CETA 78-2
CAb- A058 407)
Revised Wave Runup Curves
for Smooth Slopes
by
Philip N. Stoa
COASTAL ENGINEERING
TECHNICAL AID NO. 78-2
July 1978
eunors
DOCUMENT |
\ COLLECTION /
&
v4
Approved for public release;
distribution unlimited.
U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
+c RESEARCH CENTER
330 Kingman Building
Fort Belvoir, Va. 22060
National Technical Information Service
ATTN: Operations Division
5285 Port Royal Road
Springfield, Virginia 22151
The findings in this report are not to be construed as an official
epartment of the Army position unless so designated by other
ithorized documents.
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
READ INSTRUCTIONS
REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM
T. REPORT NUMBER 2. GOVT ACCESSION NO, 3. RECIPIENT'S CATALOG NUMBER
CETA 78-2
4. TITLE (and Subtitle) 6 5. TYPE OF REPORT & PERIOD COVERED
Coastal Engineering
Technical Aid
REVISED WAVE RUNUP CURVES FOR SMOOTH SLOPES
7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s)
Philip N. Stoa
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK
AREA & WORK UNIT NUMBERS
Department of the Army
Coastal Engineering Research Center (CEREN-CD) F31234
Kingman Building, Fort Belvoir, Virginia 22060
V1. Neaa ee Ement Die Ae AND ADDRESS 12. REPORT DATE
ae © t e a July 1978
oastal Engineering esearch Center Bi, 13 Sa
Kingman Building, Fort Belvoir, Virginia 22060 242)
PLP CLO
14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report)
UNCLASSIFIED
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SCHEDULE
DISTRIBUTION STATEMENT (of this Report)
Approved for public release; distribution unlimited.
DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)
- SUPPLEMENTARY NOTES
KEY WORDS (Continue on reverse side if necessary and identify by block number)
Breakwaters Runup
Coastal engineering Scale effects
Coastal structures Wave runup
ABSTRACT (Continue am reverse sides if neceseaary and identify by block number)
Results of previous tests of monochromatic wave runup on smooth structure
slopes were reanalyzed. The runup results for both breaking and nonbreaking
waves are presented in a set of curves similar to but revised from those in
the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal
Engineering Research Center, 1977). The curves are for structure slopes
fronted by horizontal and 1 on 10 bottom slopes. The range of values of
(continued)
DD, een 1473 ~—s Ep Tion oF 1 NOV 65 IS OBSOLETE UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
d,/H, was extended to d,/H} = 8; relative depth (d,/H}) is important even
for d,/H) > 3 for waves which do not break on the structure slope.
A flow chart is given to assist in choosing the proper figure and in inter-
preting the results when applied to untested bottom slopes (i.e., bottom slopes
flatter than 1 on 10).
Also given are example problems and a curve for scale-effect corrections.
2 UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
PREFACE
This report describes a means of determining wave runup on coastal
structures having uniformly sloping, smooth surfaces. The report is
based principally on small-scale test results and analyses of Saville
(1956) and Savage (1959) as reanalyzed by Stoa (1978). The work was
conducted under the coastal engineering research program of the U.S. Army
Coastal Engineering Research Center (CERC).
The technical guidelines presented in this report supersede the design
runup curves for smooth slopes given in the Shore Protection Manual (SPM)
(U.S. Army, Corps of Engineers, Coastal Engineering Research Center,
1977). The revised runup curves given here include a wider range of
relative depth, d,/H3. These results are based on experiments using
regular waves. Ahrens (1977a, 1977b) presented methods for estimating
runup and overtopping, respectively, from irregular waves based on results
of regular wave testing.
The report was prepared by Philip N. Stoa, Oceanographer, under the
general supervision of Robert A. Jachowski, Chief, Coastal Design
Criteria Branch.
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
II
CONTENTS
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
SYMBOLS AND DEFINITIONS .
INTRODUCTION.
RUNUP CURVES. : j
1. Smooth Structure Fromeed by Homimoncall lpoceen :
2. Smooth Structure Fronted by 1 on 10 Bottom Slope
and Zero Toe’ Depth (d, = 0). :
Smooth Structure Fronted by 1 on 10 Bottom Slope
and Toe Depth Greater than Zero (d, > 0)
Ww
MAXIMUM RUNUP .
