Technical Note: On the formulation of Jonswap spectrum

Technical Note: On the formulation of Jonswap spectrum SUBRATA K. CI-IAKRABARTI Marine Research and Development, CB1 Industries Inc., Plainfield, 1L 60544, USA

The purpose of this note is to derive the relationship of the significant wave height and average period of a JONSWAP (Joint North Sea Wave Project) spectrum with its peak period and the peakedness parameter. The JONSWAP spectrum, developed by Hasselman et al. 1'2 describes the growth of waves in a fetch-limited sea. The JONSWAP wave energy density spectrum is written as follows: S(f)= ~-~

4 \fo /

J

Similarly, the peak frequency fo ma,y_.~e related to the zero crossing frequency, fz (defined as ~ o ) by fz = 1.4fo

Thus, the form of the P - - M spectrum may be modified (e.g. by Bretschneider) into a two-parameter spectrum by substituting equation (3) in the P - - M spectrum as: S(f)-

7 e x p l - O ' - f o) 212oaf~ol

(1) where o = o a = 0.07 for f<~fo and o = o b = 0.09 for f > f o , g = acceleration due to gravity, f = wave frequency, f0 = peak energy frequency (= 1[To, To = peak period) and 7 = peakedness parameter. The quantity, t~ is equivalent to the Phillips' constant and depends on the fetch parameter. In the absence of this parameter in a design case, it is taken as c~ = 0.0081 similar to the case of P - - M spectrum, a The JONSWAP spectrum has a sharper peak than the P - - M spectrum to account for the effect of the restricted fetch. The value of 7 may usually vary from 1 to 7 with average value of 3.3 found for the North Sea. Note that the form of the spectrum in equation (1) reduces to the P - M spectrum for 7 --- 1 and higher the value of 7, the sharper is the peak. Lee and Bales 3 modified the JONSWAP spectrum into a two-parameter spectrum by replacing ,v by /3 and derived an expression for/3 as follows: (Hs]

(4)

5Hs2916 fsexp

--4

(5)

The quantities, H s and fo (or, equivalently, Tz = 1/fz from equation (4)) may be considered here as the two parameters in describing the modified P - M spectrum. Similar forms have been suggested by the ISSC and ITTC committees. The JONSWAP spectrum may also be considered as a two-parameter spectrum in terms of 7 and fo if t~, o a and o b are taken as constants with values prescribed earlier. However, in a design case, usually the significant height and average period of a random wave are specified. Unfortunately, the moments, rn n of the JONSWAP spectrum may not be obtained simply in a closed form and the values of 7 and fo are calculated numerically by trial and error from equation (1) using H s and Tz. A detailed analysis of

56"

1"375

/3 = 16.942 \ ~ - ~ ]

48"

(2)

However, it is inconsistent in that the spectrum does not provide the significant height, H s used in equation (2) to obtain the value of/3. The P - - M spectrum is a one parameter spectrum. For example, if fo is specified, the form of the spectrum is known. The quantity, fo, in the P - - M spectrum may be related to the significant wave height, H s by integrating S 0 0 between f = 0 and oo and noting that H s = 4x/~o, co~ = 0.161 g/H s

40" bLtJ taJ h. .. -r

32'

24"

(3)

Where COo=27rfo and m n = nth moment of the energy spectrum, Y =7

m n = .ifns(f)

df

0

Received September 1983. Discussion closes September 1984. 0141-1187[841030175-02 $2.00 © 1984 CML Publications

8

7

~

i'i

13

I;

?7

To ; SEC.

Figure 1. Comparison o f "empirical v~ numerical values o f H s vs. To for 7 = 7

Applied Ocean Research, 1984, Vol. 6, No. 3

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Technical note

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which has an error of less than 0.5% for P - - M spectrum (equation (3)) and To = 1.4014Tz

~-~1.3-

~o...~. o

1.2-

~

0,~.,~,.~

Legend --

Eq. 7

o

Actual Hs= 30 FT.

11-

1.0 y Figure 2. Comparison o f empirical vs. numerical values o f To/Tz vs. 3"forHs= 3 O f t (9.1 mJ

(9)

with an error of about 0.1% (equation (4)). The correlation of H s and To between the present equation, equation (6), and the actual values from equation (1) for 7 = 7 and H s = 10-50 ft is shown in Fig. 1. Note that the correlation is excellent. Other values of 7 give equally good comparison. Similarly, equation (7) is correlated in Fig. 2 for an H s = 30 ft and 7 = 1-7. Note, however, that the quantity To/Tz is insensitive to the value o f H s, so that Fig. 2 may be considered good for all H s. Equations (6) and (7) may be used to arrive at H s and Tz from To and 3' instead of the more complicated calculation using equation (1). Alternatively, if H s and Tz are given for a particular sea state to be represented by the JONSWAP spectrum, then equations (6) and (7) may be solved for To and 3' by a simple iterative technique without involving the more complicated form, e.g. equation (1).

REFERENCES the relationship among these four parameters showed that H s and Tz may be related to To and 3' by the following two polynomial equations: H s = (0.11661 + 0.01581 7 - 0.0006572) T02

(6)

and To = (1.49 - - 0 . 1 0 2 7 + 0.014272 - 0 . 0 0 0 7 9 3 ,2) Tz (7) From the above equations, for 7 = 1 H s = 0.1315r~

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1 Hasselman, K. et al. Measurement of wind-wave g~owth and swell decay during the Joint North Sea Wave Project (JONSWAP), Deutsehen Hydrographisehen Zeitsehrift, Ergan. zungschefi 1973, 13, No. A 2 Hasselman, K. et al. A parametric wave prediction model, Journal of Physical Oceanography 1976, 6,200 3 Lee, W. T. and Bales, S. L. A modified JONSWAP spectrum dependent only on wave height and period, David Taylor Naval Ship Research and Development Center, Report No. DTNSRDC/ SPD-0918-01, May, 1980 4 Pierson, W. J. and Moskowitz, L. A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii, Journal of Geophysical Research 1964, 69 (24), 5181