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ETA 77-2
Prediction of Irregular Wave Runup

by
John Ahrens

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COASTAL ENGINEERING CLESTION /

TECHNICAL AID NO. 77-2

July 1977

Approved for public release;
distribution unlimited.
U.S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING
RESEARCH CENTER
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F READ INSTRUCTIONS
REPORT DOCUMENTATION PAGE ee EAD NSE RU CTIONS
1. REPORT NUMBER 2. GOVT ACCESSION NO.) 3. RECIPIENT'S CATALOG NUMBER
CETA 77-2

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

Coastal Engineering
Technical Aid

6. PERFORMING ORG. REPORT NUMBER

PREDICTION OF IRREGULAR WAVE RUNUP

7. AUTHOR(s)

John Ahrens

10. PROGRAM ELEM
AREA & WORK U

NT, PROJECT, TASK

9. PERFORMING ORGANIZATION NAME AND ADDRESS E
NIT NUMBERS

Department of the Army
Coastal Engineering Research Center (CERRE-CS)
Kingman Building, Fort Belvoir, Virginia 22060
11. CONTROLLING OFFICE NAME AND ADDRESS
Department of the Army
Coastal Engineering Research Center
Kingman Building, Fort Belvoir, Virginia 22060
14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)

BL AZ)

12. REPORT DATE
July 1977

13. NUMBER OF PAGES
14

15. SECURITY CLASS. (of thie report)

UNCLASSIFIED

DECL ASSIFICATION/ DOWNGRADING
SCHEDULE

15a,

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

18. SUPPLEMENTARY NOTES

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

Design techniques Waves
Irregular wave runup Wind-generated waves
Runup

20. ABSTRACT (Continue on reverse side if necesaary and identify by block number)
A technique for predicting the runup distribution of wind-generated waves
is presented which extends the traditional method of runup prediction for
regular waves (constant period and height) presented in the Shore Protection
Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research
Center, 1975). The basic assumption is that runup has a Rayleigh distribution
of the type commonly associated with wave heights. This technique appears

conservative but reasonable for predicting extreme values of runup for design
(continued)

FORM
DD , aes 1473 EDITION OF t NOV 65 IS OBSOLETE UNCLASSIFIED
ea SP = eer So rade ce Be
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SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)

purposes. A method of correcting runup for slope roughness and porosity,
which is easier to apply than the method presented in SPM, is also presented.
Example problems are given using these techniques for both plain and composite
slopes.

2
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SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)

PREFACE

This report describes a technique for estimating the runup distribu-
tion of wind-generated waves, extending the method of runup prediction
for waves of constant height and period presented in the Shore Protection
Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research
Center, 1975). A method of correcting runup for slope roughness and
porosity, which is easier to apply than the SPM method, is also presented.

This report was prepared by John Ahrens, Oceanographer, Coastal Struc-
tures Branch, under the general supervision of Dr. R.M. Sorensen, Chief,
Coastal Structures Branch.

The author acknowledges the useful suggestions by Dr. R.M. Sorensen
and R.A. Jachowski, Chief, Coastal Design Criteria Branch, for improving
the manuscript.

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.

JOHN H. COUSINS

Colonel, Corps of Engineers
Commander and Director

CONTENTS

Page
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) ....... #5
SNA XOUTS ANID) ADESSO ES) Game) Geaetlcies salllo: cm a hate gE MGs 'o OO 6
ee UN ORODUGIUON Mare ccmmrenrcetcanch serine Sth <P tones) cc. Cou lfoy te) emt er PREM me 7
II PREDICTION OF THE DESIGN SIGNIFICANT WAVE RUNUP ........ 7
TT ASIC (GS Wste! ONO IDUGIUUIION 4 6 5 6 slo oval Gila wd Bl 6 6 tc 8
VE RAMP DEEDES TGN PROBREMGM cc Sect is fots cemige ote Gas aun s uenis eee
WAPSUMMAT Gm: Ras pasate cit ay gett eo: aa o habitnes. we Smee hos Sy Asiy sch eels Mee ee
TREERATURE GEDED: cc's S0"r ct a cy (ane Gem eceney TENCE eget scan meme emt
TABLE
Values of r for various slope characteristics .........-. 9
FIGURE
Relative runup, Rp/Rs ; or relative wave height, Hp/Hsg , asa
function of the probability of exceedance, p ...........- 10

