Irregular wave runup on smooth slopes

u cS Army
Coast. Enq. Res-Ctr,
CETA

CETA 81-17

Irregular Wave Runup on Smooth Slopes

by
John P. Ahrens

COASTAL ENGINEERING TECHNICAL AID NO. 81-17

DECEMBER 1981

WHO!

DOCUMENT
COLLECTION

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~Sterine 8°

Approved for public release;
distribution unlimited.

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COASTAL ENGINEERING

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Us Fort Belvoir, Va. 22060

No, Sl

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this material

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The findings in this report are not to be construed
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IRREGULAR WAVE RUNUP ON SMOOTH SLOPES

AU THOR(s)

John P. Ahrens

10. PROGRAM ELE
AREA & WORK

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and identify by block number)

Irregular waves Wave rundown
Smooth plane slopes Wave runup

20. ABSTRACT (Continue on reverse side if necesaary and identify by block number)

The results of several laboratory studies have been used to develop a
method to estimate the wave runup and rundown on plane, smooth slopes caused
by irregular wave action. Curves and equations are presented which can be
used to compute the 2-percent runup, significant runup, mean runup, and
approximate lower limit of rundown. A procedure is suggested for adapting
the smooth-slope results to wave runup on rough and porous slopes. Example
problems illustrate the use of the material presented.

FORM
DD | jan 73 1473 ~~ EDITION OF 1 NOV 65 1S OBSOLETE UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

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PREFACE

This report presents a method for estimating the magnitude and distribution
of wave runup and rundown on plane, smooth slopes caused by irregular wave
action. Within the method's range of applicability it supersedes Section
7.212, "Irregular Waves," of the Shore Protection Manual (U.S. Army, Corps of
Engineers, Coastal Engineering Research Center, 1977); CETA 77-2 "Prediction
of Irregular Wave Runup" by John P. Ahrens; and CETA 78-2 "Revised Wave Runup
Curves for Smooth Slopes" by Philip N. Stoa. It also supersedes the parts of
CETA 79-1 "Wave Runup on Rough Slopes," by Philip N. Stoa, which estimate wave
runup on rough and porous slopes by adjusting the runup for similar wave con-
ditions on smooth slopes using a rough-slope correction factor.

This report was prepared by John P. Ahrens, Oceanographer, under the gen-
eral supervision of Dr. R.M. Sorensen, Chief, Coastal Processes and Structures
Branch, Research Division.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th Congress,
approved 31 July 1945, as supplemented by Public Law 172, 88th Congress,
approved 7 November 1963.

TED E. BISHOP
Colonel, Corps of Engineers
Commander and Director

VI

APPENDIX
A

B

CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI).
SYMBOLS AND DEFINITIONS. . . «. «© 2 «© © «© © e © @ © © © ©

TNERODU GION ey eulethathie: ploustottotie = of edb athie. io fietiviets eo! tell +e sel entiontie
IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES .... «© « « »
IRREGULAR WAVE RUNDOWN . . 2. «© «© «© © © © © © © © © © © © © @
APPLICATION OF RESULTS TO ROUGH AND POROUS SLOPES. .... -»
EXAMPIEHPROBGEMS ciive cls hey eypeieeienye ehle fe ie: s te «8 eule aieine
SUMMARY. . . » «© »

PRTRRAEURE GREED Se ctceeet tahoe) et M ister is se! ss 6 ee 8 eee

RUNUP SCALE-EFFECT CORRECTION FACTOR, k, FOR SMOOTH SLOPES

RUNUP REDUCTION FACTOR, r, FOR VARIOUS TYPES OF ROUGH
AND POROUS STRUCTURES. . . 2. «© « © «© © © © © © © © © © ow

RUNUP SCALE CORRECTION FACTOR, k, FOR VARIOUS TYPES OF
ROUGH AND POROUS STRUCTURES. ... . . «© © «© © © «© «© © «© «© «

TABLES

1 Regression coefficients for runup parameters R2/Hg, R,/Hs,
eunidy Restle py seteimce Mon cease Nog mrad Wecmem ef uc) URI ect Pu al te). sh ek bin ca? ete oy

