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1 ECOH/YG No On the Role o Spectral Width and Shape Parameters in Control o Wave Height Distribution Yoshimi GODA and Masanobu KUDAKA ( Distributions o individual wave heights approximately ollow the distribution regardless o the bandwidth o requency spectra, even though Longuet-Higgins (95) derived the theory on the condition o narrowband spectrum. The spectral width parameter does not aect the wave height distribution, but it is merely a measure o the minuteness o data sampling interval relative to mean wave period or waves in the sea. A spectral shape parameter introduced by Rice (945) or wave envelope amplitude is a better measure o describing a slight departure o wave height distribution rom the. Several sets o previous wave simulation data and ield measurement records have been re-analyzed to veriy the above indings. Key Words: Wind waves; signiicant wave; Wilson s ormulas; minimum duration 95 Longuet-Higgins (95) H max H /3 H (99) Goda (97) (974) H /3 m () 5% H 4 η = rms m () / η rms m () n = n mn = ) d () ) () 96 (wave rider)
2 () H m (3) T (4) T T = (3) / ( m/ m ) T = (4) m/m ) = A : m min D exp[.5 =.95,.8m 5 4 exp[.5 ] ) = A + α(.5 ) exp : =.3, = 6. min max max ] = (.5[(.5 ) ] ) (6) (7) () H / 3 η rms 5 NOWPHAS Rice (945) (5) (7) (Goda 97) F ) = A : 5 min M exp[.5 =.5, max 4 ] =.. (5) Normalized Spectral Density, ) / m.. M- M-4 M-6 F-4 M-7 M-8 M Normalized Frequency, / FM Normalized Spectral Density, ) / m.. D- D- D-3 D Normalized Frequency, / D F PiersonMoskowitz Bret-schneider
3 p =. max =. 6 A m M m.5,.5,.5,,, 5.8 Normalized Spectral Density, ) / m Normalized Spectral Density, ) / m.. γ = γ = 3.3 γ = γ = Normalized Frequency, /.. J m = 3 m = 5 m = m = Normalized Frequency, / W D p =. p =.4 α =.,, 5, 5 4 F M D 9698 TOSBAC π t /(5 max ) 987 JONSWAP Wallops (Goda 988) (8), (9) J ) = A : σ =.7 or : 5 min W exp[.5 =.6, ) = A : m min max 4 ] γ = 6. exp[ ( ) / σ ] <, σ =.9 or exp[ ( m/4) =.6, max 4 ] = 6. JONSWAP (J ) γ.5,, 3.3, 5, 7,, 4, 8 Wallops (W ) m 3, 4, 5, 6, 8,, 4, 8 m = 5 J γ = J W max = 6 p t = T p / FFT 496 (= 34.3T p ) FFT (8) (9) 3
4 5 Probability Density o Wave Height, p (H /ηrms) M-5 Spectr. (m = ) ε = Wave Height Ratio, H /ηrms M(m = ) Probability Density o Wave Height, p (H /ηrms) F-4 Spec. (m = 5) Wave Height Ratio, ε =.73 H /ηrms F(m = 5) m = M m = 5 F D-4 H /3 /η rms 4. Probability Density o Wave Height, p (H /ηrms) D-4 Spec. (α = 5) ε = Wave Height Ratio, H /ηrms D(α = 5) 4 (974) 5 /4 (Goda 983) 3 (NOWPHAS) (974)
5 t H /3 T /3 (m) (s) (m) (s) , m 5 8 s 3 Goda (983) 5º 6ºº6º 7, 9, km 5 7 JONSWAP γ m 6.7 s ( 985) (NOW-PHAS) ( 985) 985. m 88 t H /3 T /3 (m) (s) (m) (s) NOWPHAS m t = 5 s 4 (5 s) c = /3 Hz FFT 5 η rms 5 cm 9 5