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST)
UNITS OF MEASUREMENT

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

————— EEE

Multiply by To obtain
eines ess sss SS OO OS 0000S —Qmawjnww\nw_w,xs—
inches 25.4 millimeters

2.54 centimeters
Square inches 6.452 Square centimeters
cubic inches 16. 39 cubic centimeters
feet 30. 39 centimeters
0.3048 meters
square feet 0.0929 Square meters
cubic feet 0.0283 cubic meters
yards 0.9144. meters
Square yards 0.836 square meters
cubic yards 0.7646 cubic meters
miles 1.6093 kilometers
square miles 259.0 hectares
knots 1.8532 kilometers per hour
acres 0.4047 hectares
foot-pounds 1.3558 newton meters
millibars 1.0197 x 1073 kilograms per square centimeter
ounces 28.35 grams
pounds 453.6 grams
0.4536 kilograms
ton, long 1.0160 metric tons
ton, short 0.9072 metric tons
degrees (angle) 0.1745 radians
Fahrenheit degrees 5/9 Celsius degrees or Kelvins!

ITo obtain Celsius (C) temperature readings from Fahrenheit (F) readings,

use formula: C = (5/9) (F -32).
To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15.

SYMBOLS AND DEFINITIONS

water depth

water depth at toe of structure

wave height

equivalent deepwater wave height

wave height associated with a particular probability of exceedance
significant wave height

probability of exceedance

wave runup associated with a particular probability of exceedance

design significant wave runup; i.e., runup caused by a wave with
significant height and period

runup correction factor for roughness and porosity

wave period

PREDICTION OF IRREGULAR WAVE RUNUP

by
John Ahrens

I, INTRODUCTION

This report provides a technique for predicting the runup of irregu-
lar wind-generated waves of the type observed in most coastal waters.
No guidance for prediction of irregular wave runup is currently available
in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers,
Coastal Engineering Research Center, 1975) or other Corps publications.
Therefore, this interim guidance is provided until results of the Coastal
Engineering Research Center (CERC) laboratory and field studies of the
runup of irregular waves are available.

The approach is consistent with and parallels the SPM development of
runup prediction for regular waves of constant period and amplitude.
Although quite simple, this approach gives a reliable estimate of the
extreme values of irregular wave runup. Because of the method of formu-
lation, the runup estimates are conservative in that they yield somewhat
higher extreme values than would be predicted by more complex prediction
techniques; e.g., Saville, McClendon, and Cochran (1962). A conservative
approach is desirable because there are little field or laboratory data to
substantiate any prediction technique for irregular wave runup.

II. PREDICTION OF THE DESIGN SIGNIFICANT WAVE RUNUP

The technique for regular wave runup prediction in the SPM (Sec. 7.21)
is used to predict the runup for the design significant wave, That value
is dependent on slope roughness and porosity and an easier method to
determine this influence is given in this report (Sec. III). This adjusted
significant wave runup is then used to predict the irregular wave runup
distribution (see Sec. III).

The first step in calculating the design significant wave runup is to
select the design wave or waves. Selection is based on many factors
(discussed in the SPM, especially Sec. 7.12), but ultimately the problem
can be reduced to selection of a design significant wave height and period.
The runup predicted by SPM for a wave with the design significant height
and period will be referred to as the design significant runup, Rg.

Using the equivalent deepwater significant height and period for the
design wave, the runup for this wave on a smooth slope can be calculated
as shown in SPM (Sec. 7.21). When the runup is corrected for scale effects
(see Fig. 7-13 of the SPM), it is the R, for a smooth, impermeable
structure.

If the structure being designed is not smooth, roughness and porosity
may be accounted for by using the SPM method (Sec. 7.21) or by applying a

correction factor. The roughness and porosity correction factor, r, is

the ratio of the runup on a rough permeable or rough impermeable slope
divided by the runup on a smooth impermeable slope; i,e.,

R. (rough slope)
pl eee ee (1)
R, (smooth slope)

A compilation of r values used for selecting r is shown in a
Table from Battjes (1974). The table includes an entry by McCartney and
Ahrens (1975). The values of r as determined by these sources are
relatively consistent.