2 Values of the runup parameters for example probleml........

1 Irregular wave
plane, smooth

2 Irregular wave
plane, smooth

3 Irregular wave
plane, smooth

4 Irregular wave
plane, smooth

5 Irregular wave
plane, smooth

6 Irregular wave

FIGURES

runup
slope

runup
slope

runup
slope

runup
slope

runup
slope

runup

parameters versus wave
of 1 on 1,'ds/H, > 3.

parameters versus wave
of 1 on 1.5, ds/Hs > 3

parameters versus wave
of don) 2, do/Hs, > 3).

parameters versus wave
of I on 2.5, ds/Hs > 3

parameters versus wave
of 1 on 35 dg/Hg = 3) e

parameters versus wave

plane, smooth slope 1 on 4, ds/Hg > 3...

steepness

steepness

steepness

steepness

steepness

steepness

7 Rs/Hs versus the surf parameter for 3 < ds/Hg

8 Rdgg/Hg versus the surf parameter

11

- 10

14

CONVERSION FACTORS, UeS- CUSTOMARY TO METRIC (SL) UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted to
metric (SL) units as follows:

Multiply

inches

by
2524

To obtain

millimeters

square inches
cubic inches

feet

square feet
cubic feet

yards
Square yards

cubic yards

miles
square miles

knots
acres

foot-pounds

millibars
ounces

pounds

ton, long
ton, short

degrees (angle)

Fahrenheit degrees

2254
66452
16.39
30.48
0.3048
0.0929
0.0283
0.9144
0.836
0.7646

1.6093
259.0

1.852
0.4047

1.3558

1.0197
28235

453.6
0.4536

1.0160
0.9072
0.01745

By

x 1073

centimeters
square centimeters
cubic centimeters
centimeters
meters
Square meters
cubic meters
meters
square meters
cubic meters

kilometers
hectares

kilometers per hour
hectares
newton meters

kilograms per square centimeter

grams

grams
kilograms

metric tons
metric tons
radians

Celsius degrees or Kelvins!

1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings,
use formula: C = (5/9) (F -32).

To obtain Kelvin (K) readings, use formula:

Ko= (5/9) Ce =32)\ +2273. 15).

Rdog

SYMBOLS AND DEFINITIONS
water depth at the toe of the slope or structure on which runup occurs
acceleration of gravity, 32.2 feet per second squared
significant wave height at the toe of the structure
runup correction factor for scale effects
deepwater wavelength, Lo = gTp*/2n
mean runup

significant runup, i.e., average runup of the highest one-third of
wave runups

2-percent runup, i.e., elevation above the stillwater level exceeded by
2 percent of the runups

98-percent rundown, i.e., depth below the stillwater level that is just
greater than 98 percent of the rundowns

rough-slope runup correction factor, ratio of rough~slope runup to
smooth-slope runup, all other conditions the same

period of peak energy density of the wave spectrum

significant wave period, i.e., average period of the highest one-third
of waves

angle formed between the slope of the structure and the horizontal

surf parameter, & = [@iz/ta)'!2 cot 6]7!

IRREGULAR WAVE RUNUP ON SMOOTH SLOPES

by
John P. Ahrens

I. INTRODUCTION

This report provides guidance on the magnitude and distribution of wave
runup and rundown elevations caused by irregular wave conditions similar to
those occurring in nature. The results presented are for plane, smooth struc-
tures with relatively deep water at the toe of the structure. For these con-
ditions this report supersedes earlier guidance in Section 7.212 of the Shore
Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering
Research Center, 1977) and Ahrens (1977) which indicate that irregular wave
runup has a Rayleigh distribution. Within the range of test conditions this
report also supersedes Stoa (1978a) and the parts of Stoa (1979) which esti-
mate wave runup on rough and porous slopes by adjusting the runup on a smooth
slope by a correction factor. The range of test conditions covered in this
report is discussed in the next section.

II. IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES

Three sources of data were used in establishing the methods presented in
this report: van Oorschot and d'Angremond (1968), Kamphuis and Mohamed (1978),
and Ahrens (1979) which discussed data recently collected at the Coastal Engi-
neering Research Center (CERC). The conditions considered are a structure
with a plane, smooth slope fronted by a horizontal bottom offshore. The water
depth at the toe of the structure is relatively deep, i.e., 3 < dg/H, < 12,
where d, is the water depth and Hg the significant wave height at the toe
of the structure. When there is relatively deep water at the toe of the struc-—
ture the offshore slope of the bottom has little influence on the wave condi-
tions and therefore little influence on the wave runups. This lack of influence
indicates that the runup results presented can be applied to situations where
there is an offshore slope. Since the water depth also has little influence on
wave runup for conditions when dg/H§ > 8 (Stoa, 1978a), where Hj is the deep-
water, unrefracted wave height, Stoa's finding suggests that the results of this
study should be good for dg/H, > 12.

Three runup parameters were chosen to characterize the runup distribution
caused by irregular wave conditions, i.e., the mean runup, R, the significant
runup, R,g, and the 2-percent runup, Rj. The significant runup is the aver-
age runup of the highest one-third of wave runups and the 2-percent runup is
the elevation exceeded by 2 percent of the wave runups.

Figure 1 shows trend-line curves for R2/Hg, Rg/Hg, and R/Hg for a plane,
smooth slope of 1 on 1. These parameters are plotted as a function of the
irregular wave steepness parameter, Hs/gTp’, where T is the period of peak
energy density of the wave spectrum and g_ the acceleration of gravity. The
approximate relationship between Tp and the average period of the significant
waves, Ts, is given by Goda (1974) as

Tp = 5 (@)5) qs (1)

Denotes + 1.0 std. dev.
about trend line

Figure 1. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope of 1 on 1, d,/Hs >.3.

Figures 2, 3, 4, 5, and 6, which are similar to Figure 1, show trend lines
for slopes of 1 on 1.5, 1 on 2, 1 on 2.5, 1 on 3, and 1 on 4, respectively.
The trend lines in Figures 1 to 5 are all of the general form

H fu. \?
Rx = Cy + Co =. ar C3 a
Hg ety 8Tp

(2)

where Rx represents Ro, Rg, or R, and C,, Co, and C3 are dimensionless re-
gression coefficients. In some cases Cy or C3 is zero; if C3 is zero the
trend line is straight.

Since a calculator or a-computer may be more convenient for calculating
the runup parameters than using the figures, Table 1 provides a tabulation of
the regression coefficients, along with some statistical parameters which can
be used to evaluate how well the curves fit the data. The standard deviation
is the standard deviation of the data about the trend-line curves and is shown
in Figures 1 to 6 to give an indication of the magnitude of the scatter about
the curves. The coefficient of variation is the standard deviation divided by
the mean value of Ry/Hg. Using the coefficient of variation to determine the
percent scatter indicates that Rg/Hg can usually be estimated within the range
of +5 to 10 percent about the trend-line curves; Ro/H, and R/Hg can be esti-
mated within the range of +10 to 15 percent about the curves.

ee

Boe
5
er

LTA

Runup/Hs
~m
on

: He
(ae
: aS
0.0 = =
2x10” 4x10" 6x10 Bx10°
H,/gTp

Figure 2. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope of 1 on 1.5, davis > Se

Denotes + 1.0 std. dev.
about trend line

Runup/Hs
Mm

nae
ae
Ez
ca
Ee
pars
an
aie a
ie
oS

Figure 3. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope of 1 on 2, d,/H, > 3.

} Denotes + 1.0 std. dev.
about trend line

2
~
a5

Runup/H,
| cy
: ie

Cey
oY aa

Bc
coe eis

Figure 4. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope of 1 on 2.5, d,g/H, > 3.