6 .8.63 m η rms m Normalized Spectral Density, ) / m 5 3 Aug.'98 6 Sep.' Frequency, (Hz) = / 5 Hz m = < /3 Hz Longuet-Higgins (95) CartwrightLonguet-Higgins (956) () ε / ε = [ m /( m m )] : ε () 4 ε Goda (97) Longuet-Higgins (975) ν () / ν = [ mm/ m ] : ν () () Spectral Width Parameter, ν Spectral Width Parameter, ν J-series W-series M-series F-series D-series Approx. ν (/ )ε () Spectral Width Parameter, ε νε Nagoya (Single Spec) Rumoi Yamase-domari Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Approx Spectral Width Parameter, ε νε () Approx. () M D ε () 6
7 () ν ε <.8() ε.8ν H /3 /η rms ε H /3 /η rms ()4.4 Wave Height Ratio, H/3 /ηrms J-series W-series M-series F-series D-series Spectral Width Parameter, ε H /3 /η rms ε F, M D ε H /3 /η rms 4.4ε M3 D3.7 JONSWAP γ ε Wallops ε H /3 /η rms ε Wave Height Ratio, H/3 /ηrms Nagoya (Single Spec) Nagoya (Multi Spec) Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Spectral Width Parameter, ε H /3 /η rms ε ε H /3 /η rms ε ε H /3 /η rms H /3 /η rms ν () ν H /3 /η rms ν νjonswap γ Wallops ε H /3 /η rms 7
8 ν νh /3 /η rms Wave Height Ratio, H/3 /ηrms J-series W-series M-series F-series D-series Spectral Shape Parameter, ν H /3 /η rms ν Wave Height Ratio, H/3 /ηrms Nagoya Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Spectral Width Parameter, ν H /3 /η rms ν 5 () () max = / t t m n max n = S ) d : = t/ () ( max 5 max m4 = () ε = max m 4 ε < ε max p max t ε t /T Spectral Width Parameter, ε Simulation Nagoya (Single Spec) Nagoya (Multi Spec) Rumoi Yamase-Domari Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Relative Sampling Interval, t /Tmean ε Spectral Shape Parameter, κ (T) t /T Simul.(m = 5) Simul.(m = 4) Nagoya Rumoi Yamase-Domari Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Relative Sampling Interval, t /Tmean ν t /T ε 8
9 t /T M m = ν t /T ν max H /3 /η rms H /3 /η rms H /3 /η rms Goda 98399:9.4 Wave Height Ratio, H/3 /ηrms Rumoi Yamase-domari Tomakomai Kanazawa Mutsu-ogawara Kami-kawaguchi Naka-gusuku Wave Nonliearity Parameter, Π/3 H /3 /η rms Π /3 (4) Π /3 3 Π = ( H / L )coth (π h / L ) (4) / 3 / 3 A A L A (Airy) H /3 /L A H /3 L A /(πh) 3 Π /3.H /3 /η rms Π /3. Longuet-Higgins (98), Tayun (983) ε, ν Rice (945) r = ρ + λ ρ = m λ = m )cosπ ( ) τ d )sin π ( ) τ d (5) τ Rice (945) Kimura (98) Battjesvan Vledder (984),Longuet-Higgins (984) Tayun (98) 9
10 Rice (945) τ = T / r r ν Forristall (984)(5) Tayun (98) (5)(6) (99) Forristall (984) κ (T ) κ ( T ) = m = m + m + m )cosπ ( ) Td )cosπ Td )sin π ( ) Td )sin π Td (6) H /3 /η rms κ (T ) H /3 /η rms κ (T ) T (4) T H /3 /η rms JONSWAP Wallops M.3.98 H /3 /η rms D-3 D-4 H /3 /η rms κ (T ) Mean H /3 /η rms κ (T ) Wave Height Ratio, H/3 / ηrms J-series W-series M-series F-series D-series Mean Spectral Shape Parameter, κ (T) H /3 /η rms κ (T ) 3 3 H / 3 / η rms = κ.385κ κ (7) κ (T ) 4 H /3 /η rms
