Title: Prediction of wave forces from nonlinear random sea simulations
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097549/00001
 Material Information
Title: Prediction of wave forces from nonlinear random sea simulations
Physical Description: xiv, 167 leaves. : illus. ; 28 cm.
Language: English
Creator: Hudspeth, Robert Turner, 1940-
Publication Date: 1974
Copyright Date: 1974
Subject: Ocean waves   ( lcsh )
Wave-motion, Theory of   ( lcsh )
Fluid dynamics   ( lcsh )
Civil and Coastal Engineering thesis Ph. D   ( lcsh )
Dissertations, Academic -- Civil and Coastal Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 156-166.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097549
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000580741
oclc - 14080847
notis - ADA8846


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under rs arn dir, i ric if .e sici s i ff C coI t l prroce;-es. 'Pr:ie ssor

K. G. lie'n, ieparit ent if iil and 'oatjl Ln ine'r inn ,

e ir I zs h3l rper r-.n ani e pr.:' r i'.J 1, r ir u i. his; et: t in Ve

e'.p ri..r lce iinri Irc.i-led e of a.e .i h. drorn cihjri -rnorin'r, ,j.

ins ight in is. rtarce Irn redu Ir ;z .:.-iF, p :.: pr'bcl.er. tr' r:l -

tr Vll tractable clutLi ui has in ncirei ri.c rin a3rp i rat i r.

Prof-.: s or 0.. H. 'hein, ri, ['epiartm ert .Ft Ca i l jnr Ci:; .tsl

Fngin re.rirn anrd Di r.:ctor Coa.'t l En nier ring L3aborar .r',',

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ey-'s pt.r i nal facility' for tr an; form' irn the coir.pl. r .-,: atlier.at i

of stoC:hals ti prC:I.'.: s. and random funcT tiora into: erin: neerini

app icat ion.

rroTfces.=cr -'. P. Luehr, Department of Mathnat ics S-ri'ed

on the coijTlittee during the c i r.' .talCe. and provided rni.:h-

needed aTsistarnce ini undJerst.in.Iinr the manth rm.atric l ph' .iC

of boldi arv ; ilue problens Protfe .cr ii. Uinlu t a, El ep'rtn ent

of i-' il an-.I Iastal Fncineer-ing, ia respcFn ible for ladin'

Me to :i.rcira important re frnnc.:: ind for pro idinrg ?lter-

nate method: of solutions. s- ,.ill t, come c.bv;iou tor the

reader, I have Idr an hC3ea il.' froci the pub I i S.-t lC ni ,:if t r.

L. E. E b riT in ii addition t:, nIirn rouiJ r; r .jr-n l discus i:':ns

With him. I am indebte d to Dr. Le anr fr making trhc-:s

exlchain es, i - m pi iss. Thc rinu -ric l t tit ,_t : il

result .hiic:i are presente.d .- re otrtaned from a co.-mput-r

prograni p rovided b .- Lr. Bor.eman.

This' tud'.' .i cc-sponsore:d r.,' he '.'oaist l Lroinincering

Research Centcr, DepP rtmrent of thie Irm'., rd t.*, 3 Ic.int

Petrci--un Indui tr.' t j..'e Force Project. Dr. F. Hki5 .M1I.?t'

Product ion Compan..'. s-cri d A a Project '., ;er for the petr .:-

leum Indii tr' rand I hi' benefited irii nisi l.' fr ,m aan.' la -

c:us.iion with him i ich ha',i stimrulated .m int r- r.t in this

topic. Dr. J. H. chaui., Cihairpers.on, [icpartr..e.t :.f C 'iil

and Coasr.al Enginee ring, and ProfeC s4 r E. D. S an- iter,

Departmer t of Ci 1I and jCoast3l [EnFinriering, ivere moist Fen. r-

ous in providing fiurds for cc'np.itc r t ime .and in Fro vidin ri

c-cunse;I1i n as i i tance.

fiMr Evel..T Hill did an c exceptional Jlo of t'pinr the

original rough draft as uell a1 pi.'in meticulous atr.ention

to the details and regulations require-d for completion dc. crce

.*rk and f.--r en:jring crhar. I aet thes'e r-quirt entr *.n time.

In t.pring trhrc .:ri d.1 t't, Ir;. Hill as chc. rfu.ll* and 3bl.-

3S :.i.ted h.y T. ; rI lvn ... rr i .:.r. Tera .3ra nei z r a nr 1 3 is Car' l

Miller. Th e '- ll ent i preicpae fi gure; w~ ere .ir fted t.

1 i. Fe rris '. tepjr, and '1 [ini -'c Frank, frT qu.nt l." urdr r

*:ritical ti re :-rn;tr'iints. The f ri jl .:.:.F' "i t pcJ 1i th

r'i sarl able r'- irc jn. : aurac. cn rL Ti -i- ri the nurri~ber of

-equ ati O:rn ir -lir e t'. r11' Ei i beth : ' e .

Finall. '. .ut qu tc po~ 1t. 1" r-.:. t Eiportpnitl ', I ;r!jt -

fuill. gi~. :nL le,:. Cl if i. tIarn: e, C asfiTi rarde r 3nd :riT ttc lr i

crf r.. fi l: .. graduate s tu,-nts : i -.slnr -. i th [,r. ,eanr ,

al i ,. mar Ig.-. t :- s.:. irehti find ti h time anl Lh r. te rest t:.

Ic rid as i :-anc:e a'n.l i'iJp[-p.rt hen the -ere require r l t-e m: st .

f Ar EI. L L F *' *-. 1 E J r _

C H.li l;l': . . . . . . . .. . ll

LIS:T *F T.-AELE . . . . . . . . . .. iii

LIST 'IF FICUlPE . . . . . . . . . 1',

A E.L .TPA CT . . . . . . . . . . . X':11


] I[ T F:''lT['L T I '. .. . .... . ... .. 1

1. -lated F 1 e . . . .
2. Linear'r iLip.- Tr..:.- i r an. F [f ir
W,,e -lifle rrt j I.,n rrc
w >e -Wi; e I t. r -I r ti C.,I. . . . . .
3. imr la] inlg a1 F r,: c:. t th F r' cit l
Line ir Fil e r Tre:hriiqui . . . . I

2 TH E '"-R i' CF fPA D I.i! J ,'1jL I rt. F .; '. .E Ei .

I1. Pa- ni 'r.. . . . . . . . . 1.
1 P a3r.-C'iT l 1 L _ffC Tc n i alI E.u a:t ic.,. . . . 1
2. In t r: uct i n t. i F r i .: Fri c:.i i ,n .. . -
3. rri. irin ce Funcr. n . . . . . .
J-. Joril ine ar .at3\.* Ia -.' I r 'l t ''..
l',rrcr rc. I: r' .ndJ F rburbha icn r '',r .r .
3. ihe Li.pcrT u,. . . . . . . . J'
r. ecor P- rr urt 3ti:r ird:r F. c r r . . 5-

3 APFLI TI'.r. OF i: N'IL I rNE P.1. ji:,'. *,E \
SI LI LIlAT i U;: ; . . . . . . .. . 5

1. F.:ur i.:r ri'c ,'pF. r. ir 'i. -,im 'i r '.. th-i
Ranp jdL ie a': jr.. ... . . . . .
2. Er:st [ouriI' r Tran- f rri I FFT . . .
3. The Fri r'chne r -:pec ,r rI -1 ir.d1 the
IhillFip Equi hlibr iuT : ct r u. . . . .. o
4. Hurricrane Cirl.a D ta l ecptecrner
S- 10, 1911 . . .. ... . .
CiTo r f e Surfa c l t- it: t :r .
D.. Digital Lin a rr er . . . .. . 10
Cc np a i : n of F're .- ire FI-.rc:ez . ... . 1-1

TABLE 'OF :C'IJTFIJT7 Icontir.'ic-.i


I4 :L' L 5I 'il*l; ,i AL' FE. ilECOrVlE'Jl li'i i .' . . ... i.

1. r random C- ii l s .:'r. l i'I :.rrecr:[ r
eco:.n. i' r turJ'b.rti n I rdj . . . . i
a. %4\'w I, riIL:.ld -'ri "' ure F.:-rc. n, C .,
rri r c.i F'jI i I 'or ,r.u.J b .' Li i t l al
Linear F later i tth "'Ctre[ r :hem"
c rti .:.al C OT jinate . . . . . . 13
P. iF'eiom.im r .J t r, for i-l.li t ri al
A.p[_l1 ic: it or. . . . . . . . 1.9
.* F' E :'.

I THE F:I.IR F':iDILI T rl"',L: i N "I R Gi",.U :I.J
V \ F. i ATE . . . . . . . . . .. 142

P LiLPh'.1.TF iUL .TIl: i FILTE .. . . . . . 14S

C EFFECT iF THE HORi: lTi'L _... TIAL ?LF A \. T i,"'
BETrv.'LLi i.,'iE. -TI' F 1' [ IJ T' L lE'JTED FILINr
ti'lj THE E', LLi-i T ie IF FP IS L ui L ','FE iCOEFFI-
CI ENT .. . . ... . . . . . .

LiVTT ,'F fEFENES .E .............. .. .. 15E

B i:E":f.P i i l ..f ..TH . .................. 1"


r ble Pc.ge

2.1 t te c of C.nL.- rcnce ~ jrj Final 'l1. Frr.:.r
C.onir'p ed for t-h. Lb t Le : t : r- F i t
t-:. iea_;urec. :p-[ectr a ft'rm Hurricane CarlI'I
fc r . . . . . . . . . 1

3.2 Ch arr:teT i tir c i: Hurri i,. Carl R-.c. rjd ; .. 7.

3.3 limpuli:e i' p`Fjr e C.-: ffi ci: nts for IIc-ri :. ,r. i l
\'el :. it. Fiel l .th retch d : rt i :
C o.rdin r t- Ih = 9 t'. i . . .... .. . 11

3.41 Dr4 C and Ili, l ri _,. rInerti a Force i.:'- 'i l-
.itent- fur R-ui tant Prc = re F,:rce .: .. . 114

C..5 .ni.par i c.n Ec t -,een 'ia.-urei rnd J ii, ul ate 1
Fr-. ui F:r-c s at f- :ot Lri ii .: .mm e r
f ei t i- ion fr.:.i H!i ri c r- I:3rl . . . . 1:

3.6 Stat.r i 1 -i ,: f re a'- ure J nl '- imui ate I
're:- z r.: Fo, rce <[ ectra fi'rOn hurrIcar,:
Carla . . . ... . . . . . . 1 1i


P'I Ci

iri:i -.rij rle i lsured pec.-t rum and Prc r chnp idier
Sprctrrum of c iu l \ c i rlir: rid E. ; t Le t-
.Squiare. Fit t.. Pri a Fr.'qu.:nc\ v tor '1, = 3115
forc i .ec. r, r.,. 6 t.) j5 . . . . . . s

k2 ,,: J r L iiT'r-.: F-r.:ctri- r m J i'ret C l t i:hneid. r
Sinectrum c.f qu-il L arian:'e arind Fe it L.ast-
,quar--a Fit t... r L. Frequeni c fa: r' t.l = 30
for Fec, rd .,:. N ir, :.' 1 . . . .

