Part C | 30.2
670 Part C Coastal Design
der Meer [30.47] formulation. The structure configuration for P D 0:4 was assumed and never tested by van
der Meer [30.26].
The acceptable level of damage is parameterized by
the dimensionless damage factor S which is the ratio
of the area eroded to the square of the nominal rock
diameter, that is, S D A e =.D
2
n ). The zero damage level
is S D 2 (for cot ˛ D 1:5, 2, and 3) and S D 3 (for cot
˛ D 4, 5, and 6). Failure levels (for a 2D n50 armor-layer
thickness) are when the secondary layer is exposed beneath the armor layer. These failure levels are S D 8
(cot ˛ D 1:5, 2), S D 12 (cot ˛ D 3) and S D 17 (cot
˛ D 4, 5 and 6). The van der Meer [30.47] equations
can be used to develop damage (fragility) curves for
a given design, stable rock size D n by calculating S as
the wave heights increase above the design wave height.
See [30.3] for intermediate damage levels S.
The surf similarity parameter m brings the mean
wave period T m into the formulation. For normal, single peaked spectrums, T m 0:82 T p , where T p is the
spectral peak wave period.
Finally, the number of waves N z is a variable in
the van der Meer [30.47] formulation. For short storms
lasting 47 h, N z D 2500 is recommended for mean
wave periods from 5 to 10 s. The maximum N z is 7500
giving 1530 h storms for waves in the 515 mean period range.
The development of spreadsheets with the structure
slope angle ˛ as the key design variable greatly facilitates the application of these equations and tables for
design of the stable size, or weight (mass) of the armor
layer.
There are many variables, coefficients, and exponents in the original [30.26, 47] formulations as seen
in (30.21)–(30.23). Graduate students at the Delft Technical University and engineers at Delft Hydraulics (now
Deltares) have been conducting research investigation
over the past 1520 years to further understand and
improve the original vdM formulas. The following summarizes some past and recent developments as found in
the literature with more (probably) to follow.
The publication by van der Meer [30.26] was his
Ph.D. dissertation at the Technical University of Delft.
In it, van der Meer [30.26] acknowledges the earlier
work of Thompson and Shuttler [30.50] as the starting point for this basic research using irregular, random
waves in the laboratory. He recognized that the coefficient 6:2 (defined as coefficient C pl for the plunging
breaker (30.21)) and coefficient 1:0 (defined as coefficient C s for the surging breaker (30.22)) were stochastic, mean values with standard deviations of mean of 0:4
and 0:08, respectively. Hence, the coefficient in the transition (30.23) separating both formulas was C pl =C s D
6:2. van der Meer [30.26] also recognized that waves
moving into shallow water would be depth-limited in
height and suggested using the Rayleigh distribution
ratio of H 2% =H s D 1:4 to adjust the C pl and C s coefficients and to use H 2% in the formulas. Finally, van der
Meer [30.26] called for more research to understand the
permeability parameter P and stone shape (roundness)
on static stability.
It has been recognized that most of the over 300
tests conducted by van der Meer [30.26] were with relatively deep water at the toe of the structure. Additional
tests (over 200) of rock stability with shallow foreshores
have been conducted at Delft Hydraulics [30.51], Smith
et al. [30.52] as reported by van Gent et al. [30.53].
These additional tests demonstrated that:
1. Less scatter was evident if the mean wave period T m
was replaced by T m1:0 , the spectral wave period to
find the surf similarity parameter m1:0 .
2. Less scatter was evident if the ratio of H 2% =H s is
employed because of depth-limited wave breaking.
3. The additional data reduced the standard deviations
of the mean values of the modified C pl coefficient in
the original formulations of van der Meer [30.26].
For shallow water wave conditions, van Gent
et al. [30.53] recommended using C pl D 8:4 .H s =H 2% /
and C s D 1:3 .H s =H 2% /. These values are also given in
the Rock Manual [30.3] and require/allow the user to
specify the appropriate .H s =H 2% / ratio. Note that these
modified values for C pl and C s for shallow water will
result in smaller (lighter) rock for stability than the original, deep water [30.26] values since the .H 2% =H s / ratio
will be less than 1:4. This is in keeping with shallow
water wave breaking and truncation of the wave spectrum from that of the deep water, Rayleigh distribution.
The Rock Manual [30.3] defines shallow water when
the water depth h is less than 3 H s at the toe of the structure. Van Gent and Pozueta [30.54] suggested using
H stoe D 0:7H so (deep water wave) as the shallow water
condition. No partial safety factors have yet been published for the shallow water, van der Meer formulations.
More recent research efforts have focused on the
permeability parameter P. The values originally assigned by van der Meer [30.26] have no physical
definition except the general trend that as P increases,
the structure has more pore space volume to absorb
wave energy. Verhagen et al. [30.55] used a numerical model (volume of fluid, VOF) method to calculate
wave run-up on permeable and impermeable cores and
correlated P as a function of run-up ratios and the surf
similarity parameter. Kik et al. [30.56] showed with new
laboratory experiments that the P D 0:4 configuration
in Fig. 30.12 is correct in the van der Meer [30.26] formulas. Note that P is an important variable in (30.23)
670 Part C Coastal Design
der Meer [30.47] formulation. The structure configuration for P D 0:4 was assumed and never tested by van
der Meer [30.26].