11 H /3 /η rms. H /3 /η rms = Wave Height Ratio, H/3 /ηrms 4. Simulation Caldera Sakata Mutsu-ogawara Kami-kawaguchi Naka-gusuku Kochi-inra Multi-peaks Spectral Shape Parameter, κ (T) H /3 /η rms κ (T ) 6% % H /3 η rms 4 κ =.4H /3 /η rms = Wave Height Ratio, H/3 / ηrms 4. Simulation Caldera (5) Sakata (88) Mutsu-ogawara (7) Kami-kawaguchi (37) Naka-gusuku (35) Multi-peaks (55) Kochi-inra (9) Spectral Shape Parameter, κ (T) H /3 /η rms κ (T ) (.8) (.54) (.44) (.35) (.44) (.66) (.45) (.7) (.93) (.) (.5) (.64) (.5) (.34) (.63) (.55) (.3) (.) (.44) (.64) (.5) ε ν κ (T ) H /3 /η rms (.) 39 (.53) 63 (.88) (.67) 4 (.84) 3.74 (.4) 46 (.95) 3 m L H/ 3 4. m L = 4. 6% H /3 /η rms σ H σ H /η rms σ H /η rms κ (T ) D
12 m = M σ H /η rms.39 H /3 /η rms 4.4 κ (T ) = σ H /η rms =.76% κ (T ) = H /3 /η rms = 686% Wave Height Ratio, σ H /ηrms J-series W-series M-series F-series D-series Mean M(..4)T p σ T =.5T p α = 5D.7T p.3t p σ T =.73T p D Probability Density o Wave Period, p (T / Tp) m =, Tmean =.96, κ =.9 m = 5, Tmean =.74, κ =.39 α = 5, Tmean =.47, κ = Wave Period Ratio, T / Tp Spectral Shape Parameter, κ (T) σ H /η rms κ (T ) σ H /η rms κ (T ) κ (T ) κ (T ) m = M (.8.)T p σ T =.T p m = 5 COV o Wave Period, σt / Tmean J-series W-series M-series F-series D-series Spectral Shape Parameter, κ (T) σ T /T κ (T )
13 H /3 /η rms NOWPHAS H /3 /η rms % εν, 4 ε. Rice(945) κ (T ) NOWPHAS (4) No. 86 p. + A & B. (). (985) 4 4 pp (99) 333 p. (974) 3 pp (97) 3 No., 4 p. Battjes, J, A. and van Vledder, G.Ph. (984): Veriication o. Kimura s theory or wave group statistics, Proc. 9th Int. Con. Coastal Eng., Houston, ASCE, pp H /3 /η rms κ Cartwright, D.E. and Longuet-Higgins, M.S. (956): The (T )= 4.4 κ (T )= statistical distribution o the maxima o random unction, 6 Proc. Roy. Soc. London, A. 37, pp. -3. % Forristall, G.Z. (984): The distribution o measured and simulated wave heights as a unction o spectral shape, J. Geophy. Res., 89 (C6), pp.,547-,55. H /3 /η rms Goda, Y. (97): Numerical experiments on wave statistics 3.733% with spectral simulation, Rept. Port and Harbour Res. Inst., 9(3), pp Goda, Y. (983): Analysis o wave grouping and spectra o H /3 /η rms = 5 long-travelled swell, Rept. Port and Harbour Res. Inst., Vol., No., pp % Goda, Y. (983): A uniied nonlinearity parameter o water waves, Rept. Port and Harbour Res. Inst., Vol., No. 3, pp Goda, Y. (988): Statistical variability o sea state parameter as a unction o wave spectrum, Coastal Engineering in Japan, κ (T ) JSCE, Vol. 3, No., pp Kimura, A. (98): Statistical properties o random wave groups, Proc. 7th Int. Con. Coastal Eng., Sydney, ASCE, pp
14 Longuet-Higgins, M.S (95): On the statistical distribution o sea waves, J. Marine Res., XI (3), pp Longuet-Higgins, M.S (975): On the joint distribution o the periods and amplitudes o sea waves, J. Geophys. Res., 8 (8), pp Longuet-Higgins, M.S. (984): Statistical properties o wave groups in a random sea state, Phil. Trans. Roy. Soc. London, A3, pp Rice, S.O. (945): Mathematical analysis o random noises, reprinted in Selected Papers on Noise and Stochastic Processes (Dover Pub., 954), pp Tayun, M.A. (98): Distribution o crest-to-trough wave heights, J. Waterway, Port, Coastal and Ocean Eng. Div., ASCE, 7 (WW3), pp
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