Denotes + 1.0 std. dev.
about trend line

Figure 5. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope of 1 on 3, dois oS.

10

i Denotes + 1.0 std dev.
about trend line

3.0

R
2.0 "Mis: (61
wo
a5
a
= Rs
2 ceo Hsia ast
Fy,
10 $= 0.84 €
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
2
H,/gTp

Figure 6. Irregular wave runup parameters versus wave steepness
for a plane, smooth slope 1 on 4, dg,/Hg > 3.

Table 1. Regression coefficients for runup parameters
R,/Hs, Rs/Hs, and R/Hs (see eq. 2)is

Regression coefficients
Cot 6 Cj Co C3 Std. dev. Coeff. of

variation

eae ie eee ee Ra He

1.0° 2.32 7.15 x 10! 0 0.343 0.134

2052" 9 1295 102 0 0.487 0.156

Bak lgphl afl yes aly ) 0.421 0.123

3.39 1.29 =x 102 -=1.61 x 10* 0.420 0.118

3.70 0 =1570) = <0" 0.415 0.120

4.0 3.60 -2.22 x 104 0) 0.330 0.117
R./H,

1.0) 1.34 6.61 x 10) 0 0.133 0.085

1.38 3.18 x 102 <=1.97 x 10% 0.195 0.094

1.64 3.57 * 102 =3.09 x 10* 0.136 0.059

1.94 2.79 x 102 -3.21 x 10% 0.184 0.078

2.11 1.87 x 102 -2.67 x 10% 0.190 0.081

4.0 2.52 -7.94 x 10! ) 0.122 0.053
R/Hs

1.0 0.71 1.10 x 10? -8.07 x 103 0.150 0.157

0:75) 197 x 10>) —114—x: 10* 0.143 0.119

2.0 0.93 2.42 x 10% -1.93 x 104 0.142 0.101

1.00 2.78 x 10% —3.13)< 10? 0.141 0.099

3.0 1.19 2.09 x 102 -2.96 x 10% 0.181 0.123

440° 147 725 x TO =1a70sa0F 0.127 0.085

Figure 6, for a slope of 1 on 4, is somewhat different than Figures 1 to
5 for steeper slopes. Plunging waves become the dominant breaker type on the
1 on 4 slope, indicating that wave runup can be predicted using a type of for-
mula suggested by Hunt (1959) and used by van Oorschot and d'Angremond (1968).
Figure 6 shows trend-line curves, using equation (2), for the less steep wave
conditions, i.e.,

and a Hunt-type formula is used for the steeper wave conditions, i.e.,
Hg/gTp* > 0.003 where plunging waves dominate. The Hunt-type formulas for Fig-
ure 6 are given by the equations

R

= 1.61 5 (3)
Hy

R

oa aD) 4c = 1G
s

EVE V ns (5)
Hs

where the surf parameter, €, is given by

Ee 1 ae tan 0
(Hg/L,)/2 cot @ (g/L) !/?

Lo is the deepwater wavelength given by

and cot 6 is the cotangent of the angle 9 between the structure slope and the
horizontal.

Figure 7 provides a different perspective and additional insight on the
trends to be expected for irregular wave runup. The R,/Hg curves from Figures
1 to 6 have been transferred to Figure 7 and plotted versus the surf parameter,
—&, to show the influence of breaker characteristics on runup. When € < 2.0,
most of the larger waves in the incident wave train plunge directly on the
structure and Rg/Hg decreases with increasing H,/gT 2 and increasing cot 0.
This plunging wave region is where a Hunt-type formula (Hunt, 1959) such as
equations (3), (4), and (5) is valid. When &€ > 3.5, no waves plunge on the
structure indicating a standing wave condition or surging wave region. The
influence of Hg/gTp? and cot ® on Rs/Hg is reversed for surging waves as

l2

Transition
Region Surging Region
(Standing Waves Against Structure)

Plunging Region
(Waves Plunge Directly
on Structure) Cot §=2.0

Cot@=1.0

0 10 20 30 4.0 5.0 6.0 7.0
Surf Parameter, €

Figure 7. Rg/Hg versus the surf parameter for dg/Hg > 3.

compared to plunging waves; i.e., Rg/Hg increases as Hs/gTp* increases and cot
8 increases. The reversal of influence creates a transition region, 2.0 <

— < 3.5, where there is little net influence of He/gTp* and cot @ on Rg/Hg,.

