A modified JONSWAP spectrum dependent only on wave height and period

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RESEARCH AND DEVELOPMENT CENTER

-DTNSRDC 5602/30 (2-80) (supersedes 3960/46)

DAVID W. TAYLOR NAVAL SHIP

Bethesda, Maryland 20084 by Wah T. Lee and Susan' L. Bales

Lab. v. Schiet

Technisch

A MODIFIED JONSWAP SPECTRUM DEPENDENT ONLY ON WAVE HEIGHT AND PERIOD

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DTNSRDC/SPD-0918-01

2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

.

4. TITLE (end Subtitle)

A MODIFIED JONSWAP SPECTRUM DEPENDENT ONLY ON WAVE HEIGHT AND PERIOD

_

S. TYPE OF REPORT 8 PERIOD COVERED Final

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(*)

Wah T. Lee and Susan L. Bales

S. CONTRACT OR GRANT NUMBER(e)

9. PERFORMING ORGANIZATION NAME AND ADDRESS Ship Performance Department

David W. Taylor Naval Ship R&D Center Bethesda, Maryland 20084

10. PROGRAM ELEMENT, PROJECT, TASK

AREAS WORK UNIT NUMBERS

(See reverse side) 11. CONTROLLING OFFICE NAME AND ADDRESS

Naval Material Command

Washington, D.C. 20360

12. REPORT DATE May 1980 13. NUMBER OF PAGES

27

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APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (ContinTio on rempree side if necessary and identify bY block nuashar)

JONSWAP Spectrum Htndcast

.

Spectral Ocean Wave Model (SOWM) Fetch

Wave Energy Spectral Density

20.

,

ABSTRACT. (Continue on re eeeee aide if necessary and identify by block number)

For simplicity as well as consistency with the current state-of-the-art in seakeeping performance assessment, a wave spectral formulation for fetch-limited ocean areas which is dependent only on the significant wave height

and modal wave period is desirable. A modified version of the JONSWAP

spectrum is therefore derived,, but as is the case with the usual

fetch-.

dependent JONSWAP spectrum, it contains too much energy for fetches above

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(Block 10)

Project Number 62543N

Block Numbers SF 43 421 202 and SF 43 421 001 Work Unit Numbers 1504-100 and 1500-104 (Block 20 continued)

about 40 nautical miles. _{This inconsistency is probably due to certain}

parameter relationships which are all based on least squares fits.

Therefore, a new parameter, _{a, is developed to replace the usual a}

TABLE OF CONTENTS Page LIST OF FIGURES LIST OF TABLES _iv NOTATION ABSTRACT ADMINISTRATIVE INFORMATION INTRODUCTION

MODIFICATION OF THE JONSWAP SPECTRUM 2

$ PARAMETER t5

APPLICATIONS OF THE MODIFIED JONSWAP SPECTRUM 6

VALIDATION OF MODIFIED FORMULATION ₇

CONCLUDING REMARKS

,7

REFERENCES ₉

APPENDIX - DESCRIPTION OF COMPUTER PROGRAM JONI 21

LIST OF FIGURES

1 - Description of Five Defining Parameters of the

JONSWAP Spectrum ₁₁

2 - Comparisons of JONSWAP Theoretical and Actual

Significant Wave Height and Fetch Relationships 12

3 -

Determination of 8 for Modified JONSWAP Spectrum 13

4 - Determination of Small a for Modified JONSWAP

Spectrum ₁₄

5 - Typical Modified JONSWAP Spectra for Significant

Wave Heights of 2 and 5 m 15

6 -

Typical Modified JONSWAP Spectra for Significant

Wave Heights of 3 and

_.7

m ₁₆

7 -

Modified JONSWAP Spectral Relationships Between

Modal Wave Period, Fetch, and Wind Speed ₁₇

Page

8 - Modified JONSWAP Spectral _{Relationships Between}

Significant Wave Height, Fetch, and Wind Speed . 17

9 - Comparisons Between Modified _{JONSWAP Spectra}

and Measured Spectra

18

10 - Comparisons Between Modified JONSWAP, _{Bretschneider,}

and Hindcast Wave Spectra

19

LIST OF TABLES

1 _{- Significant Wave Heights for Varying Values of}

y 20

2 - Computer Program JONI

22,

3 - Output from Program JONI ₂₄

iv

Wave frequency, cycles sec-1 or Hertz

fo Frequency corresponding to the peak of the wave spectrum,

cycles sec-1

-Acceleration due to gravity, 9.8087 m

sec2

mo Spectral moment of order zero

S (f),S (0)) Long-crested wave spectral density ordinates

Wind speed, at 10 m above ocean surface

X

aw)1/3

To

Fetch

Significant wave height, average of one-third highest double amplitudes

Modal wave period, period corresponding to the frequency of the peak of wave spectrum