3.3 I i.:,thed c t i r.eid p r ctru in ,d Lret chneider
Spectrum :.f Eqt l l\'riancr and [.r s t Lea -
'quar'e; Fit ti f'e l. Fr quI nc ': f:'i c = -e 5
i.:r P c: :rd 'i.:. . . . . . .

.1 S ff:rth.- 'eri- rred [e.c' trtr n and Pi ti chneid.er
'_r. nct rum :, Equal 'Vari ncc and e-'t Leas t-
cSuiare. Fit [, Pi :ak Frequ:n. t.:.r !I; = 31u'
fr. r Fec: rd i . . . . . .

.. T: ruljl 1 i ',
Nc.Ori-. a zc d
Linrer rind
fo r Fe-?-: rd

PI-.t-at. lit\ Di: tri ut :n= ft'i the
' ui uicJd Ie ai i: 1 tt n rnd the
hrl:, iinri r at Sinij l t .- ] I ei al i ; t i:'
jie i .8 5 1 . . . .

.. '-ijrulat i r F, r'; ih 1 1 hillt r, i tr i utiri ns. for th-
r .rnil i: d PIc ~l urcd F.al i:ati-o n and th.:
Lirni r and '.onl inEr, r fe -ilU 1 :it i:-.n r r record
rli .. ,',C. 8 r, i 1 . . . . . . . . .

3." CL'irU 1 i '..'
N..rmal i :ed
Linear and
for Pe:cord

3. c C iul ati
tlorma'l i :c d
Linear rind
f,:r F.ecord

Prr,:bai li ty Dit i ribut i..n for the
i.: asiurcd F-e l i: t io.n aid the-
.'lunli ine r -.,Tulat d F:e al : at ion
. i . . . . . . .

Prr.bh bilit- Dl trihut inors fc-r th'
'le i]urrd :e all :at i.:n and the
'.n 11 r.ae r 'ii nul at d e, al i : Er ic-n:
iJ i '" . . . . . .

3.9 Second Order Sp.:ctra C:c-mpiutEd from :sm.zsothed
leas ured Spectrun and f'rerin i'rets cti 1 lder
Spectrum for P.-c rdJ li. C o .' l . . .

Fig lJ


LIST O-IF FIGuRF.: i,c...ntinue.jl

Fi ure P ge

3. lu 'C1 .:r j Ord r : ..: : rr j : rim ijteJ r...ii, ir...-the I
I-: I I j 'p.t- t run ar .J t ruC r i r r: i:-chn i r
',i.Ctr iT f, -r .-:*.: rd ti:,. I)I ', .: 1, . . .. ... 9

3. 11 eu c .ndj Ord- r r ectrri *:..rp t.:d Fr-.-.i F ri,.: .r.h-
Ple a:ur.:d -. ctru.n ad l r:I Pr. r..: r id. r
pec r tri, fi r I, r.1: r, 11.1 ::.,2 . . . . :

.12 -ec.on '. d r rc : rra ....r.pit:d r ,:- ..:. hed
e3..ure L. rpirictrui ar, d f l t rr. 'r t c.'.T Fi r
'-:p true l ReC rd "..-. ,". '- . . .

3. 13 E[n mh:-ribl.: -ip3ri n L eti er. Ph.- a r. ed c l : -i -
tion an.d L irii and 'I. nlin-ear Fr i : 3[ icrii:

fro i P c:rd :.. .. ..:. . . . . . . *

I.1] En 4e, ble -... ip r i .-:.n [V.:t E r, 1: ,ur ,d .- i : -' -
tion and Line ar and .ihr. 1 inir fe .l i t: c. nr:r
S .iirlar..l fr ii i .... .tri :. ; 1-:3-'ur c t rujij
fr-nii eec.:-rd :..-. fir,-l .,- 1 . . . . .

3.15 Ens iT-ib.l: I r ari n t: r r i : u- .- er. i :.a:ure i : i 1

irr.ul a3 d frori 'Cr,-,rh d I leaij re r .1 :'e: r r ir
from. I .: rd . l . . . . . :

3.1 En r .- ii ic :.p i, ris-.,r -.e .ri in :ei ur j i i 3
tic n t1.,l I i e r nrd r.I -.i ir, i r !: :al L: i r t rion:
rlriiatr ?d f'ro i '_ l th.:. -, l. :-ured p.e r.,t r.
frc-m. F'e c rd i. I" . . . . 99

3 1 E ,ri- tle C-_' ,-rr ip r i ._-o P ie r e ri ,l .- *i re I 1 L:i "1
tion iii i ri, v e i rind l ir-ar F- al a .r. i.r:
timruil ite frici iB r,.t-:.hr:i..: id r ':.. .: TruIp fr
R.: cord fic . Ir '.-.; I . . . . . .. . 1. 0

3.18 Erse5 ri le .-'r.r.a i, 1 rt1 .e-rn leaiired a F .l:i -
t icn rin-J L rinc ir ari nr, li near f a.:al :a iori
Si -ul ate d fr .m r[ r :r.ri ei i : r r r, ri r:- i
P.ec... rJ tJ.I, il.. i . . . . . . . . IT

3.19 Ens-mb]e Coip. iri-:..n B. -r.t n rleras ur d ali : -
ti n N rinj i -. i tr in:...1 -orni i ri r '-: i L :a t.. r..ri
iiula 3redi fr ni Ere chne id ei 'r-pe ,:trum fi:-ii
P.ec-r. N S.i'. 2 . . . . . . ?

LI T i'F FiCIUfF E Icor.rt nuid'll

Fi gur F age

'. 2 F r.' il-. le C': Frpar ii on Pr r.* .: : n a1 'Jr d FLrei : -
ti ri 3n.J i ir.- r and L.r: i lin ar Pe al i ati ir.n
'imul ated frc 1're1 r t hn ider 1 rp-: t r if t. rr.:ma
c ,c r J r *- ... i . . . . . . . ] 0

3. L iC:mr ri..-r .- If lri : ontl I' innijt: i Fieldsj ja rd
Prer- : rn: F-rc- Co ...put-ed b,' i rtil Line ir
Fil ter TrcIri. ie fr ,-, 1.c-'c andj .:r, .:.ed
c r .ic t r-"iJi c I'."-.i . . . . . .. i7

7.22 C.'. irr i *.-.r f hl.:r :.:.r tal n rin iri ic Field; -nrd
Fr'i ure I-or.ce: i'ii.pilr j t .,' r' f i t 3 Lin aIT
F later Tec:hrl- .i from rr :c j A 'J:n ;i ei:,:d,
t r i.: rt l r i J c . . . . . . 1

S,. -,l.r' ir, .:n Ee t reen 'le u iaiJred arid P're.dic ct
Pre ure F:r ze pFectr fr:.T PeF,: rd tI .
Or i 1 ... .... ... .......... l'

..1 :C rFJriA'r :n P tc-,E n au rl red and Pre.dic:ted
Pr es ure I rce 'I,-.; r fr'T, RP .: rd tih .
i P S i. '1 . . . . .... . .. 121

,. E'.n p j ri n i rc n i lt si,.Irei arj d Fre ic rt
Pressure Force p Etr i fr:i -,cC':.rd No.
u c. e .,' 2 . . . . . . . . . . 1

n i'.-ri-p.iri :rnn 'et i., n l 't:i red nd 're-dictei
Prc :-ure F :.r e Spectra fr.ii, Pe..:r '- .
.... ',' 1 . . .. ... ... ..... ... 12

.. 2 "' u I l 'tii c e PF'r .-'.i , i l. i [ b i 't r i o r. o f
le i urej .iiid Frec d- ric d Pre sure F:,rc',
F.:.- li : r; n f ror., r c,,:,rd fi. i '. 9 1 . . 126

'. C u -ii ar i r' ..a.t li t iis r i but -.r: n .:. f
rlc'aured and Predicir-ed Pre :-'ure F.rce
Pe al i ic.n f. ir .. PF ::,ec rd .:. i',?%f 1 . . . 1

',..9: C mii t I Fr:t- b.:i:b liti.' [ii r i ut i.:.ns f
lIea.ur.d ar, d Predicred fressure Force
Re li : ati:n.ris fr:- m Rec rd .-.. u-8.6 2R . . . 12.

3. .0 Ulnull' t '.'e Prr-:t abiliE ', 1i : tribute nr, c.
F ca ;ur d arnd Predicted PFre re e Force
Real Zat i:ns from F.ecr.rd .'c. (6 .e I . . . 129

L! 'T OF FICGUZE rE i'.ct ir,nu-d

Figure Pace

C. 1 rlerir t irn of l..e -r .ff nr in:t; rur erient.d
Pi in f. r i-.. Pr.-:. :t i . . . . ... 4

C. 2 t ftect ,:.r t Fii erj.j l -s n ,1 ''.'I'.:. Staftr
In strur rtr.lej rii in c r.e ir i,: n 3,i-t ric u, r
DriL Co. f'.c ir'i t: t..i rinl d trm'i th.,
Hlor r i n l- u it r i.- n i r h l- irr. .r The..r i: e-
mriatic IL l near theor" 'elen thi . . 15

C.3 effect of i ir:ernt iioril,:. rI 1 .\ Staff
Inrtrurr..-:ntd r l ing r ir tior, uL1: rtrnce or
Inertia i':iF t icient 0: c rmirnd frr.m th'
fIorr son .luaticrn ith LIinear The-or. iri--
matLics. Ii = lir e r th or,' 3 elergrth . ... . 155

.ibt.r.T : t .. f Li.;l Ierr3t[ cii Pr: tri -d to thie Cr'i3,i ui ic't ci 1
Ct the IJni'.-r rit r,' of F lorida in ar al i.ltfiienilern t cf r he
K.e,.i i lre ri-nts for the L.i-gr e ':r ['lDocct.r *: f P hilosopr.'

PPEt liCiT [ I it' '. : F F,_ CE FT .,l' I.
I. I T il'.E '.R A'JD Cl E 'C- [CTi li.,T i 't.-


h.ohert Turner fl'.i j- pe.