III. ESTIMATING THE RUNUP DISTRIBUTION

The approach assumes that the individual wave runup elevations have a
Rayleigh distribution of the type commonly associated with the distribu-
tion of wave heights at sea. Saville (1962), Saville, McClendon, and
Cochran (1962), van Oorschot and d'Angremond (1968), and Battjes (1971,
1974), suggest that a Rayleigh distribution for runup is plausible and
probably conservative for runup caused by naturally occurring wave condi-
tions. The runup distribution is then given by:

SO (2 (1/p) Me
paras +2 ’ (2)

Rs

where Rp is the runup associated with a particular probability of exceed-
ance, p. For example, assume a desire to know the 1-percent runup, i.e.,
the elevation exceeded by 1 percent of the runups, then p = 0.01 and
equation (2) yields:

This indicates that the l-percent runup would be about 52 percent greater
than R,. A graph of equation (2) showing R,/R, versus p is given in
a Figure. Note that the percent runup (or wave height) means the percent
exceedance and not the average value for the given percent as used in
Many sources. The figure or equation (2) can also be used to determine
the percent exceedance for wave heights (i.e., Hp/Hsg) having a Rayleigh
distribution.

IV. EXAMPLE DESIGN PROBLEMS

KK Rk Ok ek koe ek) ke A ek EXIAMPIPE jl ok ee ee ee ke RD

To illustrate the application of this approach to irregular wave run-
up prediction, consider the example given in SPM (p. 7-24).

Table. Values of r for various slope characteristics
(from Battjes, 1974).

Original source Slope characteristics r
Ce sae Eee eee Smooth, impermeable | 1.0
Shankin Concrete blocks 0.90
H.L. Delft Basalt-covered stone blocks 0.85 to 0.
CERC! Gobi blocks 0.85 to 0.
Franzius Grass 0.85 to 0.
CERC One layer of rubble 0.80
(impermeable foundation)
Shankin Set stone 0.75 to 0.
Shankin Round stones 0.60 to 0.
H.L. Delft Rubble 0.50 to 0.
H.R.S. Wallingford Rubble 9.50 to 0
Shankin Broken rubble 0.50 to 0.
CERC Two or more layers of rubble 0.50
Starosolszky Tetrapods 0.50

90

90

90

80

65

60

.55

55

1added to Battjes (1974) original table (from McCartney
and Ahrens, 1975).

TO

Seas |— af oI
[Te mils ep as
So

1.6

1.5

SR <eae
r 1.3 lak SH
~
= a Foe eT ee
= adie dilsories oes [ell PSs Lowes a
= Ami i

0.7 b} il

0.6

0.5 1.0 2.0 3.0 N

Probability of A ee p eg

Figure. Relative runup, Rp/Rs or relative wave height,
H p/Hs » as a function of the probability of
exeeetunces P-

GIVEN: An impermeable structure with a smooth slope of 1 on 2.5 and sub-
jected to a design wave, H = 7 feet, measured at a gage located where
the water depth, d= 15 feet. The design period, T = 8 seconds.
Design depth at the structure toe at high water, dg = 10 feet.
(Assume no change in the refraction coefficient between the structure
and the wave gage.)

FIND:

(a) The height above the stillwater level (SWL) to which the struc-
ture must be built to prevent overtopping by the design wave.

(b) The reduction in required structure height if uniform-sized
riprap is placed on the slope.

SOLUTION: Although the given wave is not specifically defined, it is
convenient to consider the wave as the significant wave of the design
storm. In addition, instead of no overtopping by the design wave,
assume that it is required to compute the elevation overtopped by only
1 wave in 20 during the design storm; i.e., p = 0.05. In the SPM,

R, was 21.3 feet; therefore, from equation (2):

gn051 /) MAUS nea In 20)/2 = 1.22
Raves aaursn eCedey Ro 2Mitog site

and R Gee 1.22 (21.3) = 26.1 feet.

To find (b) above, the roughness and porosity factor from the Table
is estimated as:

r = 0.50
This yields:

R, = Rg(smooth) x r = 21.3 (0.50) = 10.65 feet.