Phillip's constant

A constant dependent on the significant wave height and modal wave period

Ratio of the maximal spectral energy to the maximum of the corresponding Pierson-Moskowitz spectrum

Left, right side widths of spectral peak of JONSWAP spectra

Wave frequency, radians sec-1

NOTATION

aa '71)

ABSTRACT

For simplicity as well as consistency with the current state-of-the-art in seakeeping performance assessment, a wave spectral formulation for fetch-limited ocean areas which is dependent only on the significant wave height and modal wave period is

desirable. A modified version of the JONSWAP

spectrum is therefore derived, but as is the case with the usual fetch-dependent-MSVAP spectrum. it contains too much energy for fetches above about 40

nautical miles. This inconsistency is probably due

lo certain parameter relationships which are all

based on least squares fits. Therefore, a new

parameter, B, is developed to replace the usual a

parameter, correcting for the parameter's nonuni-versality.

ADMINISTRATIVE INFORMATION

This report was prepared under the sponsorship of the Conventional Ship Seakeeping Research and Development Program, funded under Project Number 62543N and Block Number SF 43 421 202 and the Ship Performance and Hydromechanics Program, funded under Project Number 62543N and Block

Number SF 43 421 001. It is identified by Work Unit Number 1504-100 and

1500-104, respectively, at the David W. Taylor Naval Ship Research and Development Center (DTNSRDC).

INTRODUCTION

The primary goals of the Joint North Sea Wave Pro'ect (JONSWAP), which originated in 1967, were to measure the growth of waves under limited fetch conditions and to analyze attenuation of waves propagating into shallow

water, see References 1 and 2.* The fetch dependence of the measured

one-dimensional frequency spectra was investigated by parameterizing them with

an analytic function derived by least squares fit techniques. _{The resulting}

function, currently the most widely used spectrum for representing fetch-limited seas, is known as the JONSWAP spectral density equation and is given by / [i (f-f o ekp

-(270-4 f-5

ex[-

i44-41

S

= a

g2

2a2f-2

m2-sec

which reflects the five parameters of fo a, y, aa and ab shown in Figure

1. f is the wave frequency in cycles sec-1, fo is the frequency at the

spectral peak, and g is the acceleration due to gravity. In this report,

a "mean" JONSWAP spectrum has been used, so that .y is 3.3,

aa is 0.07 and

ab is 0.09. The two parameters., a and fo, are dependent on the wind speed

and fetch such that

a = 0.076

R-M2

(2) and where ;.= gX A U2 and 0 = 3.5 R-0.33, < l04

X is the fetch in nautical miles. The wind speed U is taken to be at an

elevation of 10 m above the surface and is in units of knots.

The JONSWAP spectral form represents a generalization of the Pierson-Moskowitz form by inclusion of fetch as an additional parameter to wind

speed. If a is 0.0081 and y is 1 in equation (1), the JONSWAP spectral

form is identical to the Pierson-Moskowitz form. In general, the JONSWAP

spectrum contains more peak energy than the corresponding Pierson-Moskowitzl spectrum for the same values of a and fo, see Figure 1.

MODIFICATION OF THE JONSWAP SPECTRUM

In previously completed ship seakeeping analyses, e.g., see Reference1

3, in which the ship's operations in fetch-limited waters has to be

assessed, it has been customary to apply the JONSWAP spectrum, defined bX

wind speed and fetch as given in equation (1). However, for simplicity as

well as consistency with the current state-of-the-art in seakeeping per-formance assessment, a JONSWAP expression which is dependent only on the two parameters, significant wave height and modal wave period, is also

r

desirable. In order to achieve this, the following steps have been

-,carried out, Eliminating the dimensionless frequency

fo

from equations

(3)

and (5), U =

3'5

R-13.33g fo where 1 f

0 7

= or

U = 3.5

k-C1.33g To

The fetch dependence of the dimensionless total energy in the wave spectrum is given as

2 m g

= (9)

Furthermore, another fetch empirical relationship is well represented by the following relation

M = 1.6

(10-7)R,

< 104 (10)

where

mo is the spectral moment of order zero. Significant wave height

(Zw)1/3

can be defined as

aw)1/.3

= 4 rril; or mo =

(a )1 2

and substituting into equation (9) gives

M =

(c)1/32

92 0

16 U

7 7.7!.. 77:7

Combining equations (8) and (12),

(414)1/32

92

Mo =

16(3.5

R-(3.33g To)

3

Eliminating the term Mo from equations (10) and (13) yields 1.6

(10-7)R

-r;

2 2 g 16 (3.5 k-033g To) R-0.32 (c)1/32 2603.082 g2 To4 This equation can also be written in the form

1.375 222.92 ce w /3 A

-g1.375 To2.75

Substituting 51 into equation (2)

[

222.92

aw

/3 )

"75

a = 0.076 or 16.942 (c)1/31.375

l.3750275

T

The JONSWAP equation can now be rewritten in terms of (w)1/3 and To as

.

