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CETA 82-2

Energy Losses of Waves in Shallow Water

by

William G. Grosskopf and C. Linwood Vincent

COASTAL ENGINEERING TECHNICAL AID NO. 82-2

FEBRUARY 1982

V^ HO i

DOCUMENT

Approved for public release;

distribution unlimited.

330

U.S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING

RESEARCH CENTER

Kingman Building

Fort Belvoir, Va. 22060

Reprint or republication of any of this material

shall give appropriate credit to the U.S. Army Coastal

Engineering Research Center.

Limited free distribution within the United States

of single copies of this publication has been made by

this Center. Additional copies are available from:

'National Techniaal Information Service

ATTN: Operations Division

5285 Port Royal Road

Springfield, Virginia 22161

The findings in this report are not to be construed

as an official Department of the Army position unless so

designated by other authorized documents.

I cQ

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

REPORT DOCUMENTATION PAGE

READ INSTRUCTIONS

BEFORE COMPLETING FORM

1. REPORT NUMBER

CETA 82-2

2. GOVT ACCESSION NO.

3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle)

ENERGY LOSSES OF WAVES IN SHALLOW WATER

S. TYPE OF REPORT & PERIOD COVERED

Coastal Engineering

Technical Aid

6. PERFORMING ORG. REPORT NUMBER

7. AUTHORfsJ

William G. Grosskopf

C. Linwood Vincent

8. CONTRACT OR GRANT NUMBERfe)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Department of the Army

Coastal Engineering Research Center (CERRE-CO)

Kingman Building, Fort Belvoir, Virginia 22060

A31592

n. CONTROLLING OFFICE NAME AND ADDRESS

Department of the Army

Coastal Engineering Research Center

Kingman Building, Fort Belvoir, Virginia 22060

12. REPORT DATE

February 1982

13. NUMBER OF PAGES

17

U. MONITORING AGENCY NAME & ADDRESS(II dllferent from Controlling Office)

15. SECURITY CLASS, fof thia report)

UNCLASSIFIED

16. DISTRIBUTION STATEMENT fof thIa Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (ol the abstract entered In Block 20, II different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS f Continue (

aide If neceesary and Identify by block number)

Energy spectra

Shallow-water waves

Wave height

20. ABSTRACT (Caatimie an rmv^rmm afyCa ff rmre-aaary and Identity by block number)

This report presents a m.ethod for predicting nearshore significant wave

height given the straight-line fetch length, the windspeed, and the nearshore

water depth. The prediction curves were generated by numerically propagating

offshore JONSWAP spectra shoreward while applying shoaling and wave steepness

limitation criteria to each spectral component. Example problems are included.

1473

EDITION OF Â» MOV SS IS OBSOLETE

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SECURITY CLASSIFICATION OF THIS PAGE (When Date Ent

PREFACE

This report presents a method for predicting nearshore significant wave

height given the straight -line fetch length, the windspeed, and the nearshore

water depth. The wave height prediction curves were generated by numerically

propagating offshore JONSWAP spectra shoreward while applying shoaling and

wave steepness limitation criteria to each spectral component. The report pro-

vides an alternate approach to the problem of shallow-water wave estimation.

The work was carried out under the shallow-water wave transformation program

of the U.S. Army Coastal Engineering Research Center (CERC) .

The report was written by William G. Grosskopf , Hydraulic Engineer, and

Dr. C. Linwood Vincent, Chief, Coastal Oceanography 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.

CED E. BrSHOP

Colonel, Corps of Engineers

Commander and Director

CONTENTS

Page

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) 5

SYMBOLS AND DEFINITIONS 6

I INTRODUCTION 7

II WAVE HEIGHT PREDICTION CURVES 7

III USE OF CURVES 8

IV EXAMPLE PROBLEMS 11

LITERATURE CITED 14

APPENDIX METHODOLOGY AND GOVERNING SPECTRAL EQUATIONS 15

FIGURES

1 Transformation of JONSWAP spectrum in shallow water 8

2 Dimensionless fetch versus dimensionless wave height as

a function of cI/Hq 9

3 Ratio, R, of windspeed overwater, U^, to windspeed overland, Ul,

as a function of windspeed overland, U;^ 10

4 Amplification ratio, Rt, accounting for effects of air-sea

temperature difference 11

5 Determining the fetch length of an irregularly shaped

water body in the wind direction 12

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

square inches

cubic inches

feet

square feet

cubic feet

yards

square yards

cubic yards

miles

square miles

knots

acres

foot-pounds

millibars

pounds

ton, long

ton, short

degrees (angle)

Fahrenheit degrees

25.4

millimeters

2.54

centimeters

6.452

square centimeters

16.39

cubic centimeters

30.48

centimeters

0.3048

meters

0.0929

square meters

0.0283

cubic meters

0.9144

meters

0.836

square meters

0.7646

cubic meters

1.6093

kilometers

259.0

hectares

1.852

kilometers per hour

0.4047

hectares

1.3558

newton meters

1.0197 X 10"^

kilograms per square centim

28.35

grams

453.6

grams

0.4536

kilograms

1.0160

metric tons

0.9072

metric tons

0.01745

radians

5/9

Celsius degrees or Kelvins^

^To obtain Celsius (C) temperature readings

use formula: C = (5/9) (F -32).

To obtain Kelvin (K) readings, use formula:

from Fahrenheit (F) readings,

K = (5/9) (F -32) + 273.15.

SYMBOLS AND DEFINITIONS

C^^ wave steepness limitation factor

d water depth

E energy density

Et total energy in the wave spectrum

F straight-line fetch length (for irregularly shaped water bodies, this

should be based on an average over a 24Â° quadrant)

f frequency of spectral component

fjn frequency of spectral peak

g acceleration due to gravity

Ho deepwater significant wave height

Hs significant wave height

H dimensionless wave height

Kg shoaling coefficient

L wavelength

R land-water windspeed correction factor

RT air-sea temperature difference windspeed correction factor

Tp peak wave period

Ua windspeed to be used in wave height estimation

Ua overwater windspeed corrected for wind instabilities

U^j overwater windspeed

U2 windspeed measured Z meters above land or water surface

Uj^Q 10-meter (33 foot) windspeed

X dimensionless fetch length

a Phillips equilibrium constant

Y ratio of maximal spectral energy to the maximum, of the corresponding

Pierson-Moskowitz spectrum

TT 3.14159

Og left-side width of the spectral peak

o^ right-side width of the spectral peak

41 wave steepness limitation factor

oj]^ wave steepness limitation factor

ENERGY LOSSES OF WAVES IN SHALLOW WATER

hy

William G. Grosskopf and C. Linwood Vincent

I . INTRODUCTION

The energy in ^n irregular wave train changes as the waves propagate from

deep water toward shore. Estimates of the total change in wave energy have

traditionally been made by multiplying a shoaling, refraction and friction

coefficient by an offshore significant wave height to yield the nearshore wave

height. Recent studies of wave spectra have provided a more detailed view of

the wave field and indicate that additional processes should be considered.

