Breaker depth index of a wave breaking at a coastal structure – the modified Weggel’s nomogram revisited

a, b

slope-inclination dependent parameters

c

auxiliary variable

d

thickness of the seabed

g

acceleration due to the gravity of Earth

H

wave height

$H0′$

equivalent deep-water wave height

$H0b$

deep-water breaker height

$Hb$

breaker height

$Hs$

significant wave height

$hb$

water depth at the point of wave breaking

$hs$

water depth at a coastal structure (⁠$hs≡ds$⁠)

$kb$

breaker wave number

L

wavelength

$L0$

deep-water wavelength

$L0b$

deep-water wavelength at wave breaking

$Lb$

breaker wavelength

M0, … , M8

symbols of tangency points

m

sea bottom slope inclination

T

wave period

$xp$

breaker travel distance

Z

auxiliary variable

$γb$

breaker depth index

$γbs$

breaker depth index of a wave breaking at a coastal structure

$Δ$

discriminant

$δb$

breaking wave steepness (⁠$δb≡δmax$⁠)

$δb0$

maximum deep-water wave steepness

$Θ$

bottom slope-inclination angle

$π$

mathematical constant

$τ$

dimensionless breaker travel distance

$Ωb$

breaker height index

A breaking phenomenon related to a progressive surface water wave, both regular and irregular (randomly driven by wind), can happen under any water depth conditions – deep-water waves as well as shallow-water waves can undergo breaking processes. The ability to properly assess wave breaking parameters becomes particularly important for quantitative evaluation of the influence of a single breaking wave acting on a coastal structure like a gravitational vertical-wall breakwater: (a) low-mound breakwater founded on a stone bedding of relative small thickness; and (b) high-mound composite breakwater with a concrete caisson founded on rubble-mound foundation (covered by armour stones or armour artificial concrete blocks) of relative large thickness.

Coastal engineers often use different nomograms in order to find a limiting value of a particular parameter required in a certain design process. One of the main parameters relates to the design wave height taking into account a coastal structure subject to a breaking wave attack. In this particular case, an appropriate nomogram pertains to the breaker depth index, $γbs=defHb/hs$⁠, for a progressive regular wave breaking just in front of a coastal structure, which is mainly a breakwater. A widely and commonly used nomogram of this type was published in Shore Protection Manual (SPM, 1984), as a modified version of the original Weggel’s nomogram (Weggel, 1972).

As indicated in Figure 1, the modified Weggel’s nomogram is limited along its abscissa to $hs/(gT2)=0.02$⁠. However, the practice of engineering shows that this limiting value is not always enough and it does not suit some practical requirements. In addition, the nomogram indicates that progressions of the function $γbs=f[hs/(gT2)]$⁠, obtained for different sea bottom inclinations, converge to the limiting case of horizontal sea bottom (⁠$m=0$⁠), which is indicated by a horizontal line in the whole range of the nomogram abscissa $0<hs/(gT2)≤0.02$ (see Figure 1). Due to this fact, some authors suggest assuming a constant value $γbs=0.775$ for $hs/(gT2)>0.02$⁠, thereby unnecessarily lowering the accuracy of practical assessments of the breaker depth index, $γbs$ – the acceptance of this assumption is rather questionable. A horizontal progression of the curve denoting a horizontal seabed (⁠$m=0$⁠) would also falsely indicate that the solution reflected by the modified Weggel’s nomogram does not consider the limiting wave steepness condition, which would be in contrast to the original Weggel’s nomogram illustrated in Figure 2. Moreover, the original Weggel’s nomogram (Weggel, 1972) and the modified Weggel’s nomogram (SPM, 1984) differ in the type of abscissa – the first one is dimensional, whereas the second one is dimensionless. Could this difference be the cause of the above-mentioned doubts, or is it only a visual effect of too small a value of the abscissa dimensionless limit applied to the modified Weggel’s nomogram (Figure 1) in comparison with the respective dimensional limit in the original Weggel’s nomogram (Figure 2)? The answers to these questions, and much more, will take up the rest of the present paper.

The main goal of the present study is to check, improve, and zoom in (for a wider abscissa) to the modified Weggel’s nomogram for the breaker depth index of a wave breaking in front of a coastal structure, published in Shore Protection Manual (SPM, 1984) and illustrated in Figure 1, in order to get the nomogram free of the above-mentioned suspicions and drawbacks, which could be fully accepted by coastal engineers in their design works.