SMOOTH-SLOPE SCALE-EFFECT CORRECTION.
EXAMPLE PROBLEMS.
LITERATURE CITED.
TABLES
Example runup for T = 7 seconds, constant depth, and
(Coe ene = LO 2eSiec
Example runup for T = 13 seconds, constant depth, and
CAS nav
Example runup for constant wave steepness, Bele = 0.0101.
= 16 feet.
Summary of maximum runup for different conditions
FIGURES
Definition sketch of variables applicable to wave runup .
Relative runup for smooth slope on horizontal bottom;
d,/H3 = 5
Relative runup for smooth slope on horizontal bottom;
d./HZ = 2
Relative runup for smooth slope on horizontal bottom;
apts = Be
Relative runup for smooth slopes on 1 on 10 bottom;
d,
= 0; d/a} = 3
Page
31
31
33
33
10
12
13
14
15
10
11
12
13
Relative
cl. = OS
Relative
Lepil & O«
Relative
Lop iy 2 Ov
Relative
Oni 2 Os
Relative
JO, 2. 0).
CONTENTS
FIGURES--Continued
runup for smooth slopes
; d/H> =5.
runup for smooth slopes
Cys ts
runup for smooth slopes
55 Gales 2 O26
runup for smooth slopes
Spe de HH esel.0)
runup for smooth slopes
53 d,/HS eal) Washi
runup for smooth slopes
oF d,/H) = So >
on
on
on
on
on
on
1
Flow chart for the evaluation of wave
Runup scale-effect correction factor,
on
on
on
on
on
10
10
10
10
10
bottom;
bottom;
bottom;
bottom;
bottom;
bottom;
Page
16
LY
18
IY)
20
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted
to metric (SI) units as follows:
Multiply by To obtain
inches 25.4 millimeters
2.54 centimeters
square inches 6.452 square centimeters
cubic inches 16.39 cubic centimeters
feet 30.48 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 1.852 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) 0.01745 radians
Fahrenheit degrees 5/9 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: K = (5/9) (F -32) + 273.15.
SYMBOLS AND DEFINITIONS
water depth
water depth at toe of structure
acceleration of gravity (32.2 feet per second squared or
9.81 meters per second squared)
wave height
the deepwater wave height, neglecting refraction, equivalent to
the wave height, H, measured in a given water depth
shoaling coefficient, H/H*
runup scale-effect correction factor
wavelength
deepwater wavelength; wavelength in water depth, ad, such that
d/h = O05
horizontal length of slope fronting toe of structure
runup; the vertical rise of water on structure face resulting
from wave action
wave period
bottom slope; used for the slope fronting a structure and is
different from the structure slope
structure slope; may be beach slope if runup on the beach face
is being investigated
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REVISED WAVE RUNUP CURVES FOR SMOOTH SLOPES
by
Philip N. Stoa
I. INTRODUCTION
Wave runup is the vertical distance above stillwater level (SWL)
reached by a wave incident to a structure or beach. Prediction of wave
runup on coastal structures is necessary to determine an adequate crest
elevation to prevent overtopping or to help determine the extent of
overtopping. Wave runup curves for structures with either smooth or
rough slopes have previously been presented in the Shore Protection
Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research
Center, 1977). Runup data of Saville (1956) and Savage (1959), together
with data from other reports, have been reanalyzed (Stoa, 1978). This
report presents revised smooth-slope runup curves which vary in certain
regions from those presented in the SPM. A scale-effect correction
curve is also given for application to smooth-slope runup.
Wave runup is primarily a function of characteristics of the
structure and incident wave; wave characteristics are also a function
of water depth and bottom slope. The variables are shown in Figure 1 and
aLemdetinedsas mes a GUNUpI: Te) sanclerot Structure slopes ads." water
depth; d,, water depth at toe of structure; 8, angle of bottom slope
at the structure toe; and £, horizontal length of the bottom slope
seaward of the structure toe. L and H are the wavelength and wave
height, respectively, as measured in a water depth, d. The same wave
may be described by an equivalent deepwater wave (d/L 2 0.5) for which
the dimensions would be Lg, and Hj. Lo is the deepwater wavelength
and HS is the equivalent unrefracted deepwater wave height. L, may
be determined if the wave period, T, is known (Ly = gT2/2m); this report
uses pT? as the principal measure of deepwater wavelength. Hi is used
because it avoids the problem of defining the wave height in varying
depths over a sloping bottom where the wave may already have broken. The
wave height in deep water is related to wave height in a shallower depth
by the shoaling coefficient, H/H3i or Kg. The shoaling coefficient
and wavelength, L, may be determined from Tables C-1 or C-2 in the SPM
when Lo and the required depth are known.