The 5-percent runup corresponding to Rg = 10.65 feet is

R 95 = 10.65 x 1.22 = 13.0 feet
kok ok k kk ok ok ok kok * kok * &* BXAMPLE 2 * * * * *® ROK Kk Ok kk Ok Ok &

Saville (1958) presented a method for determining Rg on composite
slopes using the results obtained for constant slopes (see SPM example,
pp. 7-33 to 7-37). The technique discussed in this report can be applied
to composite slopes and should give conservative results for [Rep al Or
if Rs is found to exceed the berm or slope break elevation. is can

be shown by using the conditions given in the SPM example.

GIVEN: A smooth-faced levee (cross section in SPM, Fig. 7-21) is sub-
jected to a design wave having a period, T = 8 seconds and an equiva-
lent deepwater height, HS = 5 feet. The depth at the structure toe,

d, = 4 feet.
Again, it is assumed that the given design height and period are the
significant values. It is also necessary to modify the question slightly

to account for the irregularity of actual surf conditions.

FIND: The elevation which will be exceeded by only 1 wave runup in 20,
R05:

SOLUTION: From example 1:

and from the SPM example, Rg =6.04 feet; therefore,

Ke = oH ore S 7/4 wecie

To determine if the above solution provides a conservative runup estimate,

estimate the runup of the 5-percent wave, H 05? i.e., the wave with a

height exceeded by only 1 wave in 20. This approach is adopted since it
is initially suspected that the largest waves cause the highest runups.
Assuming the Rayleigh distribution holds, the deepwater height of this
wave is:

H.o5 = 5 X 1.22 = 6.1 feet

and the wave period is most likely the same as for the significant wave;
i.e., T = 8 seconds, since waves higher than the significant wave usually
have a period close to the significant period. Following the procedure
given in the SPM example, the runup on the compound slope for H.95 is
5.7 feet, which is lower than Rs as determined in this example.
Although surprising, this result is logical. Because the fronting slope
is so flat (1 on 20), the larger wave breaks much farther offshore than
the significant wave and therefore loses considerably more energy before
encountering the structure. This analysis suggests that R95 = 7.4 feet
may actually be too conservative. However, another component in the
incident wave spectrum is probably responsible for the highest runups,;
e.g., the wave having the maximum breaker height in a water depth of 4
feet, the depth at the toe of the 1 on 3 slope. The maximum breaker
height for ds = 4 feet and T = 8 seconds is 5.24 feet, and following
the procedures in SPM this wave would run up about 7.4 feet, or approxi-
mately the value estimated for R.05.

I2

V. SUMMARY

This report presents a technique for estimating the wind-wave runup
distribution on structures by extending the traditional method of runup
prediction for regular waves developed in the SPM (Sec. 7.21). The
technique assumes that the runup has a Rayleigh distribution of the type
commonly observed for wave heights. Application of the technique to both
plain and compound slopes is shown by example design problems and appears
to give conservative but reasonable estimates of the runup distribution.
In situations where the offshore slope and depth cause the larger waves
to break offshore, the elevations predicted by the technique appear to
become increasingly more conservative as the percent exceedance becomes
smaller. As a practical guide for this type of situation, the runup will
probably not exceed elevations of about 1.4 to 1.5 Rg.

A method of correcting the influence of slope roughness and porosity
on wave runup, which is easier to apply than the SPM method, is also
presented.

LITERATURE CITED

BATTJES, J.A., ''Runup Distribution of Waves Breaking on Slopes," Journal
of the Waterways, Harbors and Coastal Engineering, Vol. 97, No. WW1,
Feb. 1971, pp: 91-114-

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

McCARTNEY, B.L., and AHRENS, J.P., "Stability of Gobi Block Revetment to
Wave Attack,'' TM-55, U.S. Army, Corps of Engineers, Coastal Engineering
Research Center, Fort Belvoir, Va., Oct. 1975.

SAVILLE, T., Jr., 'Wave Runup on Composite Slopes," Proceedings of the
Stath Conference on Coastal Engtneering, 1958, pp. 691-699.

SAVILLE, T., Jr., ''An Approximation of the Wave Runup Frequency Distri-
bution," Proceedings of the Etghth Conference on Coastal Engineering,
1962, pp. 48-59.