A short introduction of some important parameters of the breaking progressive regular wave, together with their limiting values important for the considered study, will be presented at first. The next part of the present paper will be devoted to a pure case of sloped sea bottom without any coastal structure (e.g. a breakwater). An analytical derivation of the breaker depth index function, $γb=defHb/hb=f[Hb/(gT2)]$⁠, assuming Miche’s limiting wave steepness condition, requires to be performed in order to check if an analytical technique, used in the present paper to draw an envelope to the family of Weggel’s lines, is a useful tool to incorporate correctly the wave steepness limit into $γb$ curves.

Afterwards, considering a more complex case of wave breaking in front of a coastal structure, the target point of the present paper will be obtained through: (a) performing derivation of a correct form of Weggel’s equation for the breaker depth index of a wave breaking above a sloped sea bottom in front of a coastal structure, $γbs=defHb/hs=f[hs/(gT2)]$ (still without considering the limiting wave steepness condition); and (b) introducing a correction procedure for implementing the limiting wave steepness condition to the breaker depth index, $γbs$⁠, based on previously stepwise-computed values of $γb=f[Hb/(gT2)]$ together with a required conversion of the dimensionless abscissa parameter.

Finally, based on the content of the present paper, a set of high-quality practical nomograms (with dimensionless axes) for accurate readout of the breaker depth index, $γbs$⁠, and a respective program code written in Fortran 77, will be elaborated and presented in the ‘Supplementary materials’ section.

When describing and analysing the breaking of progressive waves, the following two dimensionless parameters are frequently in use:

• limiting wave steepness (breaking wave steepness):

• breaker depth index:

where: $δmax$ is the limiting wave steepness (⁠$δmax≡δb$⁠) (–), $δb$ is the breaking wave steepness (wave steepness at the time of wave breaking) (–), $γb$ is the breaker depth index (–), H is the wave height (m), $Hb$ is the breaker height (wave height at the time of wave breaking) (m), L is the wavelength (m), $Lb$ is the breaker wavelength (wavelength at the time of wave breaking) (m), $hb$ is the water depth at the point of wave breaking (m). The breaker depth index, $γb$⁠, being the ratio of the breaker height to the water depth associated with wave breaking, depends on the bottom slope inclination, m, and the relative water depth: the water depth and the wave length (or wave period), because m, $hb$⁠, and T are usually known as the input to a certain design problem.

A short digression is required concerning nomenclature. In general, the term breaker index may include several definitions of wave height ratios (Goda, 2010). A detailed search of the scientific literature brings much confusion due to different names assigned to the parameter $γb$ given in Equation 2, for example, breaker depth index (Camenen and Larson, 2007; CEM, 2003; McCormick, 2010; Robertson et al., 2013), breaker index (Goda, 2010; Robertson et al., 2013), and breaker-height-to-depth index (Kaminsky and Kraus, 1993). The term ‘breaker depth index’ should not be used simply from the linguistic point of view. The breaker, being a particular form of the wave, can be characterised by the height and the wavelength (or wave period) but never can be described by a parameter like the depth. Only under the assumption that the term ‘depth’ is accepted as a poor simplification of the water depth, can the use of ‘breaker depth index’ be estimated as fully justified. Another reason to name the dimensionless wave breaking parameter $γb=Hb/hb$ simply as the breaker depth index is probably to make a clear distinction between $γb$ and another dimensionless wave breaking parameter $Ωb=Hb/H0′$ (where $H0′$ is the equivalent deep-water wave height), which is the so-called breaker height index (CEM, 2003). In spite of the above digression, and taking into account that the term ‘breaker depth index’ is already commonly used, this term – describing the ratio of $Hb/hb$ – will be also used throughout the present paper.

The breaking wave height (the breaker height), $Hb$⁠, denotes the wave height at the beginning of the wave breaking process, and the water depth at the point of wave breaking, $hb$⁠, means the water depth at a point where the wave reaches the wave height $Hb$⁠. This type of clarification is required in view of the fact that the wave breaking process has an evolving nature and extends over a certain distance $xp$ (Figure 3) where, in the middle of breaker propagation, the height of the breaking wave undergoes gradual reduction. The beginning of the wave breaking process is determined by the time and point where the height of the breaking wave is the largest and equal to the breaker height, $Hb$⁠. According to some other definitions, the beginning of wave breaking is identified with the formation of a vertical plane of the breaker front (in case of the plunging breaker) or with the appearance of a foam on top of the breaker (in case of the spilling breaker).