I ece tib.: r, '19 "-

Ch ai er :.rn r. Pr'.h rt C. ['e in
rII :.T ['epartTi.:r.t: Ci' l ari Co, atil ErIL irier rng

The ncrlin i:r butIr, jar' ,'* cilu problem for the .r,-p c -

tiLn of r an i.':F .ur f cc r ivit., ius 3c-; in an :Ceajr t* rinite

.: pth i z ,l.l',;i correct t s.; con l c.rd ir ir, perturbia tc r

par iiTete r. A rcri 1 lirn .ar -e.:.nd i. r ler in t r act i r ..n .rnel i

ohtairned ,h1ich resh ult i5 rin icnll near ;cp.ctrr l corr.;ctr i .'r to

a line ar C IuA i n :eE p. e,: trTUirP at 3 r t,.J:r.ncie: li:ch are the

urn Ian J d if f.: r ':rnc. i. of t e Irie r icting fr.-qut :r. cic; :Of the?

li in ir r F i Cis m. Th.- triv'ar i'anc:, fum ct on is .: i:un tio I 3

clo.s:d S.[ta C :ic l -iM a ir.- of tte :-c ndi r.rjer nrni ir, rlitc e

an.J the i *:: r e Fie'as-.ire or the trr ari an e fut'inctin.: I . the

le T. c , i ui. sd tc .J t.: rmiin the rainir itude .i.f the 5.econd

order r rnon r.ear c.rntrihut t io:n ..r. aln:;:ritnn i_ pr E=er .dJ d or

simulatin j time sequenc,: of rr:nlnear r-anJm surf3ce cra\vi't,

i ce.; correct to ecor I order r i.. eimrpl,-',in the tfa3 t r -urie r

r an fo, rrn.

The normal : c:umrulati '- pr.:.L. hili '. .Ii; tri utions ,f

lineji and nonlin-ii simulated ra iz tlion- are c'irp'Ired .i ulth


me a3 urT,.' hurrlicjri gnE rarte.J r. iii :31caonr- recT-,rdd during

Hurricane Carla in the Cul t" cf Ic..1ic.: by iiv' e Frrce Fr.i.' ct

II. ihe imjnul it innz af .,rthe ~r :cj fir: a s :m.; h d mca-

tured ?pec trum r. nd from a E re chnri Jde r :-p-ct ru" hain r

equal vari ancr e irid bes t ei atr. square' fit r:c the m in..rheA.

nr ai,- rcd sr.:ctr ri iF in r:rd.- r t- d.:m. .r str.at,: th c effe ct of pec -

tral -shape and pha-e arnigle : iri r r,and.:m 1 ffiul ic ion in r aditi L.r,

to es tablis .hinr r.lh: applicab ilit F the rrc :tchrceid r pr.c-

trum f.n r d.:-:in.

Sl ulti ntc i. : a e ft.-.rcc t thj ';. fc-., d.n'r1 rr-m.a ter el -

tion are predi tedJ fr-rn, b r.th re iin-ar ian n r.nl near ,irmu-

lat1 i n-. : 5ynrthesi' d "fr.,' hri rh ipecti3 i'nmiTita C i fcidj ar.:

computed by te digital line.ir filtr i te -chniqu. e riJcdi J fir j I,'

3 ic rttCAl cr:T.rdinrate I-r rr rchi func t i.:ri inrd are u:i d in

the Mc'rinon e.lu3t ii-.n iith Lean and "ap ard Idra arnd mo.difi.:

inertia c e ft't'icienr.. r.:.rriFal 1i :-ed cuFiuljatI prjot a il r..

Ji-tribut ic.n arnd : f :.rce v etpc ra c f the s iiiulated f.:.rc:-

are comipar d I. tli m 3 a'ured w3 fuorc- re1 al i : atin.: anJ .l r.r

waice force re. al i it.ori. pnreJicte t',' filt, rini: rrhc rem ]iured

Sea _uf a.ce rE a 11: at i'ln.



iNTP-L'U.l l''._-

A: tl I de ;'i n .:f :ff':-h re penrl ir rt p ile -:uproi rted true-

tur ir ie. v h'e .: 'rin j the ft:ur h ,iLi]re d fc.ot t c.tti rm .:c.nt:our into

re ir-ci s .:,f re'a ter rept h ind ac the rm [h-:,ina tical noJ.: I L t-r:h

jer -:rit. the c ,,ri pl giant nature f the piilc oil inter: cc ion

bEc. .e rc re -.: ph i ti :at d, che requi ret i rt t,-o rer orr. Jd.ri Tiic

n, jl E". ;-.f tres ;. ;truju.:turE h ibecmi rfrY" critical. F., t., r

( c.1, Edge and HIc.,cr [41 P ': r n [:'0], 'J lhotri rd P'.n:iern

[ 1, .Th ard HIj rl.:ri r. [ : i-arte n.J i h [1, ], I e rin

ir.I '-hr I1 ) ], .l-in .ijr Ind i lmanr, [ fP Perge: and "-:n i-rLn

[12), lu r aJnd il'l .:.r, [ 1] t`r.-: aj s IaI pr-.cHnt; d model

fo'r the ,i,'nri inic rei r.i:. e r p: : ria nes pi le upF orte j _I uC -

turi c t.- ran aJo f-: rc- s Tr~,. randrom for:.. i.ie d in these

Stlj.i s re e ither i t i i i erp .-.; i 'iii f I r. r Fi.urier .onr.'.nn-

S:nt i th rar. idon Ehasi-e snrls I hich 'ere uni furm l di r. ributed

ibetieer I n,l i:r 3 i ri t l ',' rri di : c :.tl, i in 1 a3.e .)f firiir t

aripl i tud The pr.rr"po e o:f rhii ; r.tuj' .i prrF'iri ed i rn n-

liine r rindim tirie :erie: re al ion of a surfac graji it,

Wave p- ctrTLuTi in in i cea, n uf fi nire depth and tr, util I :e tl.is

re ali:-tl on 4a a fi:rcinrg frLnct i:.r : 3 filter to c attain the

k:inematics reqiireid to predict press sure fo.rces. Thc: cerphas is

will be- o .nthe h sirulatiU.,n o.f rarid.i tinr: re a:' vice the

determinrrtr.in of the invariant stati.tics of rand.:m r,:ali:ations.

in Section 1, th.e most irmportint result aJnd coric.lusi- :.n

obtained b., others in related iri e- s~ aticns. toi. ird :obtarinig

second ord r pcrt irbatr ion corrections to the s: s urlface "re

briefly rel i.ed. In ';.ctir.n -, the ef -'c,:t on the r,.- i lt ir,

5e rurfacii pro le i heof h linear si upe rp.:.i ition ing a.nr the

nonline r int. r i tio-n if tl :.:. r linear '~'iCe is xc\j ined bt

means of a simplF model. in i'-cti -n the ar r, ppllc tn i of

the nonl near .sel s iurface iim ulatiorn t. obrt1 in a r ndorim pre-z-

Sure furce time sP t ries i; rri efi,' *.jrj. lined.

1. Peiart d I're ci.ij 'tC.ijJie

Tick has d.:el,:cped a i.ethod for cen rat3 irn a re aii -:.tion

-,f a nonlinear ;e i surface for d:ep ivat.?r [(I j i r, for ia'er

of finite depr 'h 125 ) b.' fo: riial ,' perturt-ing the ener,. r .p.c-

trum. For a Cau' Lan initial timirate to tho l ire r bo.urin.ir

cjalu i pr.rble the r rturtij ed ener,.' specItrunj i- 'h : r, t.o .on-

s i t .n l'.: t e r n r- L rL r *s -.f t'l-, perturtb t[ :.nr p i ra Iet er. Ti.-

en'r'c y r pc-Ct rur Ji .a then used as the n, re e he ncnlin .sr-

ities. in order to iriphic ill isril ,' thr surerp r c iti':n of

the linear ener'..' pect rum, i th tlie quadr:rt ic energy: .' spe= trum,

the ordinate valuer of the quadratic ipectrur hajd to te

greatl.- e handedd n- ar the origin I. eri in tf. t cj-. ; r inite

dept I' in c.rder that th- rmea'sure -if the qu3dratic rerturba-

ticn of the enere.: spectrum couui ti oct,-crvcd.

In his ini ial t ud.,- I'cf. Finraian ["' ]), Tick tre- '-J

finite .irplitu le ;.aves correct to second perturbation order r

in deep water and he obtained a .:crre:ction to the free s r-

face displacements iceran by the following expression:

- in(:x,ti = 1 :. : 1 t l

S = Il:=,t f l.

-*crh re the alphat-.etic l s ub--cripts- d.notE partial lifferen-

tisat .n .'it h respectZ to the irndepen.er rit a\ riaihle5 rindij, t.ed

by the suffi: arin the numeric il prefix j-din.,tes the perturh a-

tioon -r.dcr '-f the dcpe.-ndent '3ariabl.e. The last of these three

terms rerrecent: the 'la,1: irin s eries .*xparn icOr of the varia-

tion of the free Surface ibout the sCill ;.at1er level due to

the prr; erine of the wa 'c. ihe second term inr bralck.ets repre-

snorts the ccrre:tiocr due to. the local liretLic energy of the

3t.-r particle iioti.rns The firit t-:rm represents the 'ec rnl

or-ler perrturkationr ccrrcc ti-n to the vele.itr,' pctenti~ l. T-he

_ec.:cnd crder pr rturbition corrections t. t he vel-.cit. potent ial

in leer water wi: Lic\en b: Tick in the fc llc.Ioin integral

expri ;ro:s

, l> ,: ,c j : 2 ,.,',: l..,.w e xp l Il [.- I. l | [.-,'I I 1 *l f ,. }- l f

*exp I( ( utW ,t) -exp i l.',x,tilid. .' 1.

where the rncnlincar interactrinr kernel, ; L ,. i gei .en b:,

c I l, ,- 11,. II(1.3)
I ... 1... -, I ,'1 .1, I '.' 1 1

Kinsmijn [7" ] correctcd rhe -separatiorn .:,n:L.tn t orit nail pub-

lished b.' Ticl: [1 ] I in ..r.Jer chat the ieloc,:i potential

satil fi', e actl the *-quit io:n -:'f continr itY. Thin corr:c t ".on

ha- been iricorpr .jted in Eq. 11.21. The second order p rtur-

bat i.r:n correction tn : t, ce free s surface ,:c mrutu d tfTci:i Eq. l .11

bhv Tick. 'as also .1 rested v i n r i nteLral equJatiCn:

,r(':<,t) ,I', ) I l t' ..., t f'l I ,.' ,t),l d 1r .41

where the n,,rline r int r raceti :,n rrnel i c en b,

H ( *a i = 2!*J .I IL1, (u 1 11.51

i'ne of th.- most important re ults obt jined by T i-l 3.j

that the c.:ntribution to the perturbed enerrg. 'pectrum at 1

civen frcquencr,: bh th- eci:onJ ?rler porturbs ticn cr:. rre. ti'cn

to the sea suTt'3rfia e the result of th,- corr :,l .ti:.r of all

firs-t order perturhatir :n energy Jensit'ie h.,.. '-ul- an Jni d '-

ferences r ir frreqqu.ncri:- contribute to the giren frequenrc' .'f

th,: quadr ti.c aspect r ium. Th.: e F.prei s :n icri p. uted b'' T i a'-

,5 u l 1 .* l 1 1 11. '
r r I c rl ri rlr

This important result will bc *disc. i: i ed inr more J.dltas i in

Chapter 2.