The acceptable level of damage is parameterized by
the dimensionless damage factor S which is the ratio
of the area eroded to the square of the nominal rock
diameter, that is, S D A e =.D
2
n ). The zero damage level
is S D 2 (for cot ˛ D 1:5, 2, and 3) and S D 3 (for cot
˛ D 4, 5, and 6). Failure levels (for a 2D n50 armor-layer
thickness) are when the secondary layer is exposed beneath the armor layer. These failure levels are S D 8
(cot ˛ D 1:5, 2), S D 12 (cot ˛ D 3) and S D 17 (cot
˛ D 4, 5 and 6). The van der Meer [30.47] equations
can be used to develop damage (fragility) curves for
a given design, stable rock size D n by calculating S as
the wave heights increase above the design wave height.
See [30.3] for intermediate damage levels S.
The surf similarity parameter m brings the mean
wave period T m into the formulation. For normal, single peaked spectrums, T m 0:82 T p , where T p is the
spectral peak wave period.
Finally, the number of waves N z is a variable in
the van der Meer [30.47] formulation. For short storms
lasting 47 h, N z D 2500 is recommended for mean
wave periods from 5 to 10 s. The maximum N z is 7500
giving 1530 h storms for waves in the 515 mean period range.
The development of spreadsheets with the structure
slope angle ˛ as the key design variable greatly facilitates the application of these equations and tables for
design of the stable size, or weight (mass) of the armor
layer.
There are many variables, coefficients, and exponents in the original [30.26, 47] formulations as seen
in (30.21)–(30.23). Graduate students at the Delft Technical University and engineers at Delft Hydraulics (now
Deltares) have been conducting research investigation
over the past 1520 years to further understand and
improve the original vdM formulas. The following summarizes some past and recent developments as found in
the literature with more (probably) to follow.
The publication by van der Meer [30.26] was his
Ph.D. dissertation at the Technical University of Delft.
In it, van der Meer [30.26] acknowledges the earlier
work of Thompson and Shuttler [30.50] as the starting point for this basic research using irregular, random
waves in the laboratory. He recognized that the coefficient 6:2 (defined as coefficient C pl for the plunging
breaker (30.21)) and coefficient 1:0 (defined as coefficient C s for the surging breaker (30.22)) were stochastic, mean values with standard deviations of mean of 0:4
and 0:08, respectively. Hence, the coefficient in the transition (30.23) separating both formulas was C pl =C s D
6:2. van der Meer [30.26] also recognized that waves
moving into shallow water would be depth-limited in
height and suggested using the Rayleigh distribution
ratio of H 2% =H s D 1:4 to adjust the C pl and C s coefficients and to use H 2% in the formulas. Finally, van der
Meer [30.26] called for more research to understand the
permeability parameter P and stone shape (roundness)
on static stability.
It has been recognized that most of the over 300
tests conducted by van der Meer [30.26] were with relatively deep water at the toe of the structure. Additional
tests (over 200) of rock stability with shallow foreshores
have been conducted at Delft Hydraulics [30.51], Smith
et al. [30.52] as reported by van Gent et al. [30.53].
These additional tests demonstrated that:
1. Less scatter was evident if the mean wave period T m
was replaced by T m1:0 , the spectral wave period to
find the surf similarity parameter m1:0 .
2. Less scatter was evident if the ratio of H 2% =H s is
employed because of depth-limited wave breaking.
3. The additional data reduced the standard deviations
of the mean values of the modified C pl coefficient in
the original formulations of van der Meer [30.26].
For shallow water wave conditions, van Gent
et al. [30.53] recommended using C pl D 8:4 .H s =H 2% /
and C s D 1:3 .H s =H 2% /. These values are also given in
the Rock Manual [30.3] and require/allow the user to
specify the appropriate .H s =H 2% / ratio. Note that these
modified values for C pl and C s for shallow water will
result in smaller (lighter) rock for stability than the original, deep water [30.26] values since the .H 2% =H s / ratio
will be less than 1:4. This is in keeping with shallow
water wave breaking and truncation of the wave spectrum from that of the deep water, Rayleigh distribution.
The Rock Manual [30.3] defines shallow water when
the water depth h is less than 3 H s at the toe of the structure. Van Gent and Pozueta [30.54] suggested using
H stoe D 0:7H so (deep water wave) as the shallow water
condition. No partial safety factors have yet been published for the shallow water, van der Meer formulations.
More recent research efforts have focused on the
permeability parameter P. The values originally assigned by van der Meer [30.26] have no physical
definition except the general trend that as P increases,
the structure has more pore space volume to absorb
wave energy. Verhagen et al. [30.55] used a numerical model (volume of fluid, VOF) method to calculate
wave run-up on permeable and impermeable cores and
correlated P as a function of run-up ratios and the surf
similarity parameter. Kik et al. [30.56] showed with new
laboratory experiments that the P D 0:4 configuration
in Fig. 30.12 is correct in the van der Meer [30.26] formulas. Note that P is an important variable in (30.23)