Michell (1893), based on a wave crest angle of 120° from the Stokes wave theory, determined theoretically that the maximum (breaking) wave steepness, causing a deep-water wave to break, equals:

where: $δb0$ is the maximum deep-water wave steepness (–), $H0b$ is the deep-water breaker height (m), $L0b$ is the deep-water wavelength at wave breaking (m). In addition, Michell (1893) indicated the influence of non-linear effects on the deep-water wavelength at wave breaking and proposed the following relation (Hudspeth, 2006; Weggel, 1972):

where: $L0b$ is the deep-water wavelength at wave breaking (m), $L0$ is the deep-water wavelength based on the linear wave theory (Airy’s theory of sinusoidal progressive wave) (m), T is the wave period (s), g is the acceleration due to the gravity of Earth (⁠$g=9.81m/s2$⁠). By substituting Equation 4 into Equation 3 and performing some simple mathematical manipulations, one obtains another limiting dimensionless ratio for the deep-water breaking wave according to the Michell theory (Hudspeth, 2006):

where, additionally, $Hb$ is the breaker height (m). After multiplying both sides of Equation 5a by the acceleration, g, and introducing the US customary system of measurement units, the limiting deep-water breaking wave ratio is equal to (Weggel, 1972):

where, this time, $Hb$ is given in feet (ft), and T remains constantly in seconds (s).

Miche (1951), applying the kinematic wave breaking criterion and the linear wave theory, and adopting the maximum deep-water steepness (Equation 3), proposed the following universal (i.e. valid for all water depths) criterion for the maximum wave steepness (known as the Miche equation):

where: $δb$ is the maximum wave steepness (–), $Hb$ is the breaker height (m), $Lb$ is the wavelength at the time of wave breaking (m), $kb$ is the wave number at the time of wave breaking (⁠$kb=2π/Lb$⁠) (1/m), $hb$ is the water depth at the point of wave breaking (m). The numerical coefficient in Equation 6b is very similar to 0.140 given in Miche (1951) (the difference between 0.140 and $1/7$ is equal to 2%). In case of deep-water conditions (⁠$kbhb≫1$ and $tanh(kbhb)→1$⁠), Equation 6b is simplified to the following constant function:

which is identical to Equation 3. It means that the maximum wave steepness becomes the largest in case of the deep-water wave. Assuming shallow-water conditions, for which $kbhb⪡1$ and $tanh(kbhb)→kbhb$⁠, Equation 6b can be also simplified and presented as:

After some simple mathematical operations on Equation 8, one obtains the breaker depth index for shallow-water conditions (see Equation 2):

which stays in a good agreement with values obtained from laboratory experiments on wave breaking above a horizontal seabed (McCormick, 2010). This value is estimated as the upper limit of the breaker depth index, whereas typical values belong to the range 0.6–0.8 (Andersen et al., 2014). According to other researchers, the breaker depth index can take values in the range 0.8–1.2 (Sorensen, 1993), which is typical for a sloped seabed bottom. As far as wind-driven waves are concerned, the value $Hs/h≈0.5$ (where $Hs$ is the significant wave height and h is the water depth) is observed at the moment of wave breaking (Andersen et al., 2014).

Andersen et al. (2014) and Hudspeth (2006), referring to Miche’s (1951) work, report a slightly smaller value of the breaker depth index obtained for shallow-water conditions [$kh⪡1$ (Hudspeth, 2006) or $h/L≤0.05$ (Andersen et al., 2014)] and for a horizontal seabed bottom, which is:

Assuming the solitary wave theory, McCowan (1894) gave yet another value of the breaker depth index for shallow-water waves propagating over a horizontal seabed bottom, namely (EAU, 2015; Hudspeth, 2006; Robertson et al., 2013; Weggel, 1972):

although EAU (2015) recommends $Hb/hb=1.0$ for engineering practice. Yamada et al. (1968) revised the value given in McCowan (1894) and presented (Robertson et al., 2013):

To the present day of engineering practice, a nomogram for reading out the breaker depth index is being used, allowing additionally to account for different inclinations of the bottom slope. As stated in SPM (1984), this nomogram was elaborated based on an experimentally and analytically obtained solution by Weggel (1972), where the equation for breaker depth index is presented assuming the US customary system of measurement units. Introducing the International System of Units (SI), and replacing the dimensional reference parameter $Hb/T2$ with the more convenient dimensionless parameter $Hb/(gT2)$⁠, the breaker index is defined by the following linear function:

in which:

where: $γb$ is the breaker depth index (–), $hb$ is the water depth at the point of wave breaking (m), $Hb$ is the breaker height (m), T is the wave period (s), m is the bottom slope inclination (⁠$m=tanθ$⁠) (–), $θ$ is the bottom slope inclination angle (rad), a is the slope-dependent parameter (⁠$s2/m$⁠), b is the slope-dependent parameter (–).