Tick 12.1 later xt. nd:id d his results to water ot finite

depth ind pre 'vri't. d the foll,.w ir,g n.rn inra ir in terac:t .n

.ernel :

,.I t I , +' I '

i t )

here .1- = (h .an.d the Lr:ic ri,ers =. ki ) -rad I' i'l. 'l

-irte i.lutri r:- to the Ji ipe r icrn rclati:on

u. 1 = I. tanhl- .l' ] (1.9i

The i ign :- f the t ec c.n. t er T e in n Eq. 1.1 7' has b re n corr-

rec ted fr-om th ry!cpographical crrcor ir n Pc f rcnc c i 1'. Ti.:

resol. ed I r l e d E if f ultie_ i r ob ib _lin ir. i n ider.tit, b..: r eon

thie InteractEio.n ikerr.-l z ii el en b, Eq. (1 and Eq. 1 .' jsa

thIe ,at.r dcpth ppro:iache". irfinit.' (i.e., b-*= I in teri.s %.t'

[ti. dil ic ntirnui t IT in t.'per arttic *:re t~i b .' D[ir c i delti

funct ins and bi t[he n..riurniqjueniesI :t" r, erturbati. on exp3ni ions.

Tico. I2I .1 icrinstratCe a cc.Iipairi son hetcec n a liri:ar

reali:altion .*:-,iputr d front the i eiumiriann-t .'c spectrum ["21 witr

SIeccond pcrturbat ion crder r ealizatin 'r'.-t Fc r thi ar,e spFec:trurn

t'..r the c:, e ,.f an c. ce.n cf- infinite .cpth. .c real i a i : icr

L.'er iavail t -le fro r the ini t d.iepth :a': .dueu to the comi puta-

tic-nal comrpleC it ie in-i'.Ilved. The Fjl t Fu'ric r Transiform

'FFTi alg.-ri tlim [ 3 ihi -chli as ncIt c-ner r il." j iil -lc. *t

the date T1cl put.lisheJ his i.ork, h, 3 great]., f-acilitated the

applicat i.ns f oiiadr tic S s i i ulatii-.nr b-, reducing Ic1ni fi-

c'jntl, the corput '.t icnal di tf fi:ult ie t-: public sh dJ coarp.,ri-

;:.ns o:f random .lu dr'r t ic es imiula ir c tin c iw meaiij reld il e

data seem to h-e j-aailable.

In in e 'trra:rdirnir,'- th re r-art -erie-. iHas seln-anr [51,

55,j ) et forth a po, erful theo:,r- for the rinlirme iar ene r:,

trji tef r in a pec'-:t rum .. random .3ve:. Latc:r .: n o:lr.

[52,59 ,. 1 inclu.Jed tche pr:,t- ability ic i 5Tructurt .:.f the inlti i

c,.nditi,.ri, Tpr :.Ced thr.icl-h the rilonl near fr-Lc ,.,r face boundar:,

c,::.nditi :,rn a ell ai a general the r. Irlr. for the: ai c e- .

ca3ttc rinri pr.-.cessc in the :ci arni.: .. vi' uiidei PF cencl .,

[7. ,120 prtr i n- of thi:. p: erfil t a -.. ave s:att ring

theory have been appl ied to i asur .J lajn ith c..-l: lerit

a rc' ent. ihe inc : r:-dit le nmurther u f Cgeioph s ical r.r.:t lei:

whichh are ci-'.vired b. this gcenneral rh:'.r, c: f r man'. *:.pportun-

itii' fo r cmpai:. ris.o n i h mi-a urged dajt no. he.::umir.g c n i al .le.

In ene of the-s: .- rl. ; Has se imann [I s *:Lob er ted that the

quadratic cnerg r.' p .ctru i .h ich ..as cimpl:e'. cd '. T ic:l ia a n.-t

closed to tch.e- order ;" the pe rr.irl..iti.-,n pjrajieter cho'er..

Ha;s. lmarn inclnd J inr hi i. velt. ,.rpment the Ii, an 1 i:ed r'.6,d-

Lu.t of th, ini trial K- iau iia n cit r, at?-- I-i c h th. s..lut i n C:

the third ;r-ler perturh ati. -n ci:rr.: c ion [i' the ;e ia surf aTi

re .l:i1 iticon. Therefore., in ori:r r rto meaure n'nl'ine ari T i,:

b, fo rmal pe rturbation *of th.: cnr.,' p: C:t ruTi r, thi nnriln n,- ar

co.ntri-uti rns to theo Surf ,ac: r. ali jatit ri nu t '-e c .,,n-

puted fr.-in the t.-Ounirdar- .-Value pr,-blem co:-rr-.ct I: thi r.J ord r

in the perturbation param-.- ter if p rturbat iorn of the eneri:-g

ipectrumr is- to be utili:cd si the miiea .ure of the. rninlin-eari-

tie'; The bool- i.eeping prcblerni- o p.:rturb .at ior : .,panj si.:.r or. r

are ili.-cus.ed b'1, Hj se lmann et al. ( 1" and .jquationi -'hich

aid in thi bcoPlteeping aire fliv:rn. Priefiv. the procKicm i-

if the ereri '.' spectrTUmi (or, equivalent l,, the I-it, C.3 a r an 3 cC

func:t ion'i i ; fn.rmT.ll:,' perturbed in 3 p.:,csr serie; e:.pan; ion

in the tf.ll inc f:.r..s:

E(.' = E 1.,.) (1 .9 a

*,(TI = E 1 rn lt r : mr,t* t ) (r. t
, )i n li

the rel t i:.n-hip bi: t, cin the :nc: re'.' pFirturbatiori o :rder- and the

tilrle rsrl-c perturbhat i n OTrderT iniuU t tle c.Ciiipl. te and cl:.>5e,.i

An intermed.Jiate rie ii=lt ojhtaine-Jd Ky Has elmarin (54] in

s.olvinr the prT:l.:iem c:.f dete r, i n i the rate of enerec tra! n.m'er

bet e n dij cr e e c aj111porn.:rlt in raridor,,i er ir 'i ', r'avc train: in

water of finite depth include.j a nc.rilinrear nterir cticin mi7 tri'

- which ,ay .ls-. be uj-,:d to conis trnct nr:onlirin ar sea-. in hi-

r.c.er erite e .p n; i.:.r, ..f the eri erg. sipect' r for the sea :isir-

face, Ha i -el ,3rnn included all perte rbir i on orders oi, tted in

the e...pFain:s n u:cd b Ti;k.

These -studies b" Tick aind H 'i elnia nr i h e intrOduced

mr th._.d:- fo:r simiul rine r3nndo: nc-nlin ar rea3 i:3ation of surface

Cr t.' is e theree r in\'rv tie t ion related to norli near sea

-imulati-on' h iave reen directed tiwc jrd deicriblin th e iffcc ti o:f

nanlinre ri t ic e on the states tic; of the Ji tri:iut ions :.f the

realize atiron;. In .-ne of thee -tati'tical i tr:cIst g tons,

L:'.ngu:tr.-Hige in ["9 ] show;'ed that the i.rin -1hal er distribute ion

imore closely fits me jsured -ca surface rec c rding rnd cm r.: -

over, the main di tinr ct n between the I'.us-iajn and Craii -

Chilier distribute on. ar.: terms p r.pc rt ion l to the measure

of the 'l.ei cine -. Longuet-HiM gins ["9] slc. ob.taincd rh,

important result for a di..cr.-te rspectrum cf i;3ve-. first- ie iCn

by Tic ( [ 12 ] for a continuous. spr ctu, 1 of 'ate~ that the i1rad-

ratic ene-re.,' dir s ribi t ion i5 the result of the si.umma ti.:n ci

al1 discrete r, ieri co.inpirient Ihose.: cv L'jurn and diftfe'rn.:e-: in

phbics s cntribute. to the ai i .pectral freqe nrc:.'.

C3arti richt and Ln.-r, iit- Hiseins [t.r h a \. a l=. ,- r rated

the effect of the b.andJi.Jth of a -p.ctrtim or the probabt lit.,

Ji:str ution T r ," nornlinear 3e :a i ith nr.n:ero Fectiiner There-

focre, from both the S;jtai ti c l measuredd ir iimentsl and i ,rotat. L 1 -

ist i character st c 'and prc t. ability dJenr;tv friLact i n re sul t=,

the besitmea-urTe f t the n .nl inearir e and th.: r.r -' u i in

di tributl.:r, oif a s ,i ; ii.ilati nr. i the third st. t ti cal

moirent. Further rm'cr.:' it ill be rhou, n that the boiindary al :

problem i need ni. bc -al'.,ed tEo second. Fe rt r -it i :,n ord.J r nr

order ti, mea'-sur the Ic-.est order ointribut lonr to bot- h the n rn-

lincaritie. anrd non- iai us is i r. di t r i t tion.

he ni =ion f ce rtaincf crt rrurbUat in -orders in the p.Cr.i:r

erie c.pi ansiion of the ener y mi.easiire ra3r e i cloS'ie ro' -

lei .=iuil ar to th iat n-icountere in tiur bulence Itf. Latchei:,r

[4], fianpe Ji; F r.: et 0], or Limley' 1] i. Attempts to. r. -

ol the clo ire: F rAbt en as.. i.ell a' th-: i rre 'ers abi lity pjra-

dox of the initial Value problem r.euilted in a seriess of r'pe rF.

by Lenie., anrd Saffma n '], Hls eel mafnn [r.l], Si ffrnri [1!31.,

EF'nnev [ ], BE.nnev anrd New ell [9,10,11] 1e.cll )1 I llouit:

and Pennei (i1]. 'blou.I t: [- and Cir0hoi ff, Pona and L:amp. dJ

Fer i-[ (i15] 'hic .:h eloqiiir ntl .-de;crihbe ch.: ph~Y'ic5 of th.c ;e tio'

prc.blcnii, in ',erv sophisti cat d rnathem1atical ter T Irnte rnedi te

re-ult- *-Lbtaine.J in re ici inr rheir final ob.-c,:tiv:c ircluJ-

n:.nlinear inter-ct ior nat1 rice I fi:'r the Jis:c rie e ..'3'i Fpectrum

intrcr: ct ,rn l I .:r r:.n linri tn intcrE :[ ic r. I rrie ls fIr r the .r..n -

tiri, i.us a 'e F:p ct ri" irit Tra tir ns I Whi h l i a. b um-J ti) c r,-

' ruct rnon near i iiiul jt i n Th c pii suph .,ticateJ nathe-

iati:ci ut ilieJ in the1e p FIp r- c: ':,rntribute tc the su r.:,rJi n:-

ti:nr, of the pr3.ctici l a pplicati ns :f hCir Inte rrF.Cdiat re-

i l t r b I .: f = i ri u 1 -sa t n cn n .:. r 1 i r e a r : e c .