It is in this transition region that the largest values of Rg/Hg occur, prob-
ably because the most nonlinear surging waves occur in this region. Figure 7
identifies these regions and shows the runup trends. Equations (3), (4), and
(5) can be used on slopes flatter than 1 on 4 as long as plunging waves pre-
dominate, i.e., & < 2.0.

All the results in this report were obtained in relatively small-scale
laboratory studies and must be corrected for scale effects (Stoa, 1978a).
The correction for scale effects of wave runup on smooth slopes can be found
in Stoa (1978b) (shown in App. A). Example problem 1 in Section V illustrates
the method of applying this correction.

The results in Figures 1 to 7 are all presented in terms of the significant
wave height at the toe of the structure, Hg, rather than the deepwater, un-
refracted wave height, Hj. If it is desired to convert the results of this
study to deepwater conditions, Hg should be multiplied by the shoaling coef-
ficient, given in Appendix C of the SPM (U.S. Army, Corps of Engineers, Coastal
Engineering Research Center, 1977), calculated using dg and Tp to obtain an
estimate of the deepwater, unrefracted significant wave height.

IIL. IRREGULAR WAVE RUNDOWN

Irregular wave rundown is characterized by the 98 percentile rundown, Rdgg,
i.e., the rundown depth below the stillwater level which is greater than 98
percent of the wave rundowns. The irregular wave rundown parameter, Rdgg is
analogous to the runup parameter, Ro, since only 2 percent of the rundowns
are lower than Rdgg. Figure 8 shows the trend of the relative rundown,

Rdgg/H, as a function of the surf parameter, &, and the approximate upper

[5

Approx. upper limit of
dato scatter

Approx. lower limit
of data scatter

Surf Parameter, €= TLS cre

Figure 8. Rdgg/H, versus the surf parameter.

and lower limits of data scatter about the trend-line curve. The trend-line
curve for relative rundown is given by the equation

Rdgg —2.46/é&
= -2.32e (6)
He

The absolute value of relative rundown is small for small values of the surf
parameter since the plunging waves which dominate these conditions cause con-
siderable wave setup. As the surf parameter increases a standing wave develops
against the structure and the relative rundown approaches -1.75, although values
occasionally as low as -2.25 were observed. Equation (6) provides a simple way
to estimate the approximate lower limit of rundown.

There is no scale-effect correction factor specifically developed for wave
rundown, so it is recommended that the correction factor for wave runup be
applied to rundown as illustrated in example problem 2 in Section V.

IV. APPLICATION OF RESULTS TO ROUGH AND POROUS SLOPES

The results given in this report can be applied to plane, rough- and
porous-slope structures, if there is.relatively deep water at the toe of the
structure (as discussed previously in Sec. II). To apply these results it is
necessary to have a reliable estimate of the rough-slope runup correction fac-
tor, r, which is the ratio of wave runup on a rough or porous slope to the

14

runup on a smooth slope, all other conditions being the same (Stoa, 1978a).
Normally, r is determined in laboratory experiments using monochromatic wave
conditions but it appears that r factors determined in this manner can also
be applied to irregular wave conditions (Battjes, 1974). Values of r for
various types of rough and porous slopes are given by Stoa (1979) (shown in
App. B).

Often wave runup on rough slopes must be corrected for scale effects and
the correction factors are given in Stoa (1979) (shown in App. C). Example
problem 3 illustrates how the results presented in this report can be applied
to a rough and porous slope and the method of applying the rough-slope scale-
effect correction factor.