16.942 (Zw)1"751

exp--- FT -1]

S(f) - 2

f,

,-4 ,-5 ,o2 o c

1.17c

2.7c

. g WI./ T exp[1.25(fTo !ly L g ,,, T ,... o m2-sec- (19)

Equation (19) can be further rewritten to be more compatible with the usual Bretschneider spectral density formulation as well as most ship

response calculation procedures by converting f to w, the circular frequency

1

in radians sec- 1, and taking y = 3.3,

or

g1375 T02.75

5

a PARAMETER

The modified JONSWAP spectrum given by equation (20) contains more energy than it is theoretically supposed to for the same values of sig-nificant wave height and modal wave period when these two parameters are

used to define the spectrum.* Figure 2 provides an illustration of the

difference in significant wave height at the same fetch for winds of 20

and 30 knots. The solid line represents the theoretical relationship

between significant wave height and fetch for those wind speeds. The

dashed line represents the values which are actually computed from the spectral area when the given fetch and wind speed are specified in the

usual JONSWAP formulation given in equation (1). The difference between

the solid and dashed lines represents a rather noticeable increase in

significant wave height for fetches above 40 nautical miles. The

diffi-culty arises due to the fact that the absolute value of a is not universal, as was first suggested by Phillips, and also as described by Ewing in

Reference 4. While the i and X data are well defined in the linear

re-gression fits, see equations (5) and (10), there is a large scatter for

* ,

This inconsistency is also present when the usual JONSWAP formula-tion, given in equation (1), is applied and is particularly noticeable for fetches above about 40 nautical miles.

16.942 ( wc

1/3

S(w) = 1.375 g2 w-5 exp -1.25 wT 1 2 wTo -

d

3.3exp-2a m2-sec 27r (2o)

c1.375

T 2'75 2ff where a _{0.Q7 for} w _{I (2t),} T " _o or w 1 a = 0.09 for ri-T To . (22)

absolute values of i larger than 104, Consequently, the basic form of equation (2), and hence equation (18), is not universal at relatively high

waves and periods. To correct this anomaly, a new parameter, 0, is

developed to replace the a parameter given in the usual JONSWAP formulation.

The technique developed to compute a is given in the Appendix. Figures 3

and 4 are the result and provide 0 for given values of significant wave

height and modal wave period. The wave parameter ranges are deliberately

broad in anticipation of any extreme occurrences such as those reported in

the North Sea. The 0 parameter assures that the specified (input)

sig-nificant wave height, see equation (18), corresponds to that which is

calculated from the integration of the resulting spectral ordinates. The

modified JONSWAP formulation given in equation (20) can now be rewritten

as -47 S(w) = 0 g2 w-5 exp -1.25 727°1 icia 6 /

rTo

2 3._exp-2o2 2n 2 m -sec

where :0 is-a constant dependent only on the significant wave height, aw)1/3, and the modal wave period, To.

APPLICATIONS OF THE MODIFIED JONSWAP SPECTRUM

Figures 5 and 6 provide sample JONSWAP (modified) spectra for sig-nificant wave heights of 2, 3, 5, and 7 meters and a range of modal wave

periods. As is the case with the Bretschneider formulation, equation (23)

can be applied without special regard to fetch or wind speed. The user,

upon selecting the values for significant wave height and modal wave period (e.g., from historical wave statistics, climatology, etc.), deter-mines 0 from Figures 3 or 4 and thence the spectrum can be developed.

Figures 3 and 4 contain a limited but realistic range of modal wave

periods for given significant wave heights for fetch limited geographies.

Table 1 provides comparisons of significant wave heights for different,

values of y. By substituting a set of significant wave height values

(e.g., from Table 1) in the abscissa scale of Figures 3 and 4, 0 values

can be obtained for various values of y. Since a range of fetch or wind speed values used to obtain $ could be of interest in some seakeeping analyses (e.g, for specific ocean areas), Figures 7 and 8, developed

from computer program JONI as detailed in the Appendix, provide a com-parison with corresponding height and period ranges.

As with the usual JONSWAP formulation, the modified expression given

in equation (23) is for long-crested seas. The cosine squared spreading

function can be used with the spectrum but insufficient data on the directionality of wave systems in fetch-limited waters is available to verify its applicability.