The I I t ul a.t t inc J in thi= t-1u1.j *:ii r ir cr, the ct l. ure

pr:-bl im b, appz al ing t" the ri i art anc fur- t ir as i a m uire

:ft 'e c :r.. i ]a r r-ril ris 1 iti: ar-d ci rcj.. rtnt [th- i rre rs i-

bilit, paradsJ:' b', jppi l in t.:. itatic.nari a t. d erL:dic i .

Fin al..', P.-.r .,r :' i] has *J.e el:. ped a ['chrai iquc- .hi h 1i

hae- *:.n the st tistic l .di-trl buti.-,.r, .:.tf time &erie- The

pr'i.ocej-i re inr,.lvi th e [tranr:t .rr ti .:.rn t a time sL r ries- i :hich

s rna-rmi 1 .Rj tribuIte J tc:. tir.- m rle'r vhichi i g3iarL Ji -

tr it utr. d u i.r :ec i' re n tb', ;a t .:.f p.1 , r. o ia ht i.le lt i

tc.:hnl i-lu ,: i c trn : 1. att rrai tiV' tf' it' cr.mpuLtat i.nal e fi-

ci-rinc', it i rit ia3;cJ *-,n the h.Jdr.d i','r ,ic equatin-' f motion [n.

2. Lineri i-pe rp,: it i n and Jrnlinc a
e -i' a- 3'c' interaction

Wlhitham [13 1 has de e-i opd ain e-..:ceri ie number of appil

*:cti.,:r o if the i ijr in'al itchrnique for the average Lacrin-

glan which JRnonsrtrjte [tie wi ae- a .n interacti'ns.r f rno rilinear

Ji:-per.ive ,'*-ave. The important t effet.ct of both h irFpli tude anJ

fre u-enc:" di perr ion are well c.:.'ered. Ho~ ' er, a simple:

exasipl *:'f the ieparatr ie e-r'fe.:ti cf lin: r s up: rpos l: t i rin arnd of

nonline ar inter ctiCor. fot r t' Jli ;p ri s i 'i'c i,3 e .- :.ip nerntis .lich

i. r 1 ati m e free ,**' nI t heiirat ical c..rp le. i t is ir ciTrua tiC e

in *.li t trPL iisi in the re l titc -.ns ips heti .-n the th-. Etffecti..

Formula i i n -rf thi c.imr.l- te tuun larv 1.lue prrhlm, fl i or

cont r inuu, _'p.tir rurn t" f di_ i-e:rsi: jrF ac cri- a 'r,' i'e-: 1, 1 1i hb

deferr- e until CIh'ipr. r 2. Hoi i e er, the ri ult 1E the f-, rrj a

sAlut -.n in 1 in h-p 3.ter mi., te .sur m~ ri ar .: cd tr proui c a irrmpl i.

illr 1 t i ra n o-f bt:o'th th- linEir uperp I it -.r. arn,! he nt nl ir.cC r

in tr ac- ion *:,f [.P iir ,i'r dJi crete ive c..rip ninenrr.s fr-,m a : ;-

trT in .hci L ir- c -i L ne tr ind pr-opca:3at i r th pi il e .1 ti rc i -CC

ti r.. Ti-he r 'ulltir ; g s a 'urfac: pr-T:-c le ia ,' L.- repr.-: eri te-l

nlI.,tt = l.:- ,ti ..i ,tj 12.11

,h.-ere 1: i' th line ar fi r E orde r c nt ributl i- i n3 ,- i t-c

nun 1 i nie r c' ci ri r :r r i n ur i,-n. F.--r 3a pcct utr i f tf ..o

colline ar 3 i-. *:.:. ",iip-nrreit- th-. 'i-S r. -:,rd- r corritr ,but n .,

r',-p rce c :nts the l in ar iJper .- rr s* 1 1 n *r, the E. .: i' a nd tr he

--c-nd ?order coiint r riltb ti. .: r-cpresents the t.rtrihutin'i ort

non l1rine r pr.r dij.-tr_ ., f the th ..- ;r T e Il ne aTr i .pe rpo. i -

tiunin ofif the first oTr.J-.r cr:n rit-utio n : .111 ii b r. ii ne,

bri f rl'," f ir t.

2. j. Line ar Cupe rTyt. t iton

Cronn ider a siinple line pECE:trum con' isting o ori l,," tr.

spectral coponipntci whichh are s parated bh ., srmill frequent:,

diffi-Cr nti al . arnd 'by *a small ]- riunber di ffer:.nt i l

A2k. Although Ji acent spectril conmpn'nen rs ussiully, differ in

the m5-uri'tude :. f itheiT 3iplitude I .- the FpeCtruir, is not

fl r th: tL.,: a djacenr:t co. Iprt-ne-r.ts Lil te L-C:,n i.irered tO h. e

the e 5 p itudi e -q a i t.:. H- 2, in .. rJer t.o ir 1 i f.i v hi

il u : cr 3 or.

lTh .*J- p [ t1 er tVe pr fi1c rc ultELn, fr:ri th. li riz r

rI,- P.:.;it on ,f those t_ t iC i_ civen [ .f

he re the t ,c po r .:n r, t are el i er r1 .

r, = co 1 .- '[ -t -* '.' l ;12 .35 J

H H . .:.. l .7.

= 3 i ,he2.

L.. .' i l2 4 l

Sjh i tt rin E 1'. 3,b I nt: Eq. i2.2i and c:or i-.innrig gLIve

the i'.:ll.- tin fa S ili r p r-. fil.::

1._ = H cc. i .- .It-c.. i*) --:.I :o t t-r.-- .l (2.5 '

lTht- resultin_ I'.ie contains in arippl irlde ahich i5 modulated

by the .liffe rerc:es letieenr the Fphases of rh:. tiCo: compn.rents

and ihich prropa:.' tcs at the group reC lci tr., _. |lFI. The pro-

file c.f the r.c ujlt ng '.'-.i: Frc.r ap qte.s at the q.,e -peed, 7 [ I.

The resulting '.are Is, of :course the e1ll ln ,rn beat prfiie

thich is the result of the super.pos.ition c-f two. linear i.sI-

F: re s a -ves.

There are ti.-. tire jrni -pace caile's inr'oied.l in th; r at

profile givren b:. Eq. q .5 i. One is te .:s ale c.f thc envelope

modulate ir.r, .hicF i .s s ati a il inorn .dnd t.riim orall 1 h .:

pared t': the scale ofe the profile. "The rcl titl%. 1 or, gr r and

Clo,.'cr scales are relat.:-d to pr riodicities of -S ai nd ,

respecti\xe l', ihile the relatively. -i'crier r and faster scales

of peri-:di.:it es are r' latid to I and -, r:-specti el*.

2.b. J..rnline r Interactions

The f:- rn'al deri-vation o f the second order contr, tiLbuton

in Eq. (2.11 will be Riven in Chapter 2. Ho i.c i r, fo:r the

purp-ses of this illustraticron, ccnsilder the following5 two

linear \'eliocit..' putcintia i for the deep hater r i-'e co. ,mc.ren t

g e n bH Eq . : ab.

1 b c H .:p( .,-zi s ini:,t-i -I1- et:.:- a : i in- 1 .r-a l

iith the linear dispersion equation for deep waterr ic- n b:.,

Equation 1.1' indicates that the second order .:rntribu-

ticon t.-: the Inuae profile is a furctr ii-n of the tempo:ral deri\a-

ti\e of the second order \elocit., potential. UI ing a dis-

crete forc- of the second order \locitrv potential from

Charter 2 arnd sut.sti tut ng thl- appropriate deri a'ti e' :tf

Eqs. l .c.r ,it. inrc Eq. ii. 1i n th the s .in :.f the fi r t tern

c, r. cp l fr.m .1 mi n .irl t i a plus 1 es iltr in the l:. llci(ri

veci:rir j order c.nr t r h-.lt ion ':

, = h- *'c.-. : 2" -* *c..: , cc, 2iw r.- .- ,;,

.,. -. .' J. I 1
1 * .I C O. c- 2l(' .t :l .* ) -tF }

The first ct..' terms in Eq. i 1 repre ent th. nronrdisperi i 'e

it klir i. ai -: z r.: su]ting fr:ni the r : lf intc ract ons of the two

spectral c.mri .nentri t give ir n qI -.n a, .). These = tw.n ': t o',l ar,

a c-. contrTi urte to. the si nc.s *:f the spe tr'il comiipoIcnrc t

hb' jAdtji e both tn- their tr';,,i I- and .:re tt The thi r term

in Eq. 12.S .rhilch repr e er tn the result of the cross- prod-

ctr .:.f the int.:ractic-r betweeri the t',,o sr.ct ral ccio ponrent-,

pro, pa ate it the profile e Fphi r e pe for the liri ear War;ve

given b, cq. 1'2. I but ith ticji t[Ih pha'se. This .a e j .ill

c:nritrih-ute to the -i rEilrn: of the res-ilingr lir.er pro:fil.