This report presents finite-depth wave height estimation curves, given an ini-

tial JONSWAP type of offshore spectral wave condition (Hasselmann, et al., 1973)

generated over a short fetch and incorporating finite-depth steepness effects

based on a study by Kitaigorodskii, Krasitskii, and Zaslavskii (1975). These

curves represent energy changes due to shoaling and an upper limit of energy

spectral density as a function of wave frequency and water depth.

Research at the Coastal Engineering Research Center (CERC) and elsewhere

indicates steepness effects that lead to breaking in a shoaling wave field

lead to a major loss of energy in addition to that lost by bottom friction and

percolation. These effects can be incorporated into wave estimation curves in

a fashion similar to shoaling because the effects can be made a function of

depth. The effects of refraction, bottom friction, and percolation are not in-

cluded in these curves because they are site specific. The effects of bottom

friction and percolation will always be to reduce the estimated wave height.

These curves should be applied only in areas of nearly parallel bottom contours.

Consequently, refraction will also only reduce wave height.

This report presents a method for estimating the significant wave height,

Hg, given the fetch length, F, the overwater windspeed, U^ (see U.S. Army,

Corps of Engineers, Coastal Engineering Research Center, 1981), and the water

depth, d, neglecting any additional wave growth in shallow water due to the

wind. The method differs from two recently reported methods â€” Seelig (1980),

who presents a method for predicting shallow-water wave height given deepwater

wave height, Hq, peak period, Tp, and bottom slope, m, and Vincent (1981),

who presents a method for calculating the depth-limited significant wave height

based on knowledge of the deepwater wave spectrum â€” but it does not supersede

the use of these other two methods. The report provides an alternate approach

to the problem of shallow-water wave estimation given the four quantities men-

tioned above.

II. WAVE HEIGHT PREDICTION CURVES

A series of JONSWAP spectra was generated numerically in deepwater condi-

tions for varying windspeeds and fetch length, and propagated into shallow

water over parallel bottom contours. A frequency-by-frequency calculation was

made at various depths shoreward applying independently the wave steepness

limitation criterion (Kitaigorodskii, Krasitskii, and Zaslavskii, 1975) and a

shoaling coefficient to each spectral component. If the shoaled wave energy

exceeded the limiting value, the limiting value was retained. A detailed ex-

planation of the methodology involved in this computation is presented in the

Appendix. Resulting spectra at gradually decreasing depths for a given case

are shown in Figure 1. This analysis provides the wave height prediction

curves shown in Figure 2. These curves provide the nearshore significant wave

height, Hg, at a given water depth which is related to the total energy, Ej,

in the nearshore wave spectrum by

Hg = 4v^

given the fetch length, the overwater windspeed, and the deepwater wave height.

Note that in Figure 1 there is a slight shift in the wave period toward lower

frequencies as the spectrum moves into shallow water. Later work will attempt

to quantify this shift and incorporate bottom friction effects.

III. USE OF CURVES

There are certain restraints on the use of the curves which are as follows:

(1) Curves are designed to be used for fetch-limited, wind-generated

waves in deep water over short fetches, i.e., up to 62 miles (100 kilo-

meters) .

(2) This analysis includes only the wave steepness criterion and

shoaling. It does not reflect other energy losses such as refraction,

friction, or percolation (parallel bottom contours are assumed) .

(3) The fetch length, F, is strictly the straight-line fetch

unless the water body is irregularly shaped where the fetch would be

based on an average over a 24Â° quadrant.

E(tl -s)

U:49.2 fl/$|i5ni/s)

^ F:65, 600 II (20,000m)

/ \ Curve 1

/ \ Oeepwoler

1 Y JONSWAP

1 1 Spectrum

/ \ \ d=l3l.2 ll(40m)

//,\ Vv \ y'''^^^ lldOm)

/r \ \ ^^<^ d=l9.7 ri(6m)

1

/ ^\5\V\ d=l3.lll(4m)

0.2 0.3

Frequency

0.4

Figure 1. Transformation of JONSWAP spectrum in shallow water.

10^

10'

10'

1

d

/

k-'

/.

^

^

d

' 5

^^

i20

OetpBOter Wava Hql.

Peok Period

Tp= "*^^

3.5g

1

\

1

i

^

/

1

f

0.075 0.125 0.175 0.225 0-275

Oimensionless Wove Hql., H:gH,/'

Figure 2. Dimensionless fetch versus dimensionless

wave height as a function of d/Ho.

(4) To calculate the adjusted windspeed, U^^, the following pro-

cedure should be used:

(a) If windspeed is observed at any level other than 33 feet

(10 meters) windspeed on land or water, the adjustment to the 33-

foot level is approximated by:

Uio

1/7

-m^

where Uiq is the 10-meter windspeed in meters per second, Z

the height of wind measurement above the surface in meters, and

Uz the measured windspeed in meters per second. This method

is valid up to about Z = 66 feet (20 m.eters) . If the windspeed

was measured at 33 feet, Uiq = Uz*

(b) If windspeed was measured overland, correct to overwater

windspeed by

U^ = l.lUjo for F < 10 miles (16 kilometers)

U^ = RUio for F > 10 miles

where U^ is the overwater windspeed in meters per second; R

is given in Figure 3 . If windspeed was measured overwater and

adjusted to a 10-meter height, U^ = Uiq-

0.5

Windspeeds are referenced

to 10-meter level

10 15 20 26 30 35 40 kn

5 10 15 20 25 30 35 40 45 50 55 60 m/S

-I u

5 10 IS 20 25 30 35 40 45 mph

Figure 3. Ratio, R, of windspeed overwater, Uy, to windspeed

overland, Ul, as a function of windspeed overland,

Ul (after Resio and Vincent, 1976).