The runup curves are given for three different cases: (a) horizontal
bottom at the structure toe; (b) 1 on 10 sloping bottom at the structure
toe, with a zero toe depth (d, = 0); and (c) 1 on 10 sloping bottom at
the structure toe, with toe depths greater than zero (d, > Os Case ()
has, generally, the potential for the largest waves attacking the struc-
ture. A bottom slope of 1 on 10 is relatively steep for ocean coastlines,
and its occurrence would be restricted to beach faces with coarse sediments
(see Fig. 4-33 in the SPM), backshore areas subject to flooding, or some
nearshore areas. However, most bottom slopes would be flatter than 1 on
10. Experimental data for runup on structures fronted by flatter slopes
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circumstances are given in later sections.
The incident wave characteristics seaward of the toe of the bottom
slope are partly determined by the corresponding water depth and are
important in determination of runup. The methods presented in Sections
II,2 and II,3 are designed to account for the incident wave character-
istics at the toe of the bottom slope as determined in model experiments.
Natural underwater slopes are rarely so well defined; straight-line
approximations of irregular slopes should be determined by the designer.
Intersections of the straight lines will define the location of a change
in slope.
II. RUNUP CURVES
1. Smooth Structure Fronted by Horizontal Bottom.
Relative runup, R/H} for a smooth structure fronted by a horizontal
bottom is given in Figures 2, 3, and 4 for specific values of relative
depth, d,/HS. As shown by comparing the figures, relative runup on the
flatter slopes is not a function of dg/H§. However, relative runup on
the steep slopes is sensitive to depth effects; relative runup for a
given wave steepness, H}/gT~, is largest at the lowest dg/H} value.
Thus, proper consideration of depth effects must be included in design.
Relative depth values of 2 < dg/H$ < 3 may occur for structures on
horizontal bottoms, but experimental data are limited. Figure 2
(dg/HZ = 3) is recommended for cases in which d,/H3 < 3. Large d,/Hg
values may occur, for example, in reservoirs; runup determinations for
d,/HS > 8 should be based on Figure 4 (d,/Hg = 8).
2. Smooth Structure Fronted by 1 on 10 Bottom Slope and Zero Toe
Depth (ds = 0).
When dg = 0, wave conditions are determined using the depth, d, at
the toe of the 1 on 10 bottom slope. Figures 5, 6, and 7 show the results
for d/HS (mot dg/HZ) values of 3, 5, and 8 with a 1 on 10 bottom slope.
Runup on a structure fronted by a beach slope flatter than 1 on 10°
would be expected to be less than indicated in Figures 5, 6, and 7 for
comparable wave conditions. However, these figures are recommended for
use when a flatter bottom slope is present and d, = 0.
3. Smooth Structure Fronted by 1 on 10 Bottom Slope and Toe Depth
Greater than Zero (dg > 0).
Design curves for runup on a smooth structure with d, > 0, fronted by
a 1 on 10 bottom slope, are given in Figures 8 to 1l. The curves apply
to cases where the relative bottom-slope length is Cj = 0.5). “For values
Of C/ii< OS) but) tor high da/H values (agi dae 2s 5) .therrunup
values from figures for structures on horizontal bottoms (Figs. 2, 3, and
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4) should be used as upper bounds of relative runup on structures fronted
by a 1 on 10 slope with the same d,/H} value. In the case of £/L < 0.5
with low values of ide /Hoes (cae. mOLOelametc.) ut. Shouldubesexpected
that relative runup will be somewhat higher than predicted from the curves
(Figs. 8 to 11), and probably not exceeding 15 to 20 percent higher.
However, the effect of the length of a 1 on 10 bottom slope diminishes
as the structure slope decreases, and effectively ceases to be significant
for cot 6 2 4. These comments are incorporated in a flow chart (Fig. 12)
for determining which figure to use to find the runup on a structure
fronted by a sloping bottom.