The final term in Eq. i2.81 represents the result Cf the

cross pr.od .t *:f the inte reactions b Leer n the t.o,: aspect ral

C'.-.rpO r tz ,t_ hich prT.Opi t-s -it the Cr lo-.c ph-ase speed,

, -L/ il, but t. th t.ic:: the phase. The contribute ion whichh

this i-e i m-i:es to the en\reIl .pe d.-pcrnds -.fn the si. n of the

ternI in th. brac'ke.t. For ..*a i the e .pression b -ec nomes the

fol lowi r,n

H *
^ o;' C [s -.5 .'1

This a.ie i 1i9.0 u it uf pha-c with rersict to thIe en r elou :

phi3se. I t_ iTis'. i-,.a uccuLIr undc r thc nodecs of the er'%i.-.p and

its ri i rii ,3 .iccur r i er tie jr.r t ird.d: f th:' cniclope. Thi"

wavc' 3als- hai a non:.- rc vlue ,.hen aiVtraucjd i er either the

fast.-r temipor3l or shorCter ipat.ij l sc1cai citf the result rjn

pr.:.filf 1 : p ali e Fien E .. E .I 1' 2. Lurgn.et-Hic ir_: i r1.j .te rt

[I ] discuss nume rLu-.l cffc: t- which result from t[his .ae c.-T,


It can be h-bun th-t the rTight sid of the tb-.iJnrJ c.o n-

dition equation which d te rmine' the s.c.:onrd order le i t'.'

potentiial in dJ e:p ate r rcdu.ce to the teiumporal der'r iat e of

twice the i iretic ererf. Trhe secorin .:.rJEi Ci.ri ribut r i-us ri

the rie ulting 3atie profile i; en in Eq. r2. S jiabove na, tr e

Seen from Eq. 1. 1 to be tpr:pnrtir iurnii to the ternpiral delria-

tri e of trhi i. :cond rd.Er iclr.cit. potential and t.: th' ire -

tic er cr rc.'. lir ce th -:_: tI o :Q.ui ctinc in i d monrtr'rte thit rihe

kintnetic enere- plias uch .n impo rtant role in th e s.::or d

ord:r :contributit ns.u It iii of ,value to e. .j'irc t hi- term in

s.ie: detail .

For tie casc of deer. wi it r ia1es the rirht s idi f t he

combined free surifac b'c-i ndar. ci:ndit on reduce ti- t-he tem-

poral dern, atli'e of twice the itnetic en'err' which ri.. be

erpre ed b:.

2 't t ': : I [ ; : [ t I .1 01 )

SuLstitutine the appropriate spatial derivatives o-f Lq:.

(2.6a,b) into the right side vields

r -- 1 .

,: 1
I = E [[

It .,i e C i in : in prect I)rn of E. i?. .hi t hat ti he .-erti

ci: l and h'ri' :.:.nr.1 el I .-c i :..ip--'r iT a re 1:1' uc,- f i ph

a.J J, c,?r] ;:riIe ri E the "ID o0t f th- c 1' T ,.iJi T.'i ev luated at :I=0

i ; a *..:.r. i rnt i.hi .h i n ind, j r- :n t n : f t ime. Th es. r Fr rcnr

5zr.:.l In e,1 f- inri ter i: c :.-n and th tE c"'.r"rjl diTr %at ivce "f the

f i r rrer rr. Ir t-rr aclt in Fj. 2. 111 \v n i3h in .eep. Lte r

iTS t r 1ir. *r1 ir e rtA r'[1 H* r ifii lic" i .le; .* Eat ~.
t 'tittin. rj-,e ;rtrr.',rtr1 j.- Iti I deeri'1 tirc s of E .1

12.'. i,tl irrto the nr-i .ani hinr t mri or.l der at i e in Eq.

1 2. i InJ

~ .W-' I i I l -i i lr, .- *..- I ..-in'_ _t 2.1J] j

For i ,l:.:. inte r s'r- truii ,, r is c ; i si nr, that l.n1 the

Ii l'ferenca in pSheJ result in ncnr in'i-ihir,..n Crzt rituti n .ii.

rl, reo.o.- r for -t.'.kian ,:,i.r r l .1 i, r a s i1.c., =1 -1 r' Ihe

ri it i. 1' ini.-'hesz arid thief linvir ieloci t potent i e il gi 'cn

t. the Si i. fi ( '..1a ,l'. .i : s .: t r., iec-rnd n Tder.

For the in rers3r t i of a Jd ic tionjl esp:ctrun .r-f Jeep

3 v, r a ~.-., t1- co:nri n 'tuti nr fTrOm the h "i o :c nt, t e i cit t.

fielJ: i re decreased b a miltipl; ri:ti e :Con .ntant pr'oportionr

to co:-: s i .here i is the a:niuthil JiLftferenr.e et'.ccn the

dir:cctio'n of prp.Fagatio:n ;-f the t.o L.'ive compt.r entS tivein b.,

Eqs. :. .7a,b The iT ultipiicatie conrstanirc in Eq. I 2.12) no

longer mr' b' he ,c :.rc.i jnd tht c* ntrit utni,,n. fr,.nim th-,: _um.- of

the ph is.e3 of thc inter 1-t in 1i a\e cL.ri..:.nent-I no l,:.ngc r cncel.

Cc.n;cqulentl ., riir to i co rit bhrt ln to the seC crd orTer

elInc:it pot ntial tfrom -t.: Eh thr sumr; amn di t'feren-e i.f rhe

p.haj e- cLf the hK nri : rontr, l ri d rrt i : l rveic.it fiel.I is

obt ined.

The imprort nt di ffe rence heti c.:n r tri ctl. Fprrijc. :

Str:.k in interar ct ionr col ii n: r Spe.: r inter i.ti r_: 3ri

directional srectrjl irnt r'i.:t i:.n- are Je Mon- 'trarT d i nipl for

the deep jater ;iSe. F.r the case of firn t depth, ,dd tL inal

aitt bri c Factrrs: i r rico:'urite red It -i i:. detr 'r Trmi : a- 3-.

fuilress ai -a j i nP -: illu;tr tiu e .:amiple.

3. Simlnu tin aI Fo: rce.: l, the
Digital Lin ,ar Filter T chrani .-u

one of the iracr.tri al ppii:ations ftoir the i ir i'atirc.n c.f

a rjnd.iori _e a surfai': rcal i:a ti n -. rI-- ...bt.a n ao ran.]i:n f':r.:e

input for the dynamic inali.' e- .,'f pernan:nt pile- u.uppr.rt-:d

struc.:tures. i td f1i'' i hai' .J- I.le: peJ a ,m, ethl-.d.j for th.- linear

fil terin *f a sa 3 urf c.: re aii : t i.rn in r:.rd r ti o,-bt jin the

: rinenIticis require rt: .:c:rpu' te prc s.uire forces b,. th rlori rian

equation. P-.heele r [129I uijed th I tc-chnique .t co.:mput prC-_ -

sure force coe ffi cientz froT, measured hurricane re il : it 1 r

and ojh-t in4 d x\c r lot. -- rrorns bc:ticer the me j;ured and pre-

dicted pealI presurc: forceTi it ia r..inr- elei tion; along an

instrumented pl itf orm pilin .An tlher jpplicaticon ['.."] of

this techrnilque ,i ing' hurricane re al :I ti-:3 's .'ieldeJd mCean

-quare : rrort- for pr ;-ure forces c.:mriuted ov*Vr the crest

Fportir *,i the re ll : ticonn hi-h cc.m:, arr fa vi' ra:. ith mc an

quare errr.- r fr im prTs: i i r rcLs c. 'rr' ted ui.ir the e ir

ttr. ai Fi-Lrn:tirn o ; --nce these re e t ive ,' : uc.:c- ;ful

ir.-p licir i r' of the diLitil l ine -r filt r techniques to anal.'

rii i5n ri'id m iirecd i SuTtijC re"jli :ji.nt J ith finite

Sk--"TiMi the e.tensioin of lini r di:itacl filtering to designer

ihc.iild in r .-i i sirmulatcd rnonirlneir re aii :ti:cr.n 6ith non: r.:

The .he r. t' r ncri ir car :-a simil:i t i.:.n ill t de'.:l -

oped in Ch nrter 'e lected re l :atinc.ns ill te filter.- d

in Cha pter t.:- t t iin the Iinem.atic" rc Luir. tr.:. : compute

pre S.ure f': rc:,: P T:e s re tforCD c pu.d fr':-ir n'i-rnline lr

re.ili 't rn i .* the liner ,r Jig ti.il filter will he *:cmiFpared

i, th me a -: redj pre; STue f':or:e rec ':rded i nfro hurricane crn: r-

s td U.-.


THE,.,F .' Ill T jII ,j,.)'JLI.JEAP
WAVE -w.ti[ TE A,-CT II'I

_ectiion 1 :f this ch ipter briefly (. outline ic te rrT: a-

hi lity 3ssunrpti.:rn t.: t e m iirl:'. ed .i formula tirn; the random

sea pr.'ob'lci i. -ctiorl 2 Inrtr.'duce the nrotiti.:in t.- t-e used

in *iescribir.ne rhrid.rm funt.: i.:. mnd i C\:' witbh.ut pruof ,

some : f the mF l them ati: l t rins f:rm i .: be ijed in deicel:pinr

the the..r,,' f:r rirJr. j urfac.-: grait it.' i3'i'c: correct t.:.* eco:'rn

order. ?ectin r 3 deffirn the thii order 't Lt t ic.l Ir:rn rit

which 'ill be utill ed aj i i me ure ,f the n rnritne jar1t *:,f

th:- random sea simular ionri and sho:iewl thi thi thi ttistticil

lear ure is ci.:.-c it sc rio d irrd.: r in the perturir-at i-:n riri.-

eter. -ection 4 t'f:.rnmulite; the ran-rrj:. b-ond.'rr',' ville pr.t.oblem

and solver fi'or the flir.t and ecac.n-d .: rder rtiron mea 3ures ari

spectral rei.p.onse functions r,-quircd to, in late ricnr. nl ar

r nic.i e -ei re 1I :at:[ i r.n e'- e: ti: n F re turn t,: the nonr .: i rn i r

statistic al measure of the :ec,-rtnd order correction to the ;Cea

surface re al : i tion hhi c h c k a deve ilpeJ in r ectio:r 3 -ind Jia -

C.u;es its Fourier transform. Lect ionr c. oiic'utet tIe sccor:nd

order spectral derniit,% of the sei urtfa.e reali:l tio'n rnd

sh iow that it i *: c v'.':iluti,:,n *-:f "al first order spectral


1. Rindr:i [l Iffeer nt i l E.-iajt,:ri n

lran, r ph,- i l pr':ce -:s derrons rrat rant d,-m r in -[ ic ons

i which cannot t: mrode led b', ph.- ical la r th t mas be scilred

b.v dot.: r irm is c ith;e rsic .ier.er ( i33 e L s everl

e a ip l. p Is i th I ppl 1 c it: i:-4. The, r ind r. a ri Li. : .. r -, eitt r

into th, : iE.nhc m j ic r al equ.i 1cn. which arL uied t,:' ji pro...1 mri :

the rndon T i pF. icil [.proc _:s E i the er t h :.ch i'. I ti El, dependent

\varibia e *-*r 1 I the coe efficient : n the di fferential equ -

t irajns r I3 I through bo.th fn etho s In addit i-n, the randon

dererrjert .ari a t e i ia b:. e a i r iri.;,m fu.r.ct i c n ,:,f still inicther

r r.r.i.ni t' i t .r ti hrouFhI a i ltrTine pr::ces .: i .. c con i [l11O ,

Ja: i In i [ r, 1 .r .'e ri n i [ 191 1 .