V. EXAMPLE PROBLEMS

Kok KK Kk RK RK KK OK OK OR & & & EXAMPLE PROBLEM 1 * *& *& ¥ ¥ KKK KKK KARE

This example illustrates the use of the runup equation, Figures 1 to 6, and
the recommended method of interpolation between slopes.

GIVEN: A plane, smooth slope of 1 on 2.75 is subjected to irregular wave
action. The significant wave height, significant wave period, and water
depth at the toe of the structure are 6.0 feet (1.83 meters), 7.0 seconds,
and 24.0 feet (7.3 meters), respectively.

FIND: R, Rg, and Rp for the given conditions. Would there be substantial wave
overtopping if the freeboard of the structure were 20.0 feet (6.10 meters)?

SOLUTION: Since there is no figure or set of coefficients for the runup
equation (eq. 2) for a slope of 1 on 2.75 it is necessary to compute R, Rg,
and Rg for slopes of 1 on 2.50 and 1 on 3.00 and interpolate between them.
To start, calculate the period of peak (maximum) energy density, Tp, using
equation (1).

Tp = 1.05. T,.= 1.05 (7.0) = 7.35 seconds
Then compute the steepness parameter, He/elyo
H
Ea pe great eats 0.00345
Bloor S252 i-25) =

Using the above value of steepness in equation (2) with the coefficient
given in Table 1 allows the computation of R,/H,. For example, to calcu-
late Ro/H, for a 1 on 2.5 slope

Z
Bq tees 1129200200345) [-16,100(0.00345)*] = 3.64
Ss

The above value of Rj/Hg can be confirmed, using Figure 4. Therefore,

Rg = 3.64(Hg) = 3.64(6.0) = 21.8 feet (6.64 meters)

The other runup parameters Rg and R can be calculated in a similar manner,
then used for interpolation to give the values of the runup parameters for
the 1 on 2.75 slope as shown in Table 2.

15

Table 2. Values of the runup parameters for example problem 1.

cot 0 Ro/Hg Ro Rs /Hs Re R/Hy R
(£t) (£e) (ft)
2.50 3.64 21.8 5152 i be a ae
3.00 3.49 21.0 2263 14.6 1.56 9.4
25 -- 21.4! -- 14.9! -- 9.41

ltInterpolated value.

The interpolated values in Table 2 should be corrected for scale effects
to yield the required answer. The scale correction factor for a slope of 1
on 2.7/5 is 1.125 (see App. A); therefore,

Ro = 21.4 (1.125) = 24.1 feet (7.35 meters)
Rg = 14.9 (1.125) = 16.8 feet (5.12 meters)
R = 9.4 (1.125) = 10.6 feet (3.28 meters)

A freebaord of 20.0 feet falls between Ry» and Rg, so the structure
crest would not be overtopped frequently, probably by less than 10 percent
of the waves. It is, therefore, expected that the volume of overtopping
would not be great.

It is difficult to determine how high a smooth structure would have to
be to prevent all wave overtopping but a reasonable estimate would be

Rmax ~ Ro + Hs
where Rmax is the elevation of the maximum runup.
kK KK KK KK KK KOK OK KX EXAMPLE PROBLEM 2 * * ¥ & KX KX KK KK KK KARE

This example illustrates how to calculate the approximate lower limit of
rundown.

GIVEN: A plane, smooth 1 on 2.50 slope is subjected to irregular wave action.
The significant wave height, significant wave period, and water depth at the
toe of the structure are 7.0 feet (2.13 meters), 8.0 seconds, and 30.0 feet
(9.14 meters), respectively.

FIND: Rdgg for the above conditions; this is the approximate lower limit of
wave rundown.