Because there are insufficient data available for cases where bottom
slopes are flatter than 1 on 10, it is recommended that the curves given
in this report, applicable to structures fronted by 1 on 10 bottom slopes,
be used; in most cases, results are expected to give higher estimates of
R (see Fig. 12). For the larger d./H3 values (e.g., d,/H, > 2.5),
relative runup on structures fronted by gentle bottom slopes will be
equal to or less than that given in Figures 2, 3, and 4 (horizontal
bottom) for the appropriate d,/H4 value. Relative runup on structure
slopes flatter than 1 on 4 is largely unaffected by changes in bottom
slope. Relative runup on steep structures fronted by a gentle bottom
slope will be equal to or less than values given in Figures 8 and 9 but
may be slightly higher than those given in Figure 10 (d,/Hg = 1.5).
ITI. MAXIMUM RUNUP
This section discusses the maximum runup from regular waves when a
range of conditions is possible. Maximum runup from irregular waves is
not discussed, but an approach to estimation of maximum runup from
irregular waves is given by Ahrens (1977a). In his method, runup result-
ing from a significant wave is determined from design curves such as
given here, and then runup for the irregular waves is assumed to follow
a Rayleigh distribution.
Maximum runup, R, for a range of regular wave conditions, is not
necessarily associated with the maximum relative runup, R/H3. For
structures sited on horizontal bottoms, and for a gtven wave steepness,
He /enes both the maximum relative runup and the maximum dimensional
runup occur at the minimum value of d,/H%.
For structures sited on a 1 on 10 sloping bottom, maximum dimensional
runup, R, may or may not be coincident with the maximum relative runup
determined for a range of wave conditions. If depth, d,, and wave
steepness are assumed constant, then maximum relative runup occurs when
1.0 < d./HS £ 1.5, but maximum dimensional runup, R, is found when
d,/H} is a minimum (in this report, when d, > 0, then (de/H3) min = 0.6).
In cases where a bottom slope flatter than 1 on 10 is present, for a
given wave steepness, the maximum relative runup will occur for somewhat
higher d,/H} values (1.5 < d,/H} < 2.0). However, if wave height, Hj,
and wave steepness are held constant, the maximum dimensional runup, R,
will be coincident with maximum relative runup as d,/H3 varies (i.e., as
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23
d, changes). The maximums (R/HS and R) may occur at any value of d,/Hg
(including d,/Hi = 0) depending on the wave steepness being considered.
Runup ae eae Sonn occur at Hones eae AES values of d e/g (sO <
d,/HS < 1.5) for high values of eon but at low walines OE Gla/is\s
or ton values of Riots
For a gtven wave pertod and constant depth, d, (with wave steepness
varying as d,/H4 varies), maximum dimensional runup is generally not
coincident with maximum relative runup; furthermore, the maximum dimen-
Sional runup may occur at other than the minimum d,/H3 value.
The designer of a structure subject to runup will usually have a
range of wave conditions for which maximum runup must be determined. The
preceding discussion emphasizes the need to determine the maximum actual
runup by finding the runup for each of several wave conditions. Example
problem 3 (Sec. V) highlights some of the relationships discussed here
and shows the maximum runup values for different sets of initial wave
conditions.
IV. SMOOTH-SLOPE SCALE-EFFECT CORRECTION
The smooth-slope runup curves plotted in Figures 2 to 11 are based
on small-scale wave-flume tests. A limited number of large-scale tests
(Saville, 1958) indicated scale effects were present in the runup results.
Figure 13 presents values of the correction factor, k, as a function
of structure slope; the curve is modified from that given in the SPM, and
is extended over steeper slopes.
Selection of a particular structure slope may be dependent on evalua-
tion of runup on different slopes. The trends in runup on different
structure slopes are presumed correct as given by the design curves
(Figs. 2 to 11). Comparisons of runup for different structure slopes
should be based on the design curves, with the scale-effect correction
applied only to the final selected runup value.
V. EXAMPLE PROBLEMS
The following example problem solutions use Tables C-1 or C-2 in the
SPM and the applicable curves in this report.