(c) To correct for wind instabilities over fetch lengths

greater than 10 miles:

Ua= 0.71 ul-23

where U^ is the adjusted windspeed in meters per second. If

the F < 10 miles, U| = U^.

(d) To correct for air-sea temperature differences,

Ua = ^T ^A fÂ°^ F > 10 miles

U^ = Ua for F < 10 miles

where U^. is the new windspeed adjusted for the temperature dif-

ference; R-j is given in Figure 4.

1.3

1.2

^7:-=^^^

I.I

i\

Rj 1.0

0.9

1 1

Example 2

0.8

07

1 1 1 ._.I 1 1 1

-20 -15 -10 -5 5 10 15 20

Air-Seo Temperature Difference (Tqâ€” Tj) "C

Figure 4. Amplification ratio, Rj, accounting for effects of air-sea

temperature difference (Reslo and Vincent, 1976).

IV. EXAMPLE PROBLEMS

*************** EXAMPLE PROBLEM 1***************

GIVEN ; Deepwater fetch, F = 24.9 miles (40 kilometers), adjusted 33-foot (10

meter) windspeed, Ua = 65.6 feet (20 meters) per second (an example of com-

putation of the adjusted windspeed can be found in U.S. Army, Corps of

Engineers, Coastal Engineering Research Center, 1981).

FIND ; Significant wave height and peak period of the wave spectrum at depths

of 23 and 9.8 feet (7 and 3 meters).

SOLUTION ; The dimensionless fetch, x is

_ gF (9.8 m/s^) (40,000 m)

U,

(20 m/s)'

980 = 9.8 X lo2

The deepwater significant wave height and peak period are

Ho = 1.6 X 10

-3

Ua

= 1.6 X 10-3 J ^^1^1^ (20 m/s) = 2.04 mete

Tr. =

Uax ^/3 20(980) "^/^

P 3.5g 3.5(9.8)

at a depth of 7 meters

d 7

5.79 seconds

Ho 2.04

= 3.43

In Figure 2 at x = 9.8 x 10^ and interpolated between curves for d/Ho of

3 and 5, reading down for H,

H = -2^ = 0.037

Ua

Hs = 1.51 meters

At a depth of 3 meters, d/Ho = 1.47, providing an H = 0.025 or Hg = 1.02

meters. The peak period, T , and the local wavelength would increase over

that at a 7-meter depth and currently must be calculated by the tables given

in Appendix C of the Shore Protection Manual (U.S. Army, Corps of Engineers,

Coastal Engineering Research Center, 1977).

*************** EXAMPLE PROBLEM 2***************

GIVEN: The wind direction is predominantly from the southwest over the deep,

irregularly shaped water body shown in Figure 5. The windspeed to be con-

sidered is 49.2 feet (15 meters) per second measured on top of an instru-

ment shack at 13 feet (4 meters) from the ground. The air temperature when

these conditions occur is 50Â° Fahrenheit (10Â° Celsius) and the water tempera-

ture is 60Â° Fahrenheit (16Â° Celsius) .

Figure 5,

Scole in Kilometert

The fetch length for this irregularly shaped water body in the wind

direction is determined by drawing nine radials at 3Â° increments

centered on the wind direction and arithmetically averaging the

radial lengths as illustrated. The average fetch in this example

is approximately 22.2 miles (36 kilometers).

12

FIND : The significant wave height at a 16.4-foot (5 meter) depth just off the

coast near the anemometer site.

SOLUTION ; The fetch is found by averaging over a 24Â° quadrant since the body

of water is irregularly shaped. As shown in Figure 5, nine radials are

constructed at 3Â° increments and the average fetch length of 22 miles (36

kilometers) is found.

The adjusted windspeed is found following the steps outlined previously:

(a) Adjust wind from the 4-meter to the 10-meter level

/in\V7 /10\^/'^

Uio = { 7 ) ^Z " (z~) ^-"-^^ ^ -'-'^â€¢-'- â„¢^ters per second

(b) Adjust overland wind to overwater wind with R from Figure 3

U^ = RUio = 1.25(17.1) = 21.4 meters per second

(c) Correct wind for instabilities

U^ = 0.71 U^^'^^ = 0.71(21.4)^-^^ = 30.7 meters per second

(d) Correct for air-sea temperature difference with Rt from Figure 4

U^ = Rt U^ = 1.17(30.7) =35.9 meters per second

The dimensionless fetch, S, is

,2>

= II = (9.8 m/s^) (36.000 m) ^ 273 7

U? (35.9 m/s)2

The deepwater significant wave height and peak period are

Ho = 1.6 X 10-3 /lii2|0 (.33_g ^/g) ^ 3^3 meters

35.9(273.7) '3

Tp = 3 5(9 " " 8) =6.80 seconds

At a 5-meter depth

Hq 3.5 '^

In Figure 2 at x = 273.7 and A/Vlq = 1.43

gHe

H = -~ = 0.012

U?

and

H U2 (0.12) (35.9)2

Ho = = = 1.58 meters

^ g 9.8

13

LITERATURE CITED

HASSELMAITO, K., et al., "Measurements of Wind Wave Growth and Swell Decay

During the Joint North Sea Wave Project," Deutsches Hydrographisches

Institut, Hamburg, Germany, 1973.

KITAIGORODSKII, S.A., KRASITSKII, V.P., and ZASLAVSKII, M.M., "Phillips Theory

of the Equilibrium Range in the Spectra of Wind -Genera ted Gravity Waves,"

Journal of Phystoal Oceanography, Vol. 5, 1975, pp. 410-420.

RESIO, D.T., and VINCENT, C.L., "Estimation of Winds Over the Great Lakes,"

Miscellaneous Paper H-76-12, U.S. Army Engineer Waterways Experiment Station,

Vicksburg, Miss., June 1976.

SEELIG, W.N., "Maximum Wave Heights and Critical Water Depths for Irregular

Waves in the Surf Zone," CETA 80-1, U.S. Army, Corps of Engineers, Coastal

Engineering Research Center, Fort Belvoir, Va., Feb. 1980.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, "Method

for Determining Adjusted Windspeed, U^., for Wave Forecasting," CETN-I-5,

Fort Belvoir, Va., Mar. 1981.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore

Protection 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.