SAVILLE, T., Jr., McCLENDON, E.W., and COCHRAN, L.L., "Freeboard Allow-
ances for Waves on Inland Reservoirs," Journal of the Waterways and
Harbors Dtviston, Vol. 88, No. WW2, May 1962, pp. 93-124.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER,
Shore Protectton Manual, 2d ed., Vols. I, II, and III, Stock No.
008-002-00077-1, U.S. Government Printing Office, Washington, D.C.,
LOT Spel 6 Opp.

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, 1968, pp. 888-900.

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pue sseuysnoi adots 10z dnuniz 8uyzjDe1109 Jo poyze y ‘*(WdS) TenueW
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Aoy uozjOFpead dnuna jo poyjem ay Buzpueqzxe ‘sonem pazetouss-puta

JO uot{nqTzAsTp dnuniz ay} Buyjewyz4se 1oFz peqysosep sf enbzuyoe3 y

y, °*d :Xydez30TTqQTq
(Z-LL *ou S1eque9

yorieesoy Buytosuyzsug TeIseop *s'n — Pty TeIFuyDeL) “ITE : “d HL
*/L6, ‘293uUeD yoIeesey BufrseuTZuq Teyseop *s*n : “eA

‘1ToATeg 3104 — *sueiryy uyor hq / dnuni eaem ieTNZe11f Jo uoTzIOTpeag

uyor ‘sueryy

Le9 G=LL “ou Baresn” £0ZOL

“T-LL °OU PTY TeopuyseL ‘“taqueD Yyoteesey BuTiseuTSug_ Te seoD *s*n
tseftes “II “STIFL “I ‘“seAem purty *¢ ‘dnunz oven *z ‘sanem *[

*pejueseid
OsTe sz ‘poyjew yqs ey} ueyq ATdde 03 aazsea st yotym Aq TSOI10d
pue ssouy8noz adots 103 dnuniz 3uyzq2ea1109 jo poyjeu y *(WdS) Tenuey
uoF}9970Ig BTOYS 942 UT pequeseid potied pue AYystey Jueqsuod Fo saaem
oj uot OFpead dnuna jo poyjeu ayj Yuppueqxe ‘sanen pazersues-puta

JO uoT{ngya13sFp dnunz oyQ Buy ewT3se 10y peqtazosap st aenbtuyseq y

, °d :kydeazorTqrg
(Z-LL *ou t1equeg

yoivesey Zuftissuzsugq Te3seoD “S*n — PTY TeoOTUYyDeL) “TTT : “d HF
“LL6| ‘ze,UeD YoIeesey ZuTIeeuTSug TeqseoD *S*n : “eA

‘ITOATeG 340g — *suezyy uyor Aq / dnunz aaem aepTN3eaxI~ Jo uoT,OTperg

uyor ‘suelyy

L£c9 Ga Le nou. Parson” £020L

“Z-LL °OU PTY TeOTUyIe_ ‘*1eqUeD YOIeeSey BuTIeeuTZuq Te IseoD ‘s*p
tsetles “Il “STIFL “I ‘“seaem puty *¢ ‘dnunz onary *z ‘soaey ‘|

*pojqueseid
osTe sft ‘poyjzew Was ey, ueyq ATdde 03 zaztsea st yotym ‘AqTSo10d
pue sseuysnox edots 103 dnunz guzjoe1109 Jo poyjou y *(jaS) TenueW
uoT}D99301g PLOYS 9Yy} UT pequeseiad potied pue 4yZtey queqzsuood Jo soaeA
Aoz uoz zoTpead dnunz jo poyjew oy Buppueqxe ‘seaem pazetoues-puTmM

JO uoT{NGTAAsTp dnuni ayA Butjewyz3se 1IoyJ peqtassep st enbtuyoe7 y

4, ‘d :hydears0tTqta
(Z-LL "Ou $193uU99

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“/L6[ ‘tequeD YyoTReSey BuTIesuTsuq TeJseOD *S*N : “BA

‘iToATeg 310g — *sueryy uyor Aq / dnunz even azepn3er1~ Jo uozIOTpesd

uyor ‘suslyy

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