbroken waves To model this, Battjes and Groenendijk (2000) developed the composite Weibull distribution for shallow water wave heights For horizontal seabeds, this distribution should not be applied with zero slope, as evidence suggests it tends to revert to a Rayleigh distribution (Caires and Van Gent, 2012).

While this manual primarily focuses on prediction methods based on spectral significant wave height, it can be beneficial in certain situations to consider alternative wave height metrics such as the 2% wave height (H2%) or the H1/10, which represents the average height of the highest 10% of waves Therefore, the method of Battjes and colleagues provides a comprehensive approach that incorporates these different wave height definitions for more accurate wave predictions.

Groenendijk (2000) is given here The example given above with a calculated Hm0 = 3.9 m at a depth of

7 m on a 1:40 slope foreshore has been explored further in Figure 2.5 The distribution requires the root mean square wave height Hrms, which is calculated as:

H rms   2.1 where Hrms = root mean square wave height The transition wave height, Htr, between the lower Rayleigh distribution and the higher Weibull distribution (see Figure 2.5) is then given by:

One then has then to compute the non-dimensional wave height Htr/Hrms, which is used as input to

Table 2.1 of Battjes and Groenendijk (2000) to find the (non-dimensional) characteristic heights: H1/3/Hrms,

H1/10/Hrms, H2%/Hrms, H1%/Hrms and H0.1%/Hrms Some particular values have been extracted from this table and are included in Table 2.1, only for the ratios H1/3/Hrms, H1/10/Hrms, and H2%/Hrms

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Table 2.1: Values of dimensionless wave heights for some values of Htr/Hrms From Battjes and

Non-dimensional transitional wave Htr/Hrms

Figure 2.5: Computed composite Weibull distribution Hm0 = 3.9 m; foreshore slope 1:40 and water depth h = 7 m

Joint probability of waves and water levels

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Tidal and surge levels are primary metocean variables that are often analyzed as independent factors, although they can also display some dependency with scatter In addition to these, secondary variables such as wind speed, wave direction, and wind direction are also important for comprehensive metocean assessments Understanding these variables is essential for accurate modeling and forecasting of ocean conditions.

Multi-variable assessments in coastal flood studies have become standard in the UK and the Netherlands over the past 20-30 years In Dutch sea defence evaluations, the "illustration point" is calculated for each coastal location, combining stochastic values of wind speed, direction, water level, wave height, period, and direction to identify conditions most likely to cause maximum overtopping for a specific return period This approach helps determine the minimum required crest level, with various methods like Monte Carlo simulation and FORM used to evaluate joint probability conditions effectively.

Currents

Waves approaching an oncoming current, such as at a river mouth, experience increased steepness, with wave height rising and wavelength decreasing due to the current's influence Refraction caused by the current focuses wave energy toward the river mouth, creating a complex wave-current interaction Both current effects and depth refraction occur simultaneously, resulting in intricate wave patterns, especially in areas with strong currents While modeling these combined effects is more complex, advanced computational models can accurately represent their impact on wave propagation These currents significantly influence wave behavior, including wave overtopping, as discussed in Section 5.4.5.

Return periods and probability of events

The selection of an appropriate return period for a site depends on factors such as the structure’s expected lifespan, maximum wave and water level conditions, and its intended use The potential consequences of failure also influence this choice; for example, public access sites require higher standards of defense compared to farmland protection.

Further examples are given in Chapter 3

When evaluating events with a specific return period (TR), it is generally assumed that for TR ≥ 5 years, the probability of occurrence in any given year is approximately 1 divided by TR For instance, an event with a 10,000-year return period has an annual occurrence probability of 10^-4 This approach helps in understanding the likelihood of rare events and is essential for risk assessment and planning.

Over a planned lifetime of N years, the probability of experiencing a wave condition with a return period TR at least once can be modeled using the Poisson distribution This approach helps estimate the likelihood of rare, extreme wave events over the structure’s lifespan, ensuring accurate risk assessment and coastal structure design.

Figure 2.6 illustrates the encounter probability curves, showing that the probability of a marine event occurring within a return period TR varies between 1% and 95% depending on TR and N It is important to note that there is no exact interval of TR years between events, but rather a probability distribution; for example, a 100-year event has approximately a 63% chance of occurring within that period Similarly, a 1000-year event has about a 10% probability of occurring in a 100-year timeframe For comprehensive guidance on design events and return periods, refer to the British Standard Code of Practice for Maritime Structures (BS6349), the PIANC working group reports, and the Rock Manual (2007).

Uncertainties in inputs

This section highlights key input parameters such as water levels, including tides, surges, and sea level changes, which are critical for accurate coastal and structural analysis It also discusses sea state conditions at the structure's toe, emphasizing their impact on stability and design Additionally, river discharges and currents are considered essential factors influencing hydrodynamic behavior and structural integrity in marine environments Proper understanding of these parameters ensures effective risk assessment and optimal infrastructure performance.

This article assumes that all input parameters can be defined at the toe of the structure, which is essential for accurate coastal and marine engineering analyses The statistical distributions of these parameters vary across the foreshore due to differing physical processes, including refraction, shoaling, and wave breaking Understanding how these processes influence parameter distributions is crucial for reliable modeling and design Methods to account for the changes in parameter distributions caused by these physical phenomena are discussed in previous sections and other referenced sources, ensuring comprehensive and adaptable analysis strategies.

When statistical distributions or error levels are unavailable for water levels or sea state parameters, it is recommended to assume that all parameters follow a normal distribution For significant wave height (Hs) or spectral wave height (Hm0), a coefficient of variation of 5.0% can be applied, while for peak wave period (Tp) or spectral wave period (Tm-1.0), the same coefficient of variation of 5.0% is appropriate The design water level at the toe typically has a coefficient of variation of 3.0%, according to Schüttrumpf et al (2006) It's important to note that, in some cases, a range is used for design parameters; for example, a wave height with a certain return period may occur alongside a wave period within a range of 8-12 seconds, rendering the coefficient of variation on the wave period less meaningful.

Based on insights from approximately 100 international coastal engineering experts, these values were derived to address key uncertainties Although these parameters are somewhat smaller than those proposed by Goda (2000), they have been validated against real-world cases The testing confirmed that these parameters provide a reliable range of variations for practical applications in coastal risk assessment and engineering design.

It is important to note that these uncertainties are applied to significant wave height values rather than mean sea state parameters, which impacts both the type of statistical distribution and the magnitude of the standard deviation or coefficient of variation.

Wave overtopping behaviour

Tolerable mean discharges and maximum volumes

Empirical models, including comparison of structures

The new EurOtop database

The EurOtop Neural Network prediction tool

Numerical modelling of wave overtopping

Simulators of overtopping at dikes

Model and Scale effects

Uncertainties in predictions

Wave run-up

Wave overtopping discharges

Influence factors on wave run-up and wave overtopping

Overtopping wave characteristics

Overtopping discharges

Overtopping wave characteristics

Wave processes at walls

Mean overtopping discharges for vertical and very steep walls

Overtopping volumes

Overtopping velocities and distributions