BA A 2s nd Ge ee Ey ee ta ee ENP RIPON SLIEYL JL te cS) RS C2 RO RP ns RS te ep te Ne Se
GIVEN: An impermeable structure has a smooth slope of 1 on 3 and is
subjected to a design wave, H = 8 feet (2.4 meters), measured at a
gage located in a depth, d = 30 feet (9.1 meters). Design wave
period is T = 8 seconds. The structure is fronted by a 1 on 90
bottom slope extending seaward of the point of wave measurement.
Design depth at structure toe is d., = 25 feet (7.6 meters). (Assume
no wave refraction between the wave gage and structure.)
24
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(9 409)adojs ainyoniys
FO FO VO CoO - Fo me)
00'!
aeaeee cO |
oa (EEES
sea le r= b0'|
Eee =
eae: : 901
eee SSccs
a : =| 80'|
= ot |
eaeeeeee Sees e===— (0) |
3 segsseus goes cae:
: H aaa
+ - 20
— te oI
eee = ||
Eth soessSs
AEE : eS
25
FIND: The height above SWL to which the structure must be built to
prevent overtopping by the design wave.
SOLUTION: The wave height must be converted to a deepwater value.
Using the depth where wave height was measured, calculate
Gi et ob an tn lo Dieta pe lapeeee Oe
L, milan | (G2.A/AMt » So12(@)e
f= 0.08255 -
Lo
To determine the shoaling coefficient, H/H} Table C-1 in the
SPM is used, and
= 0.9406 .
H6
Therefore,
Hf eee eee ee
@ 0.9406 0.9406
nh 8 feet °(2.6 meters)
Calculate, also,
H!
EOL Ae SE SEE NE Quinto
ar $2.,.D((8)7
and, for d, = 25 feet
d 25
— = —_ = 2.94
He as)
The bottom slope is very gentle (1 on 90). Assuming that the
slope approximates a horizontal bottom, the appropriate set of
curves for d,/H} = 2.9 is Figure 2 (for dg/H{ = 3). For a 1 on
26
3 slope and
The runup, uncorrected for scale effects, is
Ke (Boil) (eis)
= (2.1) (8.5)
R = 17.9 feet (5.5 meters)
The scale-correction factor, k, can be determined from
Figure 13, and, for cot 6 = 3, the correction factor is
k = 1.12.
Thus, the corrected runup is
R = (1.12) (17.9) = 20.0 feet (6.1 meters)
xe Kk Kk kK Kk kk & * * * EXAMPLE PROBLEM 2 * * * * * * & * & * & & *& &
GIVEN: An impermeable, smooth, 1 on 2 structure is fronted by a 1 on
10 bottom slope. Toe depth for the structure is Ga 2 IO ese (G
meters), but the bottom slope extends seaward to a depth of 50 feet
(15.2 meters), beyond which the slope is approximately 1 on 100.
The design wave approaches normal to the structure and has a height
of H = 9 feet (2.7 meters) and period of T = 9 seconds, measured at
a depth of 55 feet (16.8 meters).
FIND: The height of wave runup using the appropriate set of curves.
SOLUTION: The wave height given is not the deepwater wave height; it
is measured, however, above the gentle 1 on 100 bottom slope which
approximates a horizontal surface. To determine the shoaling coeffi-
cient, K> for the location of measurement, calculate
GU Pune!
eT2/2n
ES
9
55
(5.12) (9)2
=
0.1326 .
Cll,
From Table C-1 in the SPM,
Therefore,
H 9 S
al? = = ——— = 9.82 feet (3.0 meters)
O Kg 0.9162
d
US On & 1.0 ,
He 9.82
and
He 9.82
2. = —_-—=___ = ().00377 .
ane | (B2.2) (9) ey
Because there is a steeply sloping bottom fronting the structure,
the value of &/L must be determined:
£ = (50 - 10)(10) = 400 feet (122 meters)
Next determine the wavelength in water depth of 50 feet
(the depth at the toe of the 1 on 10 slope). For
days EO as, |,
L (32.2/21) (9) 2
and from Table C-1,
=0.1585 .
|e
Therefore,
6 Lk BIS ae Bowie (96.1 meters)
ay S-osisss
28
Then,
(eS
w
HS
w|o
* |oO
sf
oO
thus,
L
L
The appropriate set of design curves is then determined;
the flow chart in Figure 12 shows that Figure 9 has the appro-
priate curves, and that the results are presumed correct at
model scales. From Figure 9, for H}/gT? = 0003538.