SOLUTION: The period of peak energy density is

Tp = 1.05(T,) = 1.05 x 8.0 = 8.40 seconds

and the surf parameter is

1 i
E= 1/2 = vf = 2.87
(Hg/Lo) cot 6 {7.0/ 132.2 x (8.4)71/20} (2.5)

16

Using this value cf §& in equation (6) gives the relative rundown, i.e.,

RaSé -2.46/E
a = -2.32e = -0.99

Ss

which can be confirmed in Figure 8. Then

Rdgg = 2(7.0)(-0:99) = 6-9 feet. (—2.10 meters)
and using Appendix A to correct this rundown for scale effects gives
Rdgg (corrected) = -6.9(1.128) = -7.8 feet (-2.38 meters)

The same scale correction factor used for runup is used for rundown.

kok Kk & kK kK kK Ok OK OK OK OK ® & EXAMPLE PROBLEM 3 * *¥ *¥ ¥ KK KK KKK KK KK

This example illustrates how the results of tests with irregular waves on
smooth slopes can be applied to situations where the structure is rough and
porous.

GIVEN: A rubble-mound breakwater is to be built with a slope on the seaward
face of 1 on 2 which will be overtopped by wave action only occasionally
under the design conditions. The design conditions include a significant
wave height, significant wave period, and water depth at the toe of the
structure of 15.0 feet (4.57 meters), 12.0 seconds, and 45.0 feet (13.72
meters), respectively. The core of the breakwater will be slightly above
the design water level, i.e., a high core breakwater.

FIND: The height at which the breakwater will only occasionally be overtopped
during the design conditions.

SOLUTION: The period of peak energy density is

Tp = 1.05(Ts) = 1.05 (12.0) = 12.6 seconds
and the steepness parameter is

s N50
7 = aa
Bie 6052. 202,16)

Using equation (2) with the coefficients in Table 1 for a plane, smooth slope
of 1 on 2 and RjH/g gives

ee

Ss

=.3.2083 + 71.879 (0.00293) = 3.42

(this value can be checked in Fig. 3) and

Ro = 3.42(15.0) = 51.3 feet (15.64 meters)

17

The runup reduction factor, r, for rubble-mound breakwaters with high
cores is 0.52 (see App. B) and the scale-effect correction factor is 1.06
(see App. C) so Rg for the breakwater is

Ro (breakwater) = 51.3(0.52) 1.06 = 28.3 feet (8.63 meters)
Rg and R are found in a similar manner to be

Rg (breakwater) = 20.0 feet (6.10 meters)

12.2 feet (3.72 meters)

R (breakwater)

These calculations indicate that if the freeboard were 28.3 feet only 2 per-
cent of the waves with a Hg = 15 feet and Tg = 12 seconds spectrum would
overtop the structure while a freeboard of 12.2 feet would allow about half
the waves to overtop. A freeboard equal to Rg, i.e., 20 feet, will satisfy
the condition of only occasional wave overtopping since about 13 percent of
the waves would be expected to overtop the breakwater.

KR KKK KKK KK KK KKK KKK KK KKK KKK KKK KK KKK KK KKK

VI. SUMMARY

Equations and curves are presented for computing three runup parameters and
one rundown parameter for plane, smooth slopes exposed to irregular wave condi-
‘tions where dg/Hg > 3. These parameters are R2, the elevation exceeded by
only 2 percent of the runups; Rs, the average runup of the highest one-third
of the wave runups; R, the mean runup of all the runups; and Rdgg, the
depth below the stillwater level which is just greater than 98 percent of the
rundown. Example problem 1 illustrates the use of equation (2) in computing
the rundowns, parameters, and the method of interpolation for runup on slopes
not specifically covered in this report. Example problem 2 illustrates the
method of computing rundown. Example 3 illustrates how the study results for
smooth slopes can be applied to rough and porous slopes, in this case to com-
pute the desired freeboard for a rubble-mound breakwater.

LITERATURE CITED

AHRENS, J.P., "Prediction of Irregular Wave Runup," CETA 77-2, U.S. Army, Corps
of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July
WOT e

AHRENS, J.P., "Irregular Wave Runup," Proceedings of the Conference on Coastal
Structures '79, American Society of Civil Engineers, Vol. II, 1979, pp.
998-1019.