VALIDATION OF MODIFIED FORMULATION

The general shape of the modified JONSWAP spectra agree well with the shape of the measured spectra reported for Argus Island in Reference

5. Several such comparisons are given in Figure 9. Furthermore, a total

of four spectra in the North Sea, hindcast by the U.S. Navy's Spectral Ocean Wave Model (SOWN), see Reference 6, were randomly selected for

comparison with the modified JONSWAP formulation. The hindcasts were also

compared with Bretschneider spectra defined using the hindcast significant

wave height and modal period. Figure 10 shows these comparisons. In

general, the modified JONSWAP spectra provide a much closer approximation to the hindcasts than do the Bretschneider spectra though secondary

spectral peaks are, of course, not well approximated due to the unimodal

restriction of the model. The modal wave period has been used as the

defining parameter of the spectra in this comparison. However, since

some hindcast spectra contained multiple peaks, average wave period (e.g., zero crossing period) may permit a better shape definition of the theoretical wave spectra.

CONCLUDING REMARKS

An expression of the mean JONSWAP spectrum dependent on the

signifi-cant wave height and modal wave period is derived. However, as is the

case with the usual fetch-dependent JONSWAP spectrum, it contains too milch

energy for fetches above about 40 nautical miles. The inconsistency

arises due to the fact that the absolute value of a is not universal,

while the f and X data are well-defined in the least squares fits

(equations (5) and (10)). Further, there is a broad scatter for absolute

values of X larger than 104.

is developed to replace the a parameter given in the usual mean JONSWAP formulation and to correct for the parameter's nonuniversality. Figures 3 and 4 permit the determination of 8 for given values of

signifi-cant wave height and modal wave period. Hence, wave energy is conserved

in the modified JONSWAP formulation.

Recent hindcast spectra from the North Sea, as well as actual wave measurements from fetch-limited areas, suggest that the modified JONSWAP

spectrum may describe wave growth conditions more realistically than the

Bretschneider spectrum in fetch-limited or shallow water conditions. In

general, it is concluded that the modified JONSWAP spectrum provides a reasonable representation of wave conditions in fetch-limited or shallow water ocean areas.

REFERENCES

Hasselmann, K. et al., "Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP)," Deutschen Hydrographischen Zeitschrift, A8, No. 12 (1973).

Hasselmann, K. et al., "A Parametric Wave Prediction Model,"

Journal of Physical Oceanography, Vol. 6, pp. 200-228 (1976).

Bales, S.L., "Ship Motion Predictions for the MONOB

1,

Operating

In the Waters Near the Bahama Islands," Report DTNSRDC/SPD-727-01 (Sep

1976).

Ewing, J.A., "Some Results from the Joint North Sea Wave Project

of Interest to Engineers," international Symposium _{on the Dynamics of}

Marine Vehicles and Structures in Waves (Apr 1974).

Lazanoff, S.M., "Wave Power Spectra from Argus Island," U.S. Naval Oceanographic Office, Report No. 0-46-64 (Dec 1964).

Cummins, W.E. and S.L. Bales, "Extreme Value and Rare Occurrence Wave Statistics for Northern Hemispheric Ship Lanes," Proceedings of

EMAX EPM MAX 11 EMAX PM

EMAX (Pierson-Moskowitz spectrum)

92

f-5

2704

Figure _{i - Description of Five Defining Parameters of the JONSWAP Spectrum}

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Figure 2 - Comparisons of JONSWP Theoretical and Actual

Significant Wave Height and Fetch Relationships

12

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MIMMIKAMMilmilmilmmeMMEMMEMEMMEm

MMILIMMEMEMMOMMAMMWAMEMMWIMMEWAMMINIEFAMEMMEWAIMMWOUMMM

IIMMOP MEMEMEMMOMtdimilliMEMEMEMMEM

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IOMMAMMIMrOMEMPidEMMIMMdMMEMEMMEMMERMEMMOMMEMMENEMOMMIMMMEMO 0 2 4 6 8 10

SIGNIFICANT WAVE HEIGHT, (w)1/3, M

Figure 3

-Determination of 0

for Modified JONSWAP Spectrum

12

14

16

MOOMMOOMMOO' MOOMOOMMOI MOO

MEOW

MOOONOMMKOOMMOOMMOKOMMOOMMOVIOMMOOMECOOMOOMOMMEMOOMOMOMIEMOONOMO MOOMOOMPAOMMOOMMOOWOOMOMMONIMOOMMEMOUOMOMOMOMOONOOOMMMOMOOOMMOOO ol. OMMOOMOIMMOMEMOOMAMOMOOMOOMMOMOOMCOOMMOCMOOMMMOOMOOMMO

* =WOW

MOOMMOIOAMMOOMENOrMOMMOMOOLMMOOMOORIOOMEOUOMMOOMOMOOMMOO

O,

monommommommommummummommunnommomionommummmotommommins nommomnnr monsmmin, "