Equations 10a–10c, eventually with certain numerical factor roundings of no practical meaning, were cited by many authors, for example, CEM (2003), Camenen and Larson (2007), Hudspeth (2006), Robertson et al. (2013), and SPM (1984). Unfortunately, heightened attention is strongly recommended due to sometimes erroneous presentation of the governing Equations 10a–10c, as it was done in Camenen and Larson (2007). It is worth nothing that Weggel’s (1972) solution should be applied only for a mildly-sloped seabed bottom (⁠$m≤0.105$ (Weggel, 1972), $m≤0.1$ (CEM, 2003; Hudspeth, 2006), or $0.02<m<0.2$ (Robertson et al., 2013)). A direct application of the above Equations 10a–10c leads to a simplified version of the nomogram for readout of the breaker depth index, $γb$⁠. This nomogram, containing a family of lines depending on the bottom slope inclination, is shown in Figure 4.

Weggel (1972) informed also of the necessity of additional inclusion of a certain limitation of the wave breaking zone formed by the transition line, being a non-linear connection between McGowan’s equation for the breaker height in shallow water (Equation 9c), Michell’s equation for the limiting wave steepness in deep water (Equation 3), and Miche’s transition equation (Equation 6b). Weggel (1972) suggested two different approaches for implementation of Miche’s transition equation for the limiting wave steepness. Unfortunately, he did not inform the reader which analytical method he applied himself.

In order to obtain coherence with the empirical lines described by Equation 10a, Weggel (1972) proposed, as the first approach, that the run of the non-linear transition line could be successfully simulated by analytical preparation of the lower envelope of linear solutions for different bottom slope inclinations. Following this idea, an analytical method of determining the function of envelope, described by Quenell (2009), has been used for the purposes of the present paper. And, thus, performing basic steps of the method, the functions for horizontal and vertical coordinates of the envelope (to which Weggel’s lines are tangential) have been found and are as follows:

in which:

The values computed from Equation 11a for a certain bottom slope inclination, m, are simultaneously the maximum values of $Hb/(gT2)$⁠, for which the linear function, given by Equation 10a, is valid. A point $M$⁠, having the coordinates ${[Hb/(gT2)]M,(Hb/hb)M}$⁠, is a tangency point between Weggel’s line for a certain bottom slope inclination, m, and the smooth curve obtained as the lower envelope for the entire family of lines considered in the solution. Going further away, individual lines go into the envelope curve. The beginning of the curved envelope is situated in point $M0$⁠, located on Weggel’s line typical for the horizontal seabed bottom (⁠$m=0.0$⁠).

If one intends to compute the coordinates of any points of the envelope, where the horizontal coordinate (abscissa) fulfils the inequality $Hb/(gT2)≥[Hb/(gT2)]M0$⁠, in the first step it is necessary to solve the non-linear Equation 11a with respect to m, and then the calculated value of m must be figured either into Equation 11b or Equation 10a. Table 1 contains both coordinates of all considered points of tangency established for individual Weggel’s lines tangent to the envelope of the entire family of lines dependent on the bottom slope inclination.

According to the above-described algorithm, the computations have been performed and the results are presented graphically in Figure 5. Analogous illustrations, containing the results obtained for a slightly smaller number of bottom slope inclinations and for the horizontal coordinate in the range $Hb/(gT2)≤0.02$⁠, can be found in CEM (2003) (the original Figures II 4-2, p. II-4–5).

According to the second approach mentioned by Weggel (1972), the limiting wave steepness, $(δb)max$ can be incorporated in Weggel’s solution for the breaker depth index, $γb$⁠, by solving analytically a system of coupled equations constituted by Equation 10a (together with Equations 10b and 10c) and Equation 6b. In order to obtain the goal, it was considered whether it would be possible to present the maximum wave steepness (Equation 6b) as a function of a horizontal coordinate traditionally used in presentations of the breaker depth index, $γb$⁠, shown in Figures 4 and 5. And, thus, taking the Miche equation for the maximum wave steepness (Equation 6b) and the wavelength equation:

together with the deep-water wavelength of the breaking wave (see Equation 4) and the deep-water wavelength:

it becomes possible to derive the following relation:

and, dividing both sides of Equation 13 by Equation 6b, finally the required relation can be presented in the following form:

For comparison purposes, the author has derived Equation 14 by presenting Miche’s maximum wave steepness, given by Equation 6b, as a function of the horizontal coordinate $Hb/(gT2)$⁠, used commonly in many presentations of the breaker depth index, $γb$⁠. The comparison of Miche’s transition line, obtained by two different analytical methods, is illustrated graphically in Figure 6, proving that the envelope technique matches almost perfectly the analytically derived Equation 14 within the range of location coordinates of the tangency points calculated using the envelope method. Therefore, the use of envelope technique seems to be fully justified in order to model Miche’s transition line with acceptable accuracy.