In an, eVent. the de.script triton *-ft a ph.'rical rTocei .:

"rand,:rm" require: an c 'ti ,rtc of the. p rob-abtlrit; c structure

'f r'h T andil tr'n':t or. nhier trh.:;e r.irnd-.ri, frun t rn..:. arc

deter rm trin .1 b'' ft' rent 1 1 e.lquat i .nr, here ire 3r. least fo:.ur

c:r..'cerCeIce .(u-~- stne hih i u FIL t Ib: ai t 'i d for ec h ini l

c:o:,nergen.: que stior in the d tr.rir .i t ic differently I elusa-

tirn icr o:...n [ 116], Ja:. in : I L l or -:, shr nn i o [Il11'.

Orne :r f tit: iToirst elli p no n.r, prcr b b i t i: rStructureS ii

the.: 3aus sian dens it.y ,anrd .1 'Aui iar pr.r..ib Ili tic field is

frequently assu1rmed tr, m -,n. i-,'.,drd%.'ri, afic problems. Ch rain

[ 3I describe: main. of the n in iciat ins of .-ssurr"ir. 2a iau- -

sian field for r.and.m flo:' problcmr in hvdr,:dj"nm3ic Pos-en-

blatt [ 11] and t.ampe de Ferire ['1 ] i. res; a pecificall I' the

prub bilistic quect ion p,-.cd I:.% the 3ausia3n field- assuIjrpt i'.n

for the randrm linea, r e-ap problem. Th.: probaih lit ic struc-

rur-' of nanr, ph 'sci i probci e Ti are fr- ijquent l 3.lassumiie to b.

kno.) Ti l thn T II 3 umi pti.-.r, c. f i ia.l.u ia3n fi:ld fc.r the- random

linear -r'.a ill be u edi heroin. 'i b 'r f ,Jt cu.. r oi rf the'

probabilis ri.- a .-ui ,ptlonir us-.: liz un ji'cidablet in crder that

the limirationr an1 i nt..rpi.t:t t ions n f thE pri-.bati 1 i i'tru -

ture of the applic' ti..ri. of the res-uit; t. hbe d.rilved Taj, b,:

ful l.." undierstoo':d.

The s5.'ution to. the r3nriler sc :a pro'blchl -i i a r.anIc-do fun: -

ticn uhich i J.:termir..d by a partial diff-rrriial equiti.r'ri

lvi-. the Berrnoul li equat i..n I in n ,.hi:ch c-rie of th:.- pi rid.:rt

variabl-le: i rth- \'eloci ty p.jt ntit i i ,hi,.: i : s! I a rar :.Ir

fun artic r. The raridurm Eioiacit, p r te-rti l ii3 c rsiirin:-d fr-om

i boundiar' 'alu. problem in h.ihich thle coeffi:i.rtx of thr.

governing' field equation anrd cf trie b-oui-idar' ,:cridi t i.:.-ii ar

deterministfic. ih.- ranl' v l Ic-.c ity potc'n ait i a r rad-:m

function oi ',f a rando-m vari bic- -called th. phase anr'l,:.. I'h,.

prc-babi 1it- deni it ,' functic.r of the rarni.,rn pha i .- rin l i1

assui be j t ro be irn.:-pendc, tl an rid id-:r.ti -call. 1, u ii f:orn. i d1 -

tributed bct een l F 3pc'uli ; ( ., -j' dE rrjritrat:" that

a sum of random furnct i'rn.r -hilch depond- on jn i-'a nric ai. l.

uniformly -.istribur ted rarnd,-m v ariable- t.rinl 'ier rapidly

toi.iard a GijL i ian di t ribution eve\ n for c c.: uhein htb.- ium

include.I relate ie\ l fe. random a riables The riT orou prirof

of th- ne ten tdcn :- tE ard Caus iasn di.tributic- b:, rand-,or

field? as the numr ber of the rand-:. m variables. inct.eas.-s .'ith-

r.it limit miT3 be f.'und in Poci-nblatt [ 11 ] L.y mcarn_ of the

Central Limit Theoren.

The three funrdamentral cornvrgenre theoremrs required in

dJeterriri. tic cIlci lu ini volve .:onEinuit., differentiation

and inteeratji, n. in rando:'m differrEtil1 cA 'jl iiu i eai-h :,f

the e thrth r flunm arn n ntjal th-oreer,: i riu t be :foin to ctnlL:'rge

III in prrobjbilit,-, i1n m inT n -,u re, I'*I in distribution,

ard i 'I in "alj c. t ure" probabtiliti 'ct. .Socng [ll] )I. Mean

-,q,,re u: lculi_ iv l] i -e iu eiz tC( deterrir,, e the c.onverLecnce of

functions and m3athe atical opcerat rs;. The linear ':lution t.

the radp.j 'ea pro:bleit uill t,. ai sunied to b, ec:ccn ',d 'er

- t -ti i on r ianj r.ou -en Oh.: w.: in the .-t tisti cu[ l m i iori, er, t rhis

a-- mpr.tir oi. a .tationari t,- anr ho.-mokene it ,' rende r the- pr:obha-

bil ic, .Jist ri utilc n i ndepi nd nt L ..f t" ci n i r n i-p.,:e, rez p ,ct li 1,l.,

let. 'Ysa-la r [51 ]i. The iasumrpt ion that rarn, n process is

-ecorii d ri d r tjti, nar, r r homo:.:; nec.us h's been .id.iel .arudt ied

and r xploited in rermi: ..f the corrcl t13 in theor. I l. enkirn

and Watts [rr 9 The terlenc toward a '-.aussian di tributi cn

O-f j s ii of randjom fin'iction i.rl-.ih are identicall. and uinl-

fc.rmlv di trihuted beti.ern I -ir, result in th. probhabilit.

r.ructure ot"f the random linear sEi proc Le" b-eirnL .-cR.pl. te l.

Jescrii:e tb.' it.= mearn and variance The complexities a3-s:.ci -

ated w ith e n ionstr.a ing "alaircnt sure" probability are dis-

cu ised b'. P~rtle t ([ 1] jnr will nro be pojstulacte for rh.

riadoo sea prc.blei. The determination of rht evolution of

the pr'bajbilt.. den it.- function for the random nonlinear

sea will also not he arrtemptcd 'cf. .j- 'inr ki 16'8s j .

The complete formulation cof the coririanci: equatrins for

the random 1r in.:ar sea protl .m using cor rel at lonr the r.- ma- be

found ir, cn'.der (12i'ij. iTe. di sci r tiuris tbetwcen .ir. ir.plift -

catirons of the so-jln ti :.n to 1le liner .:i rPr F.Im 1 fm r the

prob tbilitr eni;it: funciL 1on. iising proti:bilit th.:or:' v'er.u"

the cot-.iri ance t'ncti. ns us ine .:: rrlt l i '.r, the.: r. ajre Jenc.rn-

itrated bt .:.,,mp r ir ng t h. papers t. b rFo.:l ntlatt [111] anid i m e

de Ferir s i.- 1 i th thi: pi-per bt.. In.ecr [10u].

2. IntroJ'uction r to FarLJnd i .n i r,:ti r.n

i n. r n ('2; ci. Chs. ? 'and ] J-v l.:- ps ,i:en.:ral e. pr- --

si-r.n f.r the thr jee-dinm r ;conal nTr: ',' spectruIm ti-.r r randJom

sea surf ia'J reral i 3ti-.:r. His dei ,el5prment in :.lics the thre-:

coni:ccpt of i I' FOurier rn JI:,' :. i, I 'i t: chi ic pro c.:.c es : nj

pro.,b billty, and i.'l the h',dr:ol 'nvar i- ,; c r ti. rand- m '-ei ir.

the general .Jer t1' ic i n f i insmar "'f '_1t. "), e act ..tF rh,:

three concepts ire deiie oped ._ep rather. in .-'rdcr rhai th.

efi'tfcct: .: f ri.:ddi f cat I rD :i- f j -single c .: epr. he ..,c- iined

only inr terms of tr ne onre concept L hich it ri:,difii. TI,-

:eneral nm:odc 1 i then c.-piicitlv. dc fined in t. rmr- .:. I-.A:th 2

deter rmrinist : and a st'-..: tic i..-del in which i-the tnrc rd.:p.:r -

dencics i tetc-ee the three cO.:nci pt s are e.'mir n.d ~ in-'m-rn [''],

Ch. 1.

This procedur-. utili:ed t .' inr main [( to d.ce l.op a

model for rin enerL y spectrum oif r ndoar n ifunctiOin .ill bt-

f'lloiled in ottaininan a s.lurion for the second order :-.rtri -

bution r- a rand.lrr sea surfiC-t re.i] :ation. 1uHn r tre nriota

tion.il c.:onvrition used to de-'crihe random ifunctionis d. tffr in

-one rTepctC t froiri re icrfnernce to re t rence. In order to a\ri..

possible .-onfiu ion, as ..c 11 a.- for e ase of refe rcrnce, the

mathemnticsl expressi.ons .ind definitions to he serd in devel-

oping the corid order the;.r,' of a r3ndjbm runlinear sFa arc

given in this section.

The ;tencral Je finiton- tf'rr r arndom ft.nctironfs .and for thc

.orrel at ion th,...r, f'nctio ni t th both tinle i nd space argu-

,,eint arc c i'en b E.in rr n ["-; i'hs. 3nd 3] rin bh,' Ihillips

[ 100 '-h. 4 Fiefin tio i ns r ft.nctionsi ith Cn.rl tenpor.al

argurientr jc g Lvc\rn I,.,' Sve :hnikos [11 1) r 1fagl om [1 3 5). The

c.lutir.n del. e loped in i.cticn I :of this chapter is initi all

a sulitir ..n tr.. a ipa.tial boundlry,' r aleic probl cm. Therefor'-e,

the d fin t ionn r irTn b1elic. 3re fior function r having c'.nlc

_pati l a3rgurents.