VINCENT, C.L., "A Method for Estimating Depth-Limited Wave Energy," CETA 81-16,

U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort

Belvoir, Va., Nov. 1981.

14

APPENDIX

METHODOLOGY AND GOVERNING SPECTRAL EQUATIONS

1. Deepwater Representation of Fetch-Limited Wave Spectrum .

A spectrum of wind waves, generated in deep water for a long period of

time, is limited by the length of the fetch over which the wind is blowing.

The wind will generate a spectrum with a shape which ?ias been parameterized

by Hasselmann, et al. (1973). The parameterization, or JONSWAP spectrum, pro-

vides a functional relationship between energy and frequency as well as the

windspeed, fetch length, and width of the spectral peak:

r . ,. . -| exp ^^~^"^^

E(f) = ag2(2Tr)-'^ fS exp |- f (|-yt*J y 2a2f2 ^^_^^

and

a^ for f < fn

Ox, for f > f â€ž

where

E = energy density

F = fetch length

f = frequency of wave component

fiQ = frequency of spectral peak = 3.5g/(Uio ^ /

g = acceleration due to gravity

Ua = adjusted 10-meter windspeed

X = dimensionless fetch = gF/U^

a = Phillips equilibrium constant = 0.076 x

-0.22

Y = ratio of maximal spectral energy to the maximum of the corresponding

Pierson-Moskowitz spectrum = 3.3

CTg = left-side width of the spectral peak =0.07

a-. = right-side width of the spectral peak = 0.09

This equation provides a wave spectrum as shown in curve 1 (Fig. 1), with a

total energy equal to the deepwater significant wave height, squared over 16.

15

2. Energy Reduction in Shallow Water .

As an irregular wave train enters transitional and shallow-water depths,

the presence of the sea bottom causes changes in wave steepness which, due to

the limitation on wave steepness, lead to a loss of wave energy. Kitaigorodskii,

Krasitskii, and Zaslavskii (1975) suggest that an upper limit of energy exists

at a given frequency which is a function of depth and a:

E(f) = ag2 f-5 (2u)-'+ (A-2)

where

C^ tanh (w^ C^) = 1

d = water depth

a = 0.0081

^ = Ch{l + [2a){i Ci^/sinh(2aJh C^) ] i

OJ^j = 2TTf /d/i

This equation represents a stability limit or "limiting form criterion" on a

wave component. Kitaigorodskii, Krasitskii, and Zaslavskii used a value of

a of 0.0081 based on field data. Recent work at the U.S. Army Engineer

Waterways Experiment Station (WES) has indicated that another mechanism, non-

linear wave-wave interaction, has an equivalent effect but that a would vary

with dimensionless fetch (gF/U^) . The application of this theory is further

outlined by Vincent (1981) .

Shoaling of a wave in shallow water also changes wave energy. A shoaling

coefficient can be calculated as in the Shore Protection Manual (App. C in

U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) for

each frequency component according to linear theory:

and can be multiplied by the deepwater energy at each frequency band to obtain

a "shoaled" spectrum,

E(f) shoaled = Ks(f) E(f) deep (A-4)

3 . Determination of Shallow-VJater Energy Spectrum .

Figure A-1 is a flow chart describing the solution process used in produc-

ing the design curves presented in this paper.

16

Q>

Generate deepwater JONSWAP spec-

trum using equation (A-1) based

on windspeed and fetch length.

At each frequency compare energy from equation

(A-2) with that from (A-4) . The greatest of

the two values at each frequency band is re-

tained to form the spectrum at the particular

shallow-water depth.

( Stop )

Figure A-1. Flow chart illustrating the use of equations (A-1) to

(A-4) in generating the curves presented in Figure 2.

r-^

0) r^

> cu

a.

S -^ vo

o

3 4J

3

3 4J

-id

(U X!

-ili

0) x;

Vi g

4-t -T3 >-. iH i-i

01 S

4J -3 >, rH U

d d -0 -H CO d

01 rH

3 3 .0 M CO 3

-T3

CO V4

â€¢3

CO CO .CO) 0)

O CO

CO

.^".^â– ^o I

M '"

'ffl

-o 3

y-l -O iJ T3 ^ -H

UH -0 4J -o 4-1 â– H

nH

â€¢H 4) M t. >

â– H 0) 10 u >

O O ^

C 0) ^1 ti3 n)

U XI

CO

3 01 IH CO CO

(0

M C. CU 3 -1

00 D, 0) 3 -rl - .

B >. "-t

â€¢H (fl C QJ U 1-1 01

a &.rH

â€¢H

â– rl 01 3 0) H VH 01

S tâ€¢r^

d

to Ta 0) p 0) a

CO a -H

â€¢r4 S-l flj

^

d 00 i-t -H

,C

3 00 4-1 â– rH

.n < >

tU -H Â£ -H . . )-.

-i < >

0) â– H X; -rl . . IH

^ Cfl

tJ30)miJ-OMa)

-H CO

V43a)oiV4^ooia)o4

U 0) cu w

,H â€¢

3 cn "

00

^ QJ QJ 0] Xt >

0) ji: 3 u C 3 Â« .

s

3 en â€¢â€¢

00

^^ S 2 3-5 S .S

>> o â€¢

U 4J U -1 3 M

>, :=) .

3

14 4J 4-1 rH 3 rH

J^ CT]

â– H

X3 CO

CO 01 â– rl M .

<u .a)a)iJCt.>H

â€¢ â€¢ >

C ^ > Cl. rt -r-f OJ

3

Bxi'>O.CO^r4a) 3

0)

u va Ti

M M 3 -H 0) n) .

IH CO -3

3

00 00 3 â– H 0) CO .

tH

ccoa,au3a)

01 > rH

â– rH

BBOO-atiSOl

â– HO) < -H CO rH

OO

4-1 0)

00

d

W ^ C 3 -H .4-1

CO - -rl

3

W rH 3 3 rH . tJ

"3 U ^-i

â– " J= -2 Z 01 i " H

3 U 14H

0)

Cn CO CI â– rl

â€¢H x: -H z 0) a H

â€¢H 00

â– rl 00

s o c

^

â€¢0 u 0) (U

30c

M

â€¢a u 0) 0)

O > -H

tu 4-) u -n (U rH . .