The graph of the function in Equation 14 must be obviously treated as the most correct from the theoretical point of view, because it reflects exactly the limitation $Hb/(gT2)=0.027$ (Equation 5a), which is not the case as far as the empirically based envelope is concerned. This limiting value can be also obtained directly from Equation 14, assuming that for $γb→0$ the denominator in Equation 14 tends to infinity, and this happens when $6.054Hb/(gT2)→1$⁠, which finally denotes $Hb/(gT2)→0.0273≈0.027$⁠.

Some authors, for example, McCormick (2010) and SPM (1984), presented the governing solution (only for $0≤Hb/(gT2)≤0.02$⁠), expressing the equation for $γb$ (see Equation 10a) by its inverse:

the graphical representation of which is shown in Figure 7 for all the considered bottom slope-inclinations. One has to be careful because McCormick (2010) referred to the parameter $1/γb$ as the breaker depth index, which can be confusing because this name is already reserved to depict the parameter $γb$⁠. However, presenting the breaker index in the form of $1/γb$ (see Equation 15) seems to be quite reasonable because the reader is informed intuitively that $hb$ plays the role of the unknown parameter, appearing in the numerator of the definition equation $1/γb=defhb/Hb$⁠.

Of course, there are newer equations describing the function of the breaker depth index $γb=f(Hb/hb,m)$⁠. A very thorough and valuable review of this subject was done by Robertson et al. (2013). Only as an example of newly elaborated relations for the case of progressive regular waves, an empirical equation by Goda (1974) can be mentioned. The breaker depth index, given in the form $Hb/hb=f(hb/L0,m)$⁠, corrected in the meantime by Rattanapitikon and Shibayama (2000), is written as (Goda, 2010):

It is worth noting that Equation 16 already accounts for the limiting wave steepness condition. Camenen and Larson (2007) wrote that a procedure applied to solve Equation 16, and Weggel’s Equation 10a (together with Equations 10b and 10c) as well, requires an iterative technique. This sentence is not precise in case of Goda’s Equation 16 and is wrong in case of Weggel’s Equation 10a. Considering Weggel’s equation, Equation 10a is linear, allowing a direct computation of both parameters, either $hb$ or $Hb$⁠, depending on which of them is known. As far as Goda’s equation is concerned, everything depends on what the unknown is. If one is looking for $Hb$⁠, no iteration is needed – $Hb$ can be calculated directly from Equation 16. However, if $hb$ is treated as the unknown, it becomes necessary to use a solver for non-linear equations unless a proper nomogram is available.

Therefore, the question is whether it is possible to present Equation 10a in the form allowing one to calculate directly the breaker height, $Hb$⁠, as it is the case in Goda’s Equation 16. Of course, it is possible, following a simple transformation of Weggel’s Equation 10a:

and solving Equation 17a with respect to $Hb/hb$⁠, which leads to the following result:

and the slope inclination coefficients, a and b, stay in their unchanged forms given in Equations 10b and 10c. Unfortunately, the procedure given earlier in the present paper cannot be repeated in case of the breaker depth index in the form of Equation 17b, since this equation is a non-linear function and the envelope method by Quenell (2009) can be applied only in the case of families of lines and not curves. However, a new nomogram, presenting the breaker depth index in the form $γb=defHb/hb=f[hb/(gT2),m]$ can be easily achieved from the already prepared nomogram $γb=defHb/hb=f[Hb/(gT2),m]$ (see Figure 5) through a proper exchange of the horizontal coordinate. Thus, due to the following transformation:

one obtains the expression for a new definition of the horizontal coordinate (abscissa):

the calculation of which is based on the previous form of the horizontal and vertical coordinates, $Hb/(gT2)$ and $Hb/hb$⁠, respectively, of a certain point taken from the nomogram shown in Figure 5. The result of such action is presented in Figure 8, where the locations of tangency points are also indicated. As it can be easily checked, the run of each individual curve, for the horizontal coordinate in the range from zero to the horizontal coordinate of the tangency point of this curve, is obviously described by Equation 17b. A maximum range of applicability (due to the deep-water breaking wave limit) of the new horizontal coordinate, $hb/(gT2)$⁠, can be also calculated from Equation 18b. Thus, taking the limiting value, given by Equation 5a, and solving the non-linear equation presented in Equation 11a, one obtains $m=0.1044$⁠. The use of this value in Equation 11b implies $Hb/hb=0.3553$⁠. Finally, applying Equation 18b, it is an easy task to calculate the upper limit of applicability of the new horizontal coordinate, which equals $[hb/(gT2)]max=0.0272/0.3553=0.077$⁠; this is also indicated in Figure 8.