The runup is
R = = (HS) = (3.0) (9.82)
R = 29.5 feet (9.0 meters)
For cot @ = 2, the scale-correction factor, from Figure 13, is
k= 1.136:
Thus, the corrected runup is
R= (15186) (9.5) 2 55.5 ese (Os meeeies))
See kak Maken xT K eke x EOD Ee PROIBIEEM) Sucks kits (cece oe) ek Rp ene
GIVEN: A design geometrically similar to that in example problem 2,
where an impermeable, smooth, 1 on 2 structure is fronted by a 1 on
10 bottom slope. Toe depth for the structure is d, = 10 feet, but
the bottom slope extends seaward to a depth of 50 feet beyond which
29
the slope is approximately 1 on 100. However, a range of wave
periods and deepwater wave heights are known; el LG feet (4.9
meters).
FIND: Maximum runup for three different wave conditions: Tyg, = 7
seconds; Tmgy = 13 seconds; and constant wave steepness, Hh /gT? =
0.0101, with Tyg, = 7 seconds.
SOLUTION: For any given d,/H} value, the design curves show that
relative runup is highest for the longest wave period (or the
lowest wave steepness, ED) However, for constant toe depth,
d,, and for constant wave steepness, the largest wave height
(or lowest d./H3 value) usually results in the largest absolute
runup, R. When a sloping bottom is present, and wave period
and toe depth (dg) are held constant, the maximum runup may occur
at other than the minimum d,/H4, value. Thus, runup for a range
of d,/H5 values should be investigated.
In the following development, preliminary determinations
of runup are not corrected for scale effect. Only the final
runup, as determined for selected wave conditions and structure
slope, is corrected.
(a) The maximum wave height given is Hj} = 16 feet; for
this location, the resultant dg/Hg value is
a,
ig | 6
which is approximately the lowest value used in Figures 8 to 11.
The maximum runup may be determined by constructing a table for
varying conditions. Because the maximum wave period is less here
than in example problem 2, L is also less; thus, @/L > 0.5 and
Figures 8 to 11 may be used. Furthermore, Figure 12 indicates
that the results in Figures 8 to 11 are approximately correct, to
models scales SFOrida — OU teethwl — Seconds asand gT2 = 1,577.8
feet. Table 1 may be constructed with T held constant at 7
seconds because the maximum wave period results in the highest
relative runup for each value of d,g/Hi. The maximum runup of
23.5 feet (7.2 meters) in Table 1 does not occur for the largest
wave height because the largest waves break seaward of the
structure for the given wave period.
30
Table 1. Example runup for T = 7 seconds,
constant depth, and (H3)may, = 16 feet.
Fige) “de/Hta. One Hi/eT2 = R/H =
(ft) (ft)
So R056 AGACO. 1) OROROIG. Nusa | 22sho
9 IAO)-¢ FUOROOF 60 00634enNNN2E35.1n25850
10 1S Gee OOS SAL 18.70
11 SHON Wuit553) MAOA002 WIE ANG OMNMESEOE
1d,/H) values selected to correspond with values
in figures; d, = 10 feet.
*Rnae = 23.5 feet.
(b) For the second condition where Tyg, = 13 seconds,
the maximum runup would occur for the lowest d,/H4 value.
To check 2&/L, for d = 50 feet:
re ot, = 0.0577
i) BGS)
oS 0.1020
L
L = 490.2 feet ;
LE LOOM ian) SVONsie
Doom
Table 2 may be constructed for dg = 10 feet, T = 13 seconds,
gT? = 5,441.8 feet, and using Figures 8 to 11. Table 2 shows
that, in this case, not only is runup higher for the longer
wave period than in Table 1, but the maximum runup occurs at a
lower d,/H5 value for the maximum deepwater wave.
Table 2. Example runup for T = 13 seconds,
constant depth, and (H})may = 16 feet.
Fig. aa /i ae H!/gT? RES OR
(ft) (ft)
8 “6 1600 Omen Ben ane?
9 iO 10,00 O©.001YO S80 8.0
10 1.5 Gr O,0mne0 6:00" 2560
11 5.0 3.85 O,000Gl2 S.1S 10.8
lds = 10 feet.
OR AG West.