> -H

rH i-H U

H (U -H C ^ 4J M

Energy Losses of Waves in Shallow Water

by

William G. Grosskopf and C. Linwood Vincent

COASTAL ENGINEERING TECHNICAL AID NO. 82-2

FEBRUARY 1982

V^ HO i

DOCUMENT

Approved for public release;

distribution unlimited.

330

U.S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING

RESEARCH CENTER

Kingman Building

Fort Belvoir, Va. 22060

Reprint or republication of any of this material

shall give appropriate credit to the U.S. Army Coastal

Engineering Research Center.

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REPORT DOCUMENTATION PAGE

READ INSTRUCTIONS

BEFORE COMPLETING FORM

1. REPORT NUMBER

CETA 82-2

2. GOVT ACCESSION NO.

3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle)

ENERGY LOSSES OF WAVES IN SHALLOW WATER

S. TYPE OF REPORT & PERIOD COVERED

Coastal Engineering

Technical Aid

6. PERFORMING ORG. REPORT NUMBER

7. AUTHORfsJ

William G. Grosskopf

C. Linwood Vincent

8. CONTRACT OR GRANT NUMBERfe)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Department of the Army

Coastal Engineering Research Center (CERRE-CO)

Kingman Building, Fort Belvoir, Virginia 22060

A31592

n. CONTROLLING OFFICE NAME AND ADDRESS

Department of the Army

Coastal Engineering Research Center

Kingman Building, Fort Belvoir, Virginia 22060

12. REPORT DATE

February 1982

13. NUMBER OF PAGES

17

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17. DISTRIBUTION STATEMENT (ol the abstract entered In Block 20, II different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS f Continue (

aide If neceesary and Identify by block number)

Energy spectra

Shallow-water waves

Wave height

20. ABSTRACT (Caatimie an rmv^rmm afyCa ff rmre-aaary and Identity by block number)

This report presents a m.ethod for predicting nearshore significant wave

height given the straight-line fetch length, the windspeed, and the nearshore

water depth. The prediction curves were generated by numerically propagating

offshore JONSWAP spectra shoreward while applying shoaling and wave steepness

limitation criteria to each spectral component. Example problems are included.

1473

EDITION OF Â» MOV SS IS OBSOLETE

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SECURITY CLASSIFICATION OF THIS PAGE (When Date Ent

PREFACE

This report presents a method for predicting nearshore significant wave

height given the straight -line fetch length, the windspeed, and the nearshore

water depth. The wave height prediction curves were generated by numerically

propagating offshore JONSWAP spectra shoreward while applying shoaling and

wave steepness limitation criteria to each spectral component. The report pro-

vides an alternate approach to the problem of shallow-water wave estimation.

The work was carried out under the shallow-water wave transformation program

of the U.S. Army Coastal Engineering Research Center (CERC) .

The report was written by William G. Grosskopf , Hydraulic Engineer, and

Dr. C. Linwood Vincent, Chief, Coastal Oceanography 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.

CED E. BrSHOP

Colonel, Corps of Engineers

Commander and Director

CONTENTS

Page

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) 5

SYMBOLS AND DEFINITIONS 6

I INTRODUCTION 7

II WAVE HEIGHT PREDICTION CURVES 7

III USE OF CURVES 8

IV EXAMPLE PROBLEMS 11

LITERATURE CITED 14

APPENDIX METHODOLOGY AND GOVERNING SPECTRAL EQUATIONS 15

FIGURES

1 Transformation of JONSWAP spectrum in shallow water 8

2 Dimensionless fetch versus dimensionless wave height as

a function of cI/Hq 9

3 Ratio, R, of windspeed overwater, U^, to windspeed overland, Ul,

as a function of windspeed overland, U;^ 10

4 Amplification ratio, Rt, accounting for effects of air-sea

temperature difference 11

5 Determining the fetch length of an irregularly shaped

water body in the wind direction 12

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

square inches

cubic inches

feet

square feet

cubic feet

yards

square yards

cubic yards

miles

square miles

knots

acres

foot-pounds

millibars

pounds

ton, long

ton, short

degrees (angle)

Fahrenheit degrees

25.4

millimeters

2.54

centimeters

6.452

square centimeters

16.39

cubic centimeters

30.48

centimeters

0.3048

meters

0.0929

square meters

0.0283

cubic meters

0.9144

meters

0.836

square meters

0.7646

cubic meters

1.6093

kilometers

259.0

hectares

1.852

kilometers per hour

0.4047

hectares

1.3558

newton meters

1.0197 X 10"^

kilograms per square centim

28.35

grams

453.6

grams

0.4536

kilograms

1.0160

metric tons

0.9072

metric tons

0.01745

radians

5/9

Celsius degrees or Kelvins^

^To obtain Celsius (C) temperature readings

use formula: C = (5/9) (F -32).

To obtain Kelvin (K) readings, use formula:

from Fahrenheit (F) readings,

K = (5/9) (F -32) + 273.15.

SYMBOLS AND DEFINITIONS

C^^ wave steepness limitation factor

d water depth

E energy density

Et total energy in the wave spectrum

F straight-line fetch length (for irregularly shaped water bodies, this

should be based on an average over a 24Â° quadrant)

f frequency of spectral component

fjn frequency of spectral peak

g acceleration due to gravity

Ho deepwater significant wave height

Hs significant wave height

H dimensionless wave height

Kg shoaling coefficient

L wavelength

R land-water windspeed correction factor

RT air-sea temperature difference windspeed correction factor

Tp peak wave period

Ua windspeed to be used in wave height estimation

Ua overwater windspeed corrected for wind instabilities

U^j overwater windspeed

U2 windspeed measured Z meters above land or water surface

Uj^Q 10-meter (33 foot) windspeed

X dimensionless fetch length

a Phillips equilibrium constant

Y ratio of maximal spectral energy to the maximum, of the corresponding

Pierson-Moskowitz spectrum

TT 3.14159

Og left-side width of the spectral peak

o^ right-side width of the spectral peak

41 wave steepness limitation factor

oj]^ wave steepness limitation factor

ENERGY LOSSES OF WAVES IN SHALLOW WATER

hy

William G. Grosskopf and C. Linwood Vincent

I . INTRODUCTION

The energy in ^n irregular wave train changes as the waves propagate from

deep water toward shore. Estimates of the total change in wave energy have

traditionally been made by multiplying a shoaling, refraction and friction

coefficient by an offshore significant wave height to yield the nearshore wave

height. Recent studies of wave spectra have provided a more detailed view of

the wave field and indicate that additional processes should be considered.