The water depth parameter, appearing in the breaker depth index, $γb$ (see Equation 10a), denotes the water depth associated with the beginning of wave breaking process (see Figure 3). During the breaker propagation along the distance $xp$⁠, counting from the point of wave breaking initiation to the seaward face of a costal structure, the breaker height undergoes a certain reduction compared to the breaker height at the point of wave breaking initiation. A basic geometric relation for the water depth is as follows:

where: $hs$ is the water depth at the toe of a coastal structure (m), $hb$ is the water depth at the point of initialisation of wave breaking process (m), m is the bottom slope inclination (⁠$m=tanθ$⁠) (–), $θ$ is the angle of bottom slope inclination to the horizontal plane (rad), $xp$ is the distance of the breaker propagation (breaker travel distance or wave plunge distance) (m).

For the purpose of the next part of the analysis, both sides of Equation 19a can be divided by the breaker height in order to operate with dimensionless parameters, leading to:

or additionally, after applying the inversion to Equation 19b one obtains:

where: $γbs$ is the breaker depth index of a wave breaking just in front of a coastal structure (–). Based on some observations of the breaker travelling distance, Galvin (1968) proposed the following relation:

or (see Figure 3):

By solving the system of original Equations 9–11, given in Weggel (1972: p. 421), together with Equations 19c and 20a, Weggel (1972) presented the following solution for the breaker depth index obtained for wave breaking at a costal structure:

in which:

together with its graphical representation shown in Figure 2 (please note that the abscissa dimensional parameter is expressed in the US customary measuring units).

Unfortunately, as can be easily proved, Weggel’s solution is not completely correct because it does not allow calculations of the breaker depth index, $γbs=Hb/hs$⁠, in case of the horizontal seabed bottom (⁠$m=0$⁠) – this is due to illegal division by m in the first term of Equation 21a. How was it possible to draw the curve for $m=0$ in Figure 2? In addition, it is also impossible to calculate the value of $γbs$ for $hs/T2=0$⁠. Although, from the practical point of view, the initial value $hs/T2=0$ does not seem to be so interesting (after all, a non-zero water basin in front of a coastal structure is obviously assumed and the wave is not a solitary wave with a theoretically infinite wavelength), from the mathematical point of view, it would be worthwhile to achieve the full analytical solution, giving also a possibility to calculate the initial value of $γbs$ for individual bottom slope inclinations.

An appropriate solution of the system of coupled equations (Equations 10a–10c, 19c and 20b) leads to formulation of the following quadratic equation:

the positive root of which is given as:

in which the discriminant of Equation 22 is written as:

Thus, a sought solution to the breaker depth index of a wave breaking at a coastal structure obtains the following correct form:

where a, b, and $τ$ are given by Equations 10b, 10c, and 20b, respectively. The obtained solution, given by Equation 24, is illustrated in Figure 9 for some exemplary slope inclinations of the seabed bottom, including the limiting case of the horizontal seabed bottom (⁠$m=0$⁠).

Comparing the original nomogram by Weggel (1972) (see Figure 2) with Figure 9, or other nomograms that could be found in Massel (1992) or SPM (1984) (see Figure 1), certain differences in runs of the curves, appearing at higher values of the horizontal coordinate $hs/(gT2)$ (Figure 9) or $hs/T2$ (Figure 1) can be observed – the curves in Figure 9 intersect with each other and the curves in Figure 2 are continuously separated. Weggel (1972) stated that Equation 21a was presented in a graphical form in the nomogram shown in Figure 2. Unfortunately, it is not the truth either in the case of Figure 2 or in the case of Figure 1. It appears that an analogous form of Figure 2 can be obtained only after an additional mathematical operation, consisting in implementation of a certain correction to the solution for $γbs$⁠. This correction comes from the need of additional application of the limiting wave steepness condition given in Equation 6b.

This correction can be performed analogously to the transformation procedure previously described in the first part of the present paper where the way of construction of the nomogram for breaker depth index, $γb$⁠, is presented (see Figures 4–6). However, it would be rather a very complicated and demanding mathematical task. Therefore, it is proposed to follow an alternative and very simple course of action, which is to insert into Equations 19c and 20b formerly calculated values of $γb=defHb/hb$⁠, already used during preparation of the nomogram presented in Figure 5.