3|
(c) For the third condition, suppose that wave steepness is
expected to be most important, and that the structure is being
designed for a constant wave steepness of H}/gT* = 0.0101 and
a maximum period of 7 seconds.
Table 3 shows the characteristic relationship that the
largest runup, R, occurs for the lowest dg/H} value when
H!/gT2 and d, are constant; the largest relative runup has
lower dimensional runup. However, Table 3 does not indicate
the maximum runup to be expected on this structure for the
given conditions; Table 1 shows the maximum (uncorrected for
scale effects) to be 23.5 feet when a maximum period of 7 sec-
onds is given. Thus, care should be exercised in determining
runup for a particular structure. The results of the three
parts of this problem are summarized in Table 4, and the calcu-
lated values are corrected for scale effect based on Figure 13.
x kk kk k &k * & * * * * EXAMPLE PROBLEM 4 * * * * * * * * ¥ ¥ H % ¥ ¥
GIVEN: An impermeable structure has a smooth slope of 1 on 1.5 and is
subjected to a design wave, Hi = 5 feet (1.5 meters). Design wave
period is T = 6 seconds. The design water depth at the toe of the
structure is dg = 0.0 foot. The bottom has a 1 on 10 slope from the
structure toe to a depth, d = 15 feet (4.6 meters), at which point
the bottom slope changes to 1 on 200.
FIND: Determine runup on the structure caused by a wave train
approaching normally.
SOLUTION: The toe depth is zero, and the bottom slope is 1 on 10;
assuming that the more seaward 1 on 200 bottom slope approximates
a horizontal bottom, Figures 5, 6, and 7 are applicable, subject
to the value jof | d/H,-
Therefore, Figure 5 is applicable;
H!
ED AAD NE DAB
ae (62274) (G)F
The relative runup for a 1 on 1.5 structure slope is determined
by interpolation to be
OZ
Table 3. Example runup for constant wave
steepness, H3/gT? = 0, ONON,
Fig, MO/ee c/a ae yee =
(ft) (8) (2a)
8 0.0101 0.6 LOO 7oO 5s MD i”
9 0.0101 1.0 LOO 5.5 oes toe
10 0.0101 55) ©.0/ 4.5 Lo75 UNa/
Ia 0.0101 Sr0) §o85 SoZ MoS 5.8
1d, = 10 feet.
eile = 7 seconds.
Seon O = QB.
Roe e2Oelabeets
Table 4. Summary of maximum runup for
different conditions.
Table Wave Meseinmenn” Scale-effect
Maximum
condition correction
R k R
(Gee) (ft)
1 Constant period; BoD Is LOO AS a
T = 7 seconds
2 Constant period; 41.6 Lo LEO 47.3
T = 13 seconds
3 Constant steepness: Boas Il LISS Zoya
A/a = OONONs
Tmax = 7 seconds
lUncorrected for scale effect.
35
A_ 1.23 .
H6
Therefore,
R = (1.23) (5)
R = 6.15 feet (1.87 meters)
The scale-correction factor, k, from Figure 13, is
KS sila
The corrected runup is
R = (1.14) (6.15) = 7.0 feet (2.1 meters)
34
LITERATURE CITED
AHRENS, J., ''Prediction of Irregular Wave Runup,'' CETA 77-4, U.S. Army,
Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir,
Va., July 1977a.
AHRENS, J., "Prediction of Irregular Wave Overtopping,'' CETA 77-7, U.S.
Army, Corps of Engineers, Coastal Engineering Research Center,
Fort Belvoir, Va., Dec. 1977b.
SAVAGE, R. P., "Laboratory Data on Wave Runup on Roughened and Permeable
Slopes,"' TM-109, U.S. Army, Corps of Engineers, Beach Erosion Board,
Washington, D.C., Mar. 1959.
SAVILLE, T., Jr., "Wave Runup on Shore Structures," Journal of the Water-
ways and Harbors Diviston, American Society of Civil Engineers, Vol. 82,
No. WW2, 1956.
SAVILLE, T., Jr., "Large-Scale Model Tests of Wave Runup and Overtopping,
Lake Okeechobee Levee Sections," U.S. Army, Corps of Engineers, Beach
Erosion Board, Washington, D.C., unpublished, 1958.
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. 1978.
U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER,
Shore Protectton 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.
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