MEW * MOW

mow mom OOMOOLMOONOMMOOMIHMOOMMOMOM

nommowommunninnmn-ei;

=Emir/

Mr

tlf

ow MOW OM OM

ii

millOomorAMOMOOmmiommN OmMlion

Moor

wommormoommOmommOrm ONOord

MI

Or OOMOWA

...aMEC

IMOMU oor MOMMWOOMOMMMOOMOMFOMOOMMOOMOVA

UMW

11111 OW OMUMOMOOMMOMOOMOMMMOMMOOONOO MOMMOOU 1111! MO MWAIMMEMEMOMMEMMAMMOINOMMEMORAMMIIMMEMEMEM MOM ow I U. U. Ermonninmommomnsnommninommonnommomnimmonnommommumnimmoninnowmmommmonm MUOMOOMMEMMO PAOMEMOMOMMOMMVAMOOMMENOMOVAMOOONOMMEMIOMOOMOOMOIMOOMOSECOMooMMUMEmEMUMOOMOOMMEM AOMOOMOOMOOMMOWOOMMOMOONONEMOMOOMOOMMOOMMOOMMOMOOMOOMOHOMMOUMMOOMIOOMOOMMEMO OMOOMOOMOOMOOKOMOOMOMMOOMIAMMOMMOOOMOMOOMMOOMOLOOMOOMOOIMwoomilOMOOMUOMOOMOMMMO liwwwmwwwwwommwmw000mwwwnrnow.mwwwwwwwwwimmowimmomminimilmilimmtmemm MMMMM 0

Szoo

500" SLOO"

IC

SZIO"

ennumminommomminnEmsnommmunnnEnummomm9=,_-77.7_7=,

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,

MOMOOMPAMEMOONOMMOUOMOMMOOMWOMOOOMMOCOOMOOMOVMOMOMMEMONOOLOMOOMMONIMOMO 010' nnommumminnomminnommonnimmuminnimmrnmemmmummimmnmEnnimnommu n mmin mmonomni 74 MOMOMONNOMONNomMONIAMOONEMEMOMMOMOOMoomOOMEIMEMMONCOmminwirnmumoMOMONE WOO milOOMOvmmoMMEMOommUMEMoomMEWommomMumoomMMOOMMoolomOumomowomwoROOMoolloo 510'

wruloads dVMSNOr Pe141PoW 10

9 Hews 4o uo1leultual90 -IT an161.4

W6/1(m) '114913H 3AVM INVOIJINDIS 8 9 5 0

14 12 2 1.0 1.2 1.4 1.6 1.8 2.0 WAVE FREQUENCY, w, RAD.SEC-Figure 5

- Typical Modified JONSWAP Spectra

For Significant Wave

18 16 ₀ 1.6 1.8 0 . .4 .6 .8 1.0 1.2 1.4 WAVE FREQUENCY w, RAD SEC-I Figure 6- Typical ModifTed JONSWAP Spectra For

Significant

Wave Heights 0,1 and, 7_ m. 2.0 ID

aid /

3 To M SEC U KTS FETCH NM 3-6 3 6 64 15 3-7 3 7 31 52 7-9 7 9 113 29 7-10 7 10 67 69 7-11 .7 11 42 151 7-12 7 12 26 315

200 LU _s 100 (-) 80 cc:LU 60 40

ummannier

MI

II

"142:0°10001s01

MP

_AKirdi

rw

_NENE

!gm- r:Aidimimi

prawns

=mow

AI Mr

org

ENV Awvar,,,,diainiti

Affemium

Arai

6 8 10 12

MODAL WAVE PERIOD, Ti,, SEC

Figure 7 - Modified JONSWAP Spectral Relationships Between Modal Wave Period, Fetch, and Wind Speed

17

6 8 10 12 ₁₄

SIGNIFICANT WAVE HEIGHT,(_{E )} ,

. w 1/3

Figure

8 -

Modified JONSWAP Spectral Relationships Between

.L1J Zo.