This report presents finite-depth wave height estimation curves, given an ini-

tial JONSWAP type of offshore spectral wave condition (Hasselmann, et al., 1973)

generated over a short fetch and incorporating finite-depth steepness effects

based on a study by Kitaigorodskii, Krasitskii, and Zaslavskii (1975). These

curves represent energy changes due to shoaling and an upper limit of energy

spectral density as a function of wave frequency and water depth.

Research at the Coastal Engineering Research Center (CERC) and elsewhere

indicates steepness effects that lead to breaking in a shoaling wave field

lead to a major loss of energy in addition to that lost by bottom friction and

percolation. These effects can be incorporated into wave estimation curves in

a fashion similar to shoaling because the effects can be made a function of

depth. The effects of refraction, bottom friction, and percolation are not in-

cluded in these curves because they are site specific. The effects of bottom

friction and percolation will always be to reduce the estimated wave height.

These curves should be applied only in areas of nearly parallel bottom contours.

Consequently, refraction will also only reduce wave height.

This report presents a method for estimating the significant wave height,

Hg, given the fetch length, F, the overwater windspeed, U^ (see U.S. Army,

Corps of Engineers, Coastal Engineering Research Center, 1981), and the water

depth, d, neglecting any additional wave growth in shallow water due to the

wind. The method differs from two recently reported methods â€” Seelig (1980),

who presents a method for predicting shallow-water wave height given deepwater

wave height, Hq, peak period, Tp, and bottom slope, m, and Vincent (1981),

who presents a method for calculating the depth-limited significant wave height

based on knowledge of the deepwater wave spectrum â€” but it does not supersede

the use of these other two methods. The report provides an alternate approach

to the problem of shallow-water wave estimation given the four quantities men-

tioned above.

II. WAVE HEIGHT PREDICTION CURVES

A series of JONSWAP spectra was generated numerically in deepwater condi-

tions for varying windspeeds and fetch length, and propagated into shallow

water over parallel bottom contours. A frequency-by-frequency calculation was

made at various depths shoreward applying independently the wave steepness

limitation criterion (Kitaigorodskii, Krasitskii, and Zaslavskii, 1975) and a

shoaling coefficient to each spectral component. If the shoaled wave energy

exceeded the limiting value, the limiting value was retained. A detailed ex-

planation of the methodology involved in this computation is presented in the

Appendix. Resulting spectra at gradually decreasing depths for a given case

are shown in Figure 1. This analysis provides the wave height prediction

curves shown in Figure 2. These curves provide the nearshore significant wave

height, Hg, at a given water depth which is related to the total energy, Ej,

in the nearshore wave spectrum by

Hg = 4v^

given the fetch length, the overwater windspeed, and the deepwater wave height.

Note that in Figure 1 there is a slight shift in the wave period toward lower

frequencies as the spectrum moves into shallow water. Later work will attempt

to quantify this shift and incorporate bottom friction effects.

III. USE OF CURVES

There are certain restraints on the use of the curves which are as follows:

(1) Curves are designed to be used for fetch-limited, wind-generated

waves in deep water over short fetches, i.e., up to 62 miles (100 kilo-

meters) .

(2) This analysis includes only the wave steepness criterion and

shoaling. It does not reflect other energy losses such as refraction,

friction, or percolation (parallel bottom contours are assumed) .

(3) The fetch length, F, is strictly the straight-line fetch

unless the water body is irregularly shaped where the fetch would be

based on an average over a 24Â° quadrant.

E(tl -s)

U:49.2 fl/$|i5ni/s)

^ F:65, 600 II (20,000m)

/ \ Curve 1

/ \ Oeepwoler

1 Y JONSWAP

1 1 Spectrum

/ \ \ d=l3l.2 ll(40m)

//,\ Vv \ y'''^^^ lldOm)

/r \ \ ^^<^ d=l9.7 ri(6m)

1

/ ^\5\V\ d=l3.lll(4m)

0.2 0.3

Frequency

0.4

Figure 1. Transformation of JONSWAP spectrum in shallow water.

10^

10'

10'

1

d

/

k-'

/.

^

^

d

' 5

^^

i20

OetpBOter Wava Hql.

Peok Period

Tp= "*^^

3.5g

1

\

1

i

^

/

1

f

0.075 0.125 0.175 0.225 0-275

Oimensionless Wove Hql., H:gH,/'

Figure 2. Dimensionless fetch versus dimensionless

wave height as a function of d/Ho.

(4) To calculate the adjusted windspeed, U^^, the following pro-

cedure should be used:

(a) If windspeed is observed at any level other than 33 feet

(10 meters) windspeed on land or water, the adjustment to the 33-

foot level is approximated by:

Uio

1/7

-m^

where Uiq is the 10-meter windspeed in meters per second, Z

the height of wind measurement above the surface in meters, and

Uz the measured windspeed in meters per second. This method

is valid up to about Z = 66 feet (20 m.eters) . If the windspeed

was measured at 33 feet, Uiq = Uz*

(b) If windspeed was measured overland, correct to overwater

windspeed by

U^ = l.lUjo for F < 10 miles (16 kilometers)

U^ = RUio for F > 10 miles

where U^ is the overwater windspeed in meters per second; R

is given in Figure 3 . If windspeed was measured overwater and

adjusted to a 10-meter height, U^ = Uiq-

0.5

Windspeeds are referenced

to 10-meter level

10 15 20 26 30 35 40 kn

5 10 15 20 25 30 35 40 45 50 55 60 m/S

-I u

5 10 IS 20 25 30 35 40 45 mph

Figure 3. Ratio, R, of windspeed overwater, Uy, to windspeed

overland, Ul, as a function of windspeed overland,

Ul (after Resio and Vincent, 1976).