BATTJES, J.A., "Wave Runup and Overtopping,'' Technical Advisory Committee on
Protection Against Inundation, Rijkswaterstaat, The Hague, Netherlands, 1974.

GODA, U., "Estimation of Wave Statistics from Spectral Information," Pro-
ceedings of the Symposium on Ocean Wave Measurement and Analysts, Vol. I,
1974, pp. 320-337.

HUNT, I.A., "Design of Seawalls and Breakwaters," Journal of the Waterways and
Harbors Divtston, Vol. 85, No. WW3, Sept. 1959, pp. 123-152.

KAMPHUIS, J.W., and MOHAMED, N., "Runup on Irregular Waves on Plane, Smooth
Slope," Journal of the Waterway, Port, Coastal, and Ocean Division, Vol.
104, No. WW2, May 1978.

STOA, P.N., "Reanalysis of Wave Runup on Structures and Beaches," TP 78-2,
U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort
Belvoir, Va., Mar. 1978a.

STOA, P.N., "Revised Wave Runup Curves for Smooth Slopes," CETA 78-2, U.S.
Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir,
Va., July 1978b.

STOA, P.N., "Wave Runup on Rough Slopes," CETA 79-1, U.S. Army, Corps of Engi-
neers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1979.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore
Proteetton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1,
U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.

VAN OORSCHOT, J.H., and D'ANGREMOND, K., "The Effect of Wave Energy Spectra
on Wave Runup," Proceedings of the 11th Conference on Coastal Engineering,
American Society of Civil Engineers, 1968, pp. 888-900.

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APPENDIX B

RUNUP REDUCTION FACTOR, r, FOR VARIOUS TYPES OF
ROUGH AND POROUS STRUCTURES (Stoa, 1979)

I. VALUE OF r FOR QUARRYSTONE RUBBLE-MOUND STRUCTURE (HIGH CORE)

te =sO}D2

Quarrystone
armor layer

tee Stones thick, )
random placement

Fa ®&Oe Ola
CERES,
a“ a ot ae N
Saeko a ey
ae
Ze
h d
075< —2=11 Core "e s
$ Underlayers

22

II. VALUES OF r FOR CONCRETE ARMOR UNITS
1. Embankment.

a. Gobi Blocks.

r°~"0-93 for’ H'/k 2 or —H/K_ 476
(oh r

v ! '
(use Hj when qo/HS > 3 and H_ when d/h; <5)

34¢ in (010m) fal
roo
7 ro
! ke
Lf!
Fetevation of Gobi Block Plan View of Gobi Block

b. Stepped Slopes.

Values of r for stepped slopes.

Type of step Slope (cot 6) ri
Vertical risers 15S} 0.75
2.0 0.75
3.0 0.70
Rounded edges 3.0 0.86

lj < Hi/k, < 12 where k, is the height of
the riser.

23

7A

Embankment and Rubble Mound.

Values of r for concrete armor units.

Armor unit and Length dimension,

placement method k,

Armor-layer
thickness
(No. of units)

Tetrapod
Random |
he
Uniform
Quadripod
Random |
he
Uniform hs
Tribar
Random
Uniform

Modified cube
Random

Uniform
Uniform

Uniform

1.3 to

24

III. VALUES OF r FOR QUARRYSTONE EMBANKMENT

Slope micot 6) H/k,, Te
eS 3 to 4 0.60
2:55 3 to 4 0.63
3.5 3 to 4 0.60
5.0 3 0.60
5.0 4 0.68
5.0 5 0.72

25

APPENDIX C

RUNUP SCALE CORRECTION FACTOR, k, FOR VARIOUS
TYPES OF ROUGH AND POROUS STRUCTURES (Stoa, 1979)

Structure Type k
Quarrystone, rubble-mound breakwater 1.06

Quarrystone, riprap revetment 1.00

Concrete armor units, rubble mound
or revetment 1.03

26

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