Simultaneously, it must be remembered that the breaker depth index, $γb=defHb/hb$⁠, was presented as a function of the dimensionless parameter $Hb/(gT2)$⁠, whereas the breaker depth index of a wave breaking at a coastal structure, $γbs=defHb/hs$⁠, is given as a function of another dimensionless parameter $hs/(gT2)$⁠. A consistency of presentation of both nomograms for $γb$ and $γbs$ must be preserved. Therefore, for each value of $Hb/hb$⁠, taken from the data set of the nomogram shown in Figure 5 and inserted into Equation 19c, it is necessary to calculate a new value of the horizontal coordinate $hs/(gT2)$⁠. And thus, transforming the older form of the horizontal coordinate parameter:

a new form of the horizontal coordinate parameter $hs/(gT2)$⁠, in the nomogram of the breaker depth index of a wave breaking at a coastal structure, $γbs$⁠, must be calculated according to the following recipe:

Based on the above-presented algorithm, required computations have been made and their results are presented in the form of the sought nomogram shown in Figure 10. The influence of the limiting wave steepness condition can be easily recognised – the curve denoting a horizontal seabed surface (⁠$m=0.0$⁠) is not horizontal even for larger values of the nomogram abscissa $hs/(gT2)$⁠.

Massel (1992: p. 187) argued for the use of $Hb/hs=0.775$ for $hs/(gT2)>0.02$⁠, independently of the bottom slope inclination, m. This recommendation results from the fact that the nomogram presented in Massel (1992), being an exact copy of the nomogram given in SPM (1984), does not enable readout of the values of the breaker depth index of a wave breaking at a coastal structure, $γbs$⁠, for the horizontal coordinate $hs/(gT2)>0.02$⁠. A convergence of all the curves for larger values of $hs/(gT2)$ and the lack of consideration of the limiting wave steepness condition in Weggel’s (1972) solution (see Equations 21a and 21b) and in the low-resolution nomogram (see Figure 1) were an excuse for assuming a constant value of the breaker depth index (⁠$Hb/hs=0.775$ for $hs/(gT2)>0.02$⁠). The main solution nomogram for the breaker depth index of a wave breaking in front of a coastal structure, $γbs$⁠, obtained in the present paper, is illustrated in Figure 10, covering the entire theoretically possible range of the horizontal coordinate $0.00≤hs/(gT2)≤0.07$⁠. The results presented in Figure 10 indicate clearly that the assumption of constancy of $Hb/hs$ is a kind of simplification too large even for engineering practice.

In spite of Figure 10, for the readers’ convenience, the author has prepared additional three high-resolution nomograms, illustrated in A4 format, obtained for the first three subranges of the nomogram abscissa: $0≤hs/(gT2)≤0.01$⁠, $0.01≤hs/(gT2)≤0.02$⁠, and $0.02≤hs/(gT2)≤0.03$ – most important from the engineering practice and application point of view. All of these nomograms help to better visualise the runs of curves described by the function $γbs=f[hs/(gT2),m]$⁠. These component nomograms are accessible in the ‘Supplementary materials’ section.

A computer program code ‘BREAKER_DEPTH_INDEX’ (written in Fortran 77 programming language) has been additionally elaborated and included in the section of ‘Supplementary materials’. The programme gives the coastal engineers quite a handy tool, that is functional and fast, for computation of the breaker depth index of a wave breaking at a coastal structure. The program is based on the procedure described in the present paper and utilises a simple linear interpolation between values of the breaker depth index $γbs$⁠, obtained from the breaker depth index $γb$ (only sloped sea bottom without any coastal structure) together with appropriate replacement of the nomogram abscissa parameters.

Taking a train of regular waves in a laboratory flume as an example, it can be observed that the waves undergo shoaling effects when travelling over a sloped bottom and break at a certain depth. The breaker depth index for regular wave conditions can be found using one of many older (simpler) or modern (more accurate) equations, for example, Equation 16. However, it must be kept in mind that even under well-controlled laboratory modelling procedures, the wave breaking point changes over some distance and the breaker height varies from one wave to another. Smith and Kraus (1991), when reporting their tests on regular wave propagation, admitted that: ‘despite care in conducting the tests and use of the average value of the given quantity (i.e. over ten waves), wide scatter appeared in some quantities and must be considered inherent to the breaking process of realistic waves’.

Random waves consist of approaching waves which have different heights and break at different water depth. Therefore, under natural coastal conditions, wave breaking takes place in a relatively wide zone, named as the wave breaking zone or the surf zone. It is rather a challenging task to foresee precisely the point at which irregular waves initially break. There are many approaches to this problem existing in the professional literature. Goda’s breaking method is probably the best known method for estimating the significant wave height (⁠$Hs≡H1/3$⁠) within the surf zone. The relationship given by Goda (2010) has the following form:

where: $Hb$ is the significant height of wave starting to break, and $hbi$ is the water depth at which incipient wave breaking is observed. Using this equation, which already accounts for the limiting wave steepness condition, one can only find the approximate parameter of incipient wave breaking. Instead of the significant wave height, the root-mean-square wave height, $Hrms$⁠, can also be used for defining the breaker depth index, as demonstrated by Sallenger and Holman (1985).