0.3 0.2 0

POWER SPECTRA FROM ARGUS ISLAND 32° N

-W

-WATER DEPTH 58.5 M TIME: 12002 DATE: 17 Sep 1962 (Ew)4/3 1.4 M. To 8.57 SEC - MODIFIED JONSWAP (chi3 1.4 M To ' 8.57 SEC .0013 1.0 0.9 0111 _0.7 0.6 0.5 0.4 0.3 0.2 0.1 1

-WAVE FREQUENCY, w, RAD.SEC

1

Figure 9 - Comparisons Between Modified JONSWAP Spectral aftd-Me0S-uriii SPed-tA,

- MODIFIED JONSWAP aw)1/3 ' 2.3 M To 7.5 SEC 0 .0058

POWER SPECTRA FROM ARGUS ISLAND 32. N - 65.2* W WATER DEPTH

58.5 M TIME: 1500Z DATE: 21 Sep 1962 (Q1/3 ° 2.3 M To 7.5 SEC 0.9 0.8 0.7 C.) IJJ 0.6 C1 tZ a. 0.5 o.4 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0

25 20 15 10 0 INDCAST (MAR 1868) 6'7 " T.1 11.4 SEC MODIFIED JONSIJAP aw)//3 " 6.7 PI To 13.3 SEC ORETSCHNEIDER (c)1/3 6.7 M To 13.3 SEC

arier Dag isEa) aw)1, 4.8 .T., - 4.1 SEC MODIFIED jORSIall; 6.8m T. 11.6 SEC IDETSCICNEI DER (c)1/3 PI 11.6 sgc 19 10 I 2 .2

WAVE FREQUENCY, w, RAD SEC-1

Figure 10 - Comparisons Between Modified JONSWAP, Bretschneider, and Hindcast Wave Spectra.

Nilectrr (mg 1968) (Ewhii - 1.3 N sec MODIFIED JODSIEW (k) 3.3 N SEE DRETSCEMEIDEA (V1/3 3.3 N 9.1 SEC :8 1.0 .1-30 30 .2 1 .4 .6 .8 1.0 .2 .4 .6 .8 1.0 1.3 MINDCAST (MAR 1968) ' 8 M T-1 11.4 SEC MODIFIED JONSWAP

()1/3

8 T 19.9 SEC BRETSCNNEIDER (Ew)1/3 8 M To 18.3 SEC I .2 .6 .6 .8 1.0 1.3 25 2 15

TABLE 1 - Significant Wave Heights for Varying Valves of y. 20 Y = 1 Y = 3 Y = 3.3

y5

Y = 7

x

fel 1.6 2 2

2.2

21.4

"

,..i

3.2

3.9

4

4.4

4.8

? LP

--

4.9

5.9

6

6.7

7:3

1-:

=

La

6.5

7.8

8

8.9

9.7

La

=

lit 8.1

9.8

10 11.1 12.1

<

9.7

11.8 12 13.3 14.5

1-z

11.3 13.7 14 _15.5 16.9

a

,..) 13.0 15.7 16 17.7

_19.4;

Z

C., 14.6 17.7 18

20.0

21.8 1/1

APPENDIX

DESCRIPTION OF COMPUTER PROGRAM JONI

It has been shown in the preceding text that there is a noticeable

discrepancy between the theoretical and actual significant wave heights for

fetches above 40 nautical miles when the JONSWAP spectrum is _applied.

Therefore, a new parameter, 8, is used to replace the a parameter in

equation (20) , eg,

16.942

_(V1/31.375

s

g1.375

T0275

where (Cw/

)1, _{is the theoretical significant wave height in meters.}

3

A listing of program JONI is presented in Table 2. _{The purpose of}

the program is to calculate 8, modal _{wave period, and actual significant}

wave height under the spectral area. _{Results are plotted on Figures 3 and}

4,

with _{against the actual significant wave height,}

a

_{)1, , and the}

w /3

modal period, To. _{For example, if the significant wave height is}

_4.08

_m,

and the modal period is 8 sec, then, by reading across the intersection at

4.08

m and 8 sec, 8 is seen to be about 0.0135. _{Substituting 8 value into}

equation (23) and the modified JONSWAP spectral _{can be generated by varying}

the values of w. _{A typical output from program JONI is presented in Table}

3.

21

TABLE 2 - COMPUTER PROGRAM JONI CM*E,CMD5O00fT150.e4. CHARGE.CHwE,LCLLUI8428. FTN(T.mi1frT=4,H=4) SETCURE(INUEF4AUDR1 LGu. 0000(10000u000)00000o00 .

eRuGRAm u0N1 (INPUT4S12,OUTeUT=12'.IAPE5=INPUT,TAPEtS=OUTPUT/

dimENSIuh

S(i00).w(68)

UATA N.h1/68.30..4.0veb/ 9ATA

2 .10../S.:8O..8,.16,...i8.1.01.1.1.05,I.L091.15.1.2O94.2591.30.