(c) To correct for wind instabilities over fetch lengths

greater than 10 miles:

Ua= 0.71 ul-23

where U^ is the adjusted windspeed in meters per second. If

the F < 10 miles, U| = U^.

(d) To correct for air-sea temperature differences,

Ua = ^T ^A fÂ°^ F > 10 miles

U^ = Ua for F < 10 miles

where U^. is the new windspeed adjusted for the temperature dif-

ference; R-j is given in Figure 4.

1.3

1.2

^7:-=^^^

I.I

i\

Rj 1.0

0.9

1 1

Example 2

0.8

07

1 1 1 ._.I 1 1 1

-20 -15 -10 -5 5 10 15 20

Air-Seo Temperature Difference (Tqâ€” Tj) "C

Figure 4. Amplification ratio, Rj, accounting for effects of air-sea

temperature difference (Reslo and Vincent, 1976).

IV. EXAMPLE PROBLEMS

*************** EXAMPLE PROBLEM 1***************

GIVEN ; Deepwater fetch, F = 24.9 miles (40 kilometers), adjusted 33-foot (10

meter) windspeed, Ua = 65.6 feet (20 meters) per second (an example of com-

putation of the adjusted windspeed can be found in U.S. Army, Corps of

Engineers, Coastal Engineering Research Center, 1981).

FIND ; Significant wave height and peak period of the wave spectrum at depths

of 23 and 9.8 feet (7 and 3 meters).

SOLUTION ; The dimensionless fetch, x is

_ gF (9.8 m/s^) (40,000 m)

U,

(20 m/s)'

980 = 9.8 X lo2

The deepwater significant wave height and peak period are

Ho = 1.6 X 10

-3

Ua

= 1.6 X 10-3 J ^^1^1^ (20 m/s) = 2.04 mete

Tr. =

Uax ^/3 20(980) "^/^

P 3.5g 3.5(9.8)

at a depth of 7 meters

d 7

5.79 seconds

Ho 2.04

= 3.43

In Figure 2 at x = 9.8 x 10^ and interpolated between curves for d/Ho of

3 and 5, reading down for H,

H = -2^ = 0.037

Ua

Hs = 1.51 meters

At a depth of 3 meters, d/Ho = 1.47, providing an H = 0.025 or Hg = 1.02

meters. The peak period, T , and the local wavelength would increase over

that at a 7-meter depth and currently must be calculated by the tables given

in Appendix C of the Shore Protection Manual (U.S. Army, Corps of Engineers,

Coastal Engineering Research Center, 1977).

*************** EXAMPLE PROBLEM 2***************

GIVEN: The wind direction is predominantly from the southwest over the deep,

irregularly shaped water body shown in Figure 5. The windspeed to be con-

sidered is 49.2 feet (15 meters) per second measured on top of an instru-

ment shack at 13 feet (4 meters) from the ground. The air temperature when

these conditions occur is 50Â° Fahrenheit (10Â° Celsius) and the water tempera-

ture is 60Â° Fahrenheit (16Â° Celsius) .

Figure 5,

Scole in Kilometert

The fetch length for this irregularly shaped water body in the wind

direction is determined by drawing nine radials at 3Â° increments

centered on the wind direction and arithmetically averaging the

radial lengths as illustrated. The average fetch in this example

is approximately 22.2 miles (36 kilometers).

12

FIND : The significant wave height at a 16.4-foot (5 meter) depth just off the

coast near the anemometer site.

SOLUTION ; The fetch is found by averaging over a 24Â° quadrant since the body

of water is irregularly shaped. As shown in Figure 5, nine radials are

constructed at 3Â° increments and the average fetch length of 22 miles (36

kilometers) is found.

The adjusted windspeed is found following the steps outlined previously:

(a) Adjust wind from the 4-meter to the 10-meter level

/in\V7 /10\^/'^

Uio = { 7 ) ^Z " (z~) ^-"-^^ ^ -'-'^â€¢-'- â„¢^ters per second

(b) Adjust overland wind to overwater wind with R from Figure 3

U^ = RUio = 1.25(17.1) = 21.4 meters per second

(c) Correct wind for instabilities

U^ = 0.71 U^^'^^ = 0.71(21.4)^-^^ = 30.7 meters per second

(d) Correct for air-sea temperature difference with Rt from Figure 4

U^ = Rt U^ = 1.17(30.7) =35.9 meters per second

The dimensionless fetch, S, is

,2>

= II = (9.8 m/s^) (36.000 m) ^ 273 7

U? (35.9 m/s)2

The deepwater significant wave height and peak period are

Ho = 1.6 X 10-3 /lii2|0 (.33_g ^/g) ^ 3^3 meters

35.9(273.7) '3

Tp = 3 5(9 " " 8) =6.80 seconds

At a 5-meter depth

Hq 3.5 '^

In Figure 2 at x = 273.7 and A/Vlq = 1.43

gHe

H = -~ = 0.012

U?

and

H U2 (0.12) (35.9)2

Ho = = = 1.58 meters

^ g 9.8

13

LITERATURE CITED

HASSELMAITO, K., et al., "Measurements of Wind Wave Growth and Swell Decay

During the Joint North Sea Wave Project," Deutsches Hydrographisches

Institut, Hamburg, Germany, 1973.

KITAIGORODSKII, S.A., KRASITSKII, V.P., and ZASLAVSKII, M.M., "Phillips Theory

of the Equilibrium Range in the Spectra of Wind -Genera ted Gravity Waves,"

Journal of Phystoal Oceanography, Vol. 5, 1975, pp. 410-420.

RESIO, D.T., and VINCENT, C.L., "Estimation of Winds Over the Great Lakes,"

Miscellaneous Paper H-76-12, U.S. Army Engineer Waterways Experiment Station,

Vicksburg, Miss., June 1976.

SEELIG, W.N., "Maximum Wave Heights and Critical Water Depths for Irregular

Waves in the Surf Zone," CETA 80-1, U.S. Army, Corps of Engineers, Coastal

Engineering Research Center, Fort Belvoir, Va., Feb. 1980.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, "Method

for Determining Adjusted Windspeed, U^., for Wave Forecasting," CETN-I-5,

Fort Belvoir, Va., Mar. 1981.

U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore

Protection 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.