Additional difficulties, in the case of irregular wave environment, are brought by a proper description of the wave plunge distance, $xp$ (see Figure 3). As stated by Gourley (1994): ‘No consistent simple means of predicting this distance (wave plunge distance) is available. If scaled in terms of breaker height, $xp$ is very sensitive to the bottom slope …’. Based on laboratory experiments conducted on three different plane slope beaches, Galvin (1968) proposed the relation depicted as Equation 20a in the manuscript. Among the newer proposals it is worth noting that Smith and Kraus (1991) used the surf similarity parameter and gave two equations for the breaker depth index, distinguishing between plane and barred slopes.

Taking into account the random nature of irregular waves generation process and its multiple dependence on different factors, it must be stated that at present there is no simple and reliable design tool, for example, represented by a nomogram or an equation, to assist the design process when the question of the breaker depth index of a wave breaking at a coastal structure, $γbs$⁠, is concerned. Therefore, in the case of irregular wave conditions, it seems quite reasonable to apply the Weggel nomogram, and its modified version discussed in the present paper, only as a rough estimation of the breaker depth index, $γbs$⁠, after correlating the breaker height with one of the characteristic wave heights, for example, the significant height or the root-mean-square height, of irregular wind-driven random waves. Another possibility to treat the problem would be using Goda’s (2010) Equation 26 for $γb$ and applying the procedure described in the paper leading finally to obtain nomograms for $γbs$⁠.

A derivation of the breaker depth index, $γb=defHb/hb=f[Hb/(gT2)]$⁠, of a wave breaking above a sloped sea bottom without any structure, assuming Miche’s limiting wave steepness condition, has been presented. A very good match between $γb$ curve and the envelope to Weggel’s family of lines has been found and illustrated in Figure 6, indicating benefits of use of the analytical envelope technique in the present paper.

A preparation method to build the nomogram for the breaker depth index, $γb=Hb/hb$⁠, as a function of $hb/(gT2)$ instead of $Hb/(gT2)$⁠, has been presented. The nomogram, illustrated in Figure 8, makes it possible to find directly the water depth at the point of wave breaking, $hb$⁠, contrary to the nomogram shown in Figure 5 and presented among others in CEM (2003), where the breaker height, $Hb$⁠, is the unknown (⁠$Hb$ appears in both horizontal and vertical coordinate parameters of the nomogram) and can be found by using, for instance, iterative calculations. A theoretical maximum range of applicability of the new nomogram’s abscissa has been found to be equal $[hb/(gT2)]max=0.077$⁠.

Weggel’s (1972) equation for the breaker depth index of a wave breaking at a coastal structure founded on a sloped sea bottom, $γbs=Hb/hs$ (still without assuming Miche’s limiting wave steepness condition), has been corrected and presented as a function of dimensionless parameter $hs/(gT2)$ (see Equation 24) instead of dimensional parameter $hs/T2$ used by Weggel (1972).

Based on the set of data used in preparation of the nomogram for the breaker depth index, $γb=f[Hb/(gT2),m]$ (see Figure 5), a step-by-step algorithm has been presented in order to build the final nomogram (see Figure 10) for the breaker depth index of a wave breaking at a coastal structure, $γbs=Hb/hs$⁠, assuming Miche’s limiting wave steepness condition and the maximum theoretical range of the nomogram abscissa.

A set of high-resolution practical nomograms has been finally prepared, enabling a trouble-free reading out the breaker depth index of a wave breaking at a coastal structure, $γbs=Hb/hs$⁠, assuming the extended range of nomogram abscissa up to $hs/(gT2)=0.03$⁠, and the bottom slope inclination changing stepwise from $m=0.0$ (horizontal bottom) to $m=0.20$⁠. Thereby, a coastal engineer can easily get a value of the breaker depth index, $γbs$⁠, much more precisely than before, and the extended abscissa range enables consideration of more practical cases. In addition, based on the whole calculation procedure explained in the present paper, a short computer program (Fortran 77) has been elaborated, allowing coastal engineers on very convenient and precise computing of the breaker depth index, $γbs$ for $0≤hs/(gT2)≤0.03$⁠. Both the nomograms and the program code are available from the ‘Supplementary materials’, which also include a small calculation example just to illustrate a practical use of the nomograms.

The supplementary material for this can be found online.