2. lejS, 1..44.).. 1 .45. i .8O .1.8541 .6O, I .6894 /0,1.7+1 .80,1.854

2. Ie10.1...2.ou.E.uS.e.1uic.1.2.2098.92.:10.2.J592.4092.45.12.50. 2 2..55.2.8O.2.,=)598./09c.7i42.8(1,240548./O,e.9S.J.00.3.059.3.109. 2 J01.5.3.eo,34/

READ (59100) C4LS

dU 200 1=1,NCASE5 READ (8.101) FEILm,u

CALL JUINSt.1 (N.prtILm9U.5wh1iT01..S) SahFr = Sw1-1.43.ec CUNA=16.'4421's4.10**I.J/5 CUNO=0.0W87**1.J/,1,4.10**2.13 BETA=c0AA/CU:41 WRLTF (6-flOc) FELLm.u.Swh1.8w1F(ATO.oETA tiO

Isa

= w(J)/(2.1)

4=1./F -SF1=s(J)*3.e.7t*i..ed 160 wRITF.

(bilu-3)

(..,104(J).S(J),5tT

cAlr.L ALk2,4146 (A9A.,AkEA)

-RMS=S'..4RT(AGA) WRITF (6411..) S1-,3m=4.*Rms SI.JFT=.*RMSFT

wRI1F(6.10S) Sium;SibFT

euu

coTiNoc

sT,0-) IOU- EukmAT .(15) 101

tiltimIT (8FiU.6)

--102

I :OkIfir (1h1.*

Jur-.5m,h SPECTr.:01.1*/* FtiCh

-.

2 * NM, .400 Si-,tEu .=*rn.u.*KTS. S1t.0AliL hi. =*F.8.2* M.=*

2 c5.2* FT; MV)4L eAVL,PEk:

g'SELUNUS UETA=*F10.8///

2 4X *F.* .4X aTijN Itree 8X *S9m*(A *S.t)*/)

103

F(MmAT (3FIo.3,erio.8)

104 .FORMAT (*

Rms =ier5.2 m

Fr*)

1(.1 EUkmay

SI6.

'IT. =*1=.4 * m

* FT*)

*DECK ALGq *DECK JoNs C.

C

.6004qUTIE

JUNSwAP setCTPum FROM WINO SPEED ANU FETCH.

C

CALCI!LA1.10N iS FREuUENLY t

AND IN ME Kit: UNITS INIi1ALLY

C

10 (,1" A

PIt,-±Si.r. musrDwITL SCiUM, LEI ALPHk=.0O81 AN()'

C,

opwim4=1.0. SU9r401a1NE

'EL NOrtTCri.Nor

olmE,Isi(J14 S(

1.494( N)

UA14 PII),Ge--4-5.141j,e50.dU619.0(pjul/

UA1A JAP.MA,S16A.316/3.3i401o.UV/

UO=U*1.069/j.26

rETCHO=t0cr_l*IroDc. 22

TABLE 2 - COMPUTER PROGRAM-JONI (Continued)-;I0FETCH = Glei,E(Lilu/W/UU NDFM =. J.5/60F,LICH)**.33 em = 401qi*G/flu

T0=1/FM

. ' ' $f6ww F 4. _{soRIT.4..5*NUFETCH*uO*U0*uU*UO/G/G/LUOOUOU0.1}. .

ALPHA = :V/0/(NUreTl.m)**4'd2

e

C- E0k A PURSON-mUSA0wITZ VELTRUH, Sc.T _{ALPHA=.0081 AFTER THIS}

C---

COMMENT CARL). CON1=ALrmA*b*G/( 4*e1)**4 00 log 1=1.N . f2W1T//(d.*e1) ARG1 = -1.2',./(r/rm)**4 IF ((F-rm).L..E.Ee) 4452=2.41SluivwsI(JA*FmfFM IF ((F4m).vi.t.1;5) 4P4U2=d*S1064,SItm*FM ARG.3=-(r.-Fi-.)*(1._-rm)/AR62 If 611461 _S(I)=U 1F (ARGI .LT.

-quit.) GU To 100

uU IJ 114 ARG4 = LAP.0(4G3) (jU TO

lie

ill ARG4=q C

C- :10( A PIER'SJN-muFSU4ITZ SPECTkum, SLI GAHmA=1.0 OTER THIS

-C-

LumMET CAWO,

C

112

5(1) =

CONI/F**D*EAetAR61)*Uummm**Akb4

C

C- CONvERT SPECTRum To Ma*

_{5E. (FKum.m**e/HZ).}

C

S(I)

= sun /(e.

el)

100 _ONTINUL

RETURN ENU

SUdROWTINt ALI;k140 ( 041*,STARLIA)

*FuRTRAN ROuTINt To PEReUR,1 A LAoR$01U1AN _INIEURATIUN.

01HLNsION w(N).S(N) MN=N-2 4=1*mN, ? A=4( (M.2)-A (m) 8=W(M42.)-4(m+1) C=W(11.61)-4(!4) 10 AREA=ANEA.Amim;(

_{(m)*(3.0.6-m)/(m*C)....S(m+1)*A/(8*6+}

s(i4.?)*(e.*.4-.1.*L)/(4*im

.20

CUNT1AUL RtTUON ENO 0000600u0u00U0uU0OuJOU

50.