VINCENT, C.L., "A Method for Estimating Depth-Limited Wave Energy," CETA 81-16,

U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort

Belvoir, Va., Nov. 1981.

14

APPENDIX

METHODOLOGY AND GOVERNING SPECTRAL EQUATIONS

1. Deepwater Representation of Fetch-Limited Wave Spectrum .

A spectrum of wind waves, generated in deep water for a long period of

time, is limited by the length of the fetch over which the wind is blowing.

The wind will generate a spectrum with a shape which ?ias been parameterized

by Hasselmann, et al. (1973). The parameterization, or JONSWAP spectrum, pro-

vides a functional relationship between energy and frequency as well as the

windspeed, fetch length, and width of the spectral peak:

r . ,. . -| exp ^^~^"^^

E(f) = ag2(2Tr)-'^ fS exp |- f (|-yt*J y 2a2f2 ^^_^^

and

a^ for f < fn

Ox, for f > f â€ž

where

E = energy density

F = fetch length

f = frequency of wave component

fiQ = frequency of spectral peak = 3.5g/(Uio ^ /

g = acceleration due to gravity

Ua = adjusted 10-meter windspeed

X = dimensionless fetch = gF/U^

a = Phillips equilibrium constant = 0.076 x

-0.22

Y = ratio of maximal spectral energy to the maximum of the corresponding

Pierson-Moskowitz spectrum = 3.3

CTg = left-side width of the spectral peak =0.07

a-. = right-side width of the spectral peak = 0.09

This equation provides a wave spectrum as shown in curve 1 (Fig. 1), with a

total energy equal to the deepwater significant wave height, squared over 16.

15

2. Energy Reduction in Shallow Water .

As an irregular wave train enters transitional and shallow-water depths,

the presence of the sea bottom causes changes in wave steepness which, due to

the limitation on wave steepness, lead to a loss of wave energy. Kitaigorodskii,

Krasitskii, and Zaslavskii (1975) suggest that an upper limit of energy exists

at a given frequency which is a function of depth and a:

E(f) = ag2 f-5 (2u)-'+ (A-2)

where

C^ tanh (w^ C^) = 1

d = water depth

a = 0.0081

^ = Ch{l + [2a){i Ci^/sinh(2aJh C^) ] i

OJ^j = 2TTf /d/i

This equation represents a stability limit or "limiting form criterion" on a

wave component. Kitaigorodskii, Krasitskii, and Zaslavskii used a value of

a of 0.0081 based on field data. Recent work at the U.S. Army Engineer

Waterways Experiment Station (WES) has indicated that another mechanism, non-

linear wave-wave interaction, has an equivalent effect but that a would vary

with dimensionless fetch (gF/U^) . The application of this theory is further

outlined by Vincent (1981) .

Shoaling of a wave in shallow water also changes wave energy. A shoaling

coefficient can be calculated as in the Shore Protection Manual (App. C in

U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) for

each frequency component according to linear theory:

and can be multiplied by the deepwater energy at each frequency band to obtain

a "shoaled" spectrum,

E(f) shoaled = Ks(f) E(f) deep (A-4)

3 . Determination of Shallow-VJater Energy Spectrum .

Figure A-1 is a flow chart describing the solution process used in produc-

ing the design curves presented in this paper.

16

Q>

Generate deepwater JONSWAP spec-

trum using equation (A-1) based

on windspeed and fetch length.

At each frequency compare energy from equation

(A-2) with that from (A-4) . The greatest of

the two values at each frequency band is re-

tained to form the spectrum at the particular

shallow-water depth.

( Stop )

Figure A-1. Flow chart illustrating the use of equations (A-1) to

(A-4) in generating the curves presented in Figure 2.

r-^

0) r^

> cu

a.

S -^ vo

o

3 4J

3

3 4J

-id

(U X!

-ili

0) x;

Vi g

4-t -T3 >-. iH i-i

01 S

4J -3 >, rH U

d d -0 -H CO d

01 rH

3 3 .0 M CO 3

-T3

CO V4

â€¢3

CO CO .CO) 0)

O CO

CO

.^".^â– ^o I

M '"

'ffl

-o 3

y-l -O iJ T3 ^ -H

UH -0 4J -o 4-1 â– H

nH

â€¢H 4) M t. >

â– H 0) 10 u >

O O ^

C 0) ^1 ti3 n)

U XI

CO

3 01 IH CO CO

(0

M C. CU 3 -1

00 D, 0) 3 -rl - .

B >. "-t

â€¢H (fl C QJ U 1-1 01

a &.rH

â€¢H

â– rl 01 3 0) H VH 01

S tâ€¢r^

d

to Ta 0) p 0) a

CO a -H

â€¢r4 S-l flj

^

d 00 i-t -H

,C

3 00 4-1 â– rH

.n < >

tU -H Â£ -H . . )-.

-i < >

0) â– H X; -rl . . IH

^ Cfl

tJ30)miJ-OMa)

-H CO

V43a)oiV4^ooia)o4

U 0) cu w

,H â€¢

3 cn "

00

^ QJ QJ 0] Xt >

0) ji: 3 u C 3 Â« .

s

3 en â€¢â€¢

00

^^ S 2 3-5 S .S

>> o â€¢

U 4J U -1 3 M

>, :=) .

3

14 4J 4-1 rH 3 rH

J^ CT]

â– H

X3 CO

CO 01 â– rl M .

<u .a)a)iJCt.>H

â€¢ â€¢ >

C ^ > Cl. rt -r-f OJ

3

Bxi'>O.CO^r4a) 3

0)

u va Ti

M M 3 -H 0) n) .

IH CO -3

3

00 00 3 â– H 0) CO .

tH

ccoa,au3a)

01 > rH

â– rH

BBOO-atiSOl

â– HO) < -H CO rH

OO

4-1 0)

00

d

W ^ C 3 -H .4-1

CO - -rl

3

W rH 3 3 rH . tJ

"3 U ^-i

â– " J= -2 Z 01 i " H

3 U 14H

0)

Cn CO CI â– rl

â€¢H x: -H z 0) a H

â€¢H 00

â– rl 00

s o c

^

â€¢0 u 0) (U

30c

M

â€¢a u 0) 0)

O > -H

tu 4-) u -n (U rH . .

> -H

rH i-H U

H (U -H C ^ 4J M

1 2

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