40.

D00000000U0000u4JOUuoU0 23

TABLE 3 - OUTPUT FROM PROGRAM _JONI JONSWAt! SPFCTHuM

095

1u.472 ou0 1U3 9.6ob _.020 .111 0.970

./Ja

113 b.37d ./nu 1e7 f.P5.34 .0uu .135 1:3'd .143 6491i ../uu 121 0.614 .vnu .129 167 2.2.3 2.914 .i.uuo k.unu 115 n.712 1.1uu. .1d3 2.4b4 1.12u .111 D.233 1.2uu i4143U 4.633 297 c,Oold _J.ItsvOO 215

1.Juo

:

:=

.25127 2./0323

I::

.21Jou 2.29004 .2e3

1.4uj

.231 4.33J .16212 J....6vb/ .An3do 1.07o7d .239 1.n JO 247 ::t1):

1.0

3./2( D2 111::A .113o , 0 .133o2 L.43972 .225 D L.24u49 .263 3.8uo _.v9109 i..u17.= .211 ..i.tio

u64,

219 J.54U .u7nev .6099d

Jo

./U737o/D .U5fot. .219/1 .34457 .,Jube 41991 0.14-.bi .42423 .V.34-7 J.J..1 J7bUl .4341n ..1.3l(Jo .

J#.!01.29771

u2474 e6,v0 .u2e1J e3808 .111."060 .eLabe Jitoo 19211 .I/J13 uiou, 16b3n ,0,145.) ..i1J15 .1;10.:75 .v11./2

i 598

:=5)

::1045:

.u00/0

uould uoub3 ...Woeu uwubo .U7384 .u0/4$ .uUoc9 4

:=7:

.5 .U6112 .:Lost'el1 1.933 .3.2n0 .517 INTEG0AfEU '-'0.3

= 1.)2

:,-.

= J.3i FT

STU. WAVE 'IT. = 4.01 m =1.3.39 FT

(CALCULATED FROM INTEGRATED AREA OF SPECTRUM) .

24

.2db

J.491 1..30u

:LI:

J.3/0

.3.30/ 1. ,330

4.'00

.310 J.e2C

.318

i.1.+2

=

.326

.334

d.99e..1.06 e.v5U

2.1v0

.342 C.92e

e.Li)

.320

e.d'Do e.4u0 .328 e.73

2...eu

.326

2.7Jc.

2Juo

314

2.614

2.-inu , .3d2

e.616

d.'.-J0

.3/0

e.nb,

e.42.)

.316

nl.i

e.5Jo .4%4)

c.484

e.55o

4i4

.4e2

e.4ll2.3/1 e.coit, e.0'Do 430

2.32/

2./uu

43i e.285 2.1nU

.446

d.e44

e.714

e.2un e.uno

::4

.42

e.151

e.,uu

.410

e.liu

_e.7no

4/7 2.0,74 .1.uuu

.465

2.00u

1.u,0

cooet

J.1J 0

oSU1 ii"1.17 3.17U

.5U9

1.9bi

J.e0u

FETCH =

60.

1.4ND it-Ltu =

mouAL WAVE PER. = 5.005ECOND5

F T *

40.KV). 31u. kiwiE HT. = 3.51 M =11.21 FT

WETA=

.013510

THEORETICAL

5.1

S.FT

.0U8 1e.664

_.unu

_u.uuuuu

_U.UOUOU

.016

be.d34

_.1u0

U.LWUUu U.UUU0O

.024

. 41ariti 1.0..Wuuu U.00000

.032

.040

.04g

3i.410

e.lis

2u.4

94

.)0

.220.

sill0

U.Juutiu

u.uuvuu

u:Tulig

u.UUUOU

.026

11.932

,UUUtiv

:A:2:

.uuuuu

.uutiOU

.064

1.70o2

_.uuuuu

_UOU01.

.012

1.3.963 .-020

.uOub/

.U0/lo

.0dli

12.562

.22293

.od8

11.424

..r0

.14,40/

ucluu

_i.33o45

.4.$121 4.o4299 .032o/

_o.92o12

1.339c, 3.24423 44.90e715

12.2buen

4.uu1Je

4J.u4rde 2.37fot _/.131U1 .1.42ouJ i5./7e14

i.u1411

1u.91u13

::=

4:11:=

2831-1 2.20(.103

.49ell

D.J00/2

.1.14

4.47u55 ..1.3u7u J.77300 .uu49u .u3-269 .ou4D2 .4.14ob0

.uu-.1,

oduido .U446!.U4149

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