A rn1dmli fiunrt.c i:..rL which is a c.nrtlinu.ous fujnctil.n ? f time

:'r sp'ce i. c:;ll:'d a rarnd-.n pro:eis. A raru-dc m function h.hich

is 3 djiscre-te function of tin,:' .or space is called a random

sequence e ic f. i\:.shnit ,:o [II ] p. 11 .nr. r andl.or princess

na.'' be representedJ I:., the, fhollcw. ing I ..u ric r --ti t es narteg6 al

S' I = e.,p iF' ',i i- .s (2.1)

.h re (lT 15 a ranionm measure .i th arthcgonal nrcrments.

For th.:- processs rrprest :nteJ bv- Eq. (2.1) tc. be 3 real fiinction,

the follu wing conrplex cotnjuate Trelatlonfhip must be 3stisfied:

3L . = I.' I 'I f 2. '.l

A gool di ciisio.r. cf the d1i r irn.-tic.n t b. tl. En Foiirier-'tielt eC

InEe ral I r E elrice int er ij and Fou. river inte rail r ,' m ,

found in i aciom ( 13 : Cri. ", ce:. 9]. In EqI. i2. 1 and in

tho-c' ihi:ch fr:l.i o, the parime re Tr of thli ar llliir. t i s[.Fj3c

hiut a re pr: c rnt ltior f.,r ar. irguin rt in hhich the r .rinit' r

is rc me imaq be s1 i ilarl.' J.fin d. The 3it.:c:i jriarci fLun:-

t on for Fr:pr.cs trh rziradoim "~ a35ur: 1i i'v n biv

For a rr-oce:. ith h ir sp ati l l. h c. .rmo ,~n : I or rtcrprall,

raitioi r ir the aur.c. *tajC, ri nr.:e function J per. nd orin. :,n the

ip ati l 1,J. r l.:.r te mpo.:r'l ai r anJ ha: -,.:ctrjl r.:pr..:'_: r-

t :tinr r iver, b.' 'hinci in'c Tli .rem f. S;te r.hili (119], 'It..

i[. Sec. 11, :r aglom ( l' ), ,n. 2, Sec:. i0e :

S, ri = e.p il r d_-iri 1 .11

wheTr: th.; :pecEtral di trih-ution funct i.n, 1 1 ri related rcto

the ran io3 m me aure, i T for a ho.',mo: .n-fc :,u; Fpr-ccs bit'

E iE. r. dC I = :- r' ... i -r d 'd .V id

.'h re I fi iis the Dira: delta ,i : trih uti n IFr ie.sjar (. I'li.

This distribution mi. bt, defirnrd, in thie :sp c dJor-i r. b,

l2r i .: I-*l = J *:-x.p ii I 'l ds. I .6'1

and in the n3 e nu nt.er d.mlaoir. b.

i'2ni"'. = c p i l *;7'd. I..

Equati:.n f 2. i i pipl ies t hat '-n' r trt t spe.:r ral di t rihutinC

funct i..n i i'en 1i-r I a lir:i ielta ,i lr iihiut i.:.r iri pice, r

Eq. t 2.i i i-pl ie tho at 3a cor, ;tant f .rti -. n in the patio l3

d.-m-ir i r. i 3 a :e,:ttr3l .di :trit-ution frtirT ,rjLn citcn the r ri ic

j 1e di rT ibu ti-: ii 1 rnterpr ta ti: .-.f Eq. I .ri ar d Eq.

I_."' t' 11-i: fr ri thr -e defJ r irtiun *:fI F -uri- r cr an forr. pair

and d i-vl nrr a rjtc -;n -. rh' r- Ti-: r r.' ip be t..: n tr.he .: nc: r.

:f :t 'chartrc pr iccer and tr he n r crnc pCt .sf F:uri-r anral i.;.

i'he F-:urir r tr r. Slfirm pair fCr a r:e l 'Fpatial fu nction

tith 3 c'impl..l r.-pectral d-en it ; f'ntctir n -ill bt d rnol' ed b.

1.= 1

F i : -
: --' - ,:i.i _Xr ., 1d, (2.El-,1

in place *f the 'iener -l hl i :h ir t arjnstf'rm pFa r ( ,2o.5S,1 1]

th.: F.-.uri.:r tran: .-:.rfr pai r f. r the: ai1tc.: .-T.ari an:: tfun. cti n r

aji tile "[ c t ral der. it : n "r a hu c erIe:u- pr ces-

hii b ee ien t.r h- the cl ,:wing.

, rn' .' e ,. ,Lk'r- d! :2.'? 1
S . r .

I ir, _p 1 e p il- -r dr ("..?91:'l

The normiin con's tant require.! f(-.r the irverr i i in cit .a Fourier

trann fT:rm pair T i1i a1 1 3','-s be a. s:Ci ated iith cthe transiii rm

hic.:h J.-firne th,; ipectral d.: r it f, cr ir:,n. Th.- Dirac d.:i.t

distr rit it on l de fined b', Eq: i(.6' an,] (i ."' i ere -,ultiplie-d

b 2n n ,rd, r that th irT FourieT tTan;Tlorm wo4-uld. be pr'por-

tinal3 to unity. The impr:pc:r integral iven in Eq. (2.11 is

often 3asscciit ed i.ith. F:chnEr icf. iJlli [13J 'lj ,:-r,if the

.pectrr l Ji t ributl n furnc iocn is confined to an interr al

-n,r 6i th He r lot: if hi lk [i:. ]l.

:Con.paring the stc-.:hastic Fourier-Stieltit:. intecr-il

represent action of randor.: funct ion ii'ien b Eq. .1) .i

the Fourier intePrii rrepre;,rnt tin of an:, fuLnctrun 1 I.e.,

both determin istic and r,-nd et. rministic. f'un cti'. nsc ) which

s ti tii-es the rqci lirein ernr. for th. e.' istence nf a F :.urer

transf orm pair l'c Titch arsht (12 ) the r i r. itic. h p

bctieen the rajnilorm me iure d.-ki 'l,gic r b,' Eq. 12.1. ,an] the

Fourier aspect rum, F i 'lgivern / t 12. 1 ii ; ccr. t t c

FI' I d ik I .l i

i.e., the rsrand- m..' ure .. f the pr-..:c imul t ha e .:r it..

5imlarl,', compsaring the spect. ral reprer entatI .n 'i *f ho":-

gener-us pr'.:cess d-'i'i gi u n by Fhinchin' The-c.rem in Eq.

12.41 .irh the Fourier trans furm of the autoc.::i.ari n.:e funr:

tion, 51i l, g ien tb' Eq. 2.9 a the rc ations-hip a,:ain n-

plies that the Jd tribt ir c.r function has d-rni t.

SikI d (il i2.11I

The ms]c.r distincti -n bet. een a rand.,om function repcres ntei

b' a Fourier-Stielt. integrsl an a randr-m function repre-

serted b' a Fourier inte ral i s the e'i- nce of i Je ri .ati e

of the Jditribut ion fiu ct ion.

In jppli i:c t i .r, to ranr..ido ;ur f.,ce c tr.'ii .s '~ prc '-ler .s,

th -: r ieS trl ti'n n o ti- h. a r r is, ri- ,r, pt ion th t h.- pr:oce;s

h 3. 3 J en 'it.' tfLrnLioun iL fir .quntirl',' unirL iportrr n. .iat ris an

S'-: *.h. i. .li i cu -sc:- rh.: ine an uar r inrt: rabilitry r-.: quire-

ir,r-nt hIr t, n t c,:,:h tt, L: sn. Fi.u'jr r int.-grils in t[. rmis th..

.c alr :- *f" the Fprces

TFle ihiftin F pr-,pertv r t" the fi iric .i t. 1 dli tr t ution

lcf. P ipoul tI'J i pl i- .3n iimpi.rtant r ile in the fi lt.ri nt

pr.:-b I.b i i.'hr:l', i s CJ -' in i:h.pter r 2. 71 T-i s hif ri n ro cp-

--rt." c t'fin:d i r th I sp o: ildom in r ,'

l .T = .e. P 1, rli -~~"I 1d.12

ind L r thte .i num. r Imr in ,i '

-; ," i = -..p i -: .- i' lr n .i i' .131
S n I -

ihe ,utiC. ,iir ln nc, f3ur ct ir:n .Je finr abd3r i- onec of the

sr r. i t is.:l ne -,j_ a i h es h .i i-, ll be. r qeji'l red tc. compare tnt J t C

ch irj:ct:ri - i-ra.rjor.i per.c.-ss Th :s ; tati'tic i l mTeaur,' are

de. fin d 1.. r, in -riiemi.l e 16e ri ing op rarcr dinoIted b, the

:. :ctati:n *.nic.,], ri d dfin d b.,'

= / ,;- i 2 I -1

w ,'rI pl':) i th,: probabil y jTreni itv of the r3idJom i .iri.ialle

. The la3rlance, o i' the -second -ord': r s-tatitic l mominnt

which me- asjr.: the Jjisp er- ion iL.abit the n.-jn jnd is computEd


, = Cf(--E{ } )" = i:- ")- (1} 2'.151

The standard *Jeviati',,rn i1- the positive iquire root of the

varitance.- defined Lby

STD DEV = o.

The autocovariance ftunction given b' Eq. 12.31 for n rand.ii'

process with :err:. mean e~aluated for :cro -patial lag j 1:

equival nt to the v'ari anc..

The trivariarnic fuLnctioin if a rhir.J ord- r ttati ticja

moment defined t., the ex.pec tat ion op rator ai

I. ,71,r 1.l = ,- ( 7,.)1 (2. l' I

The v .kene-s iE a m-3.sure of the i ,i .ml tr.' o tIhe riiandom =ca

=uriface sbout the rearn normal i :e b., trhe cu'b of the stanjarj

dcvi ati :n, i.e.,

S E I ( -E( I'I .

The -.1'eTie e m c a ur ma i v bci 'valuated f r-tm th: tr ari ai nce

fiLnction for a random, proceS- with rei c. In n for :ero 'Fpa iaj

lags r T,r2.

Although higher order pol'.'=p.2ctra i,.c a uret uia. t.e iJIfined

(cf. Prillingrr and Ric.-;snblatr [( ,3ii1 onl., the :urt.a s

mnia-ure will bte evaluated herein. The kurtoc i ; ir the fourth

static ri..:al moment about the mean normal i tr the -quare of

the variance and is re ercrcnced to an equivalent normal random

process with the amc mean and variance by reducing the

quotiIcnt b ; i.e.,

.= ( .19)

1 N

3. Trn an jri:e inrct ion

The tatis t ic l nr.Ir- nt h .l:h g i L s the be r. n: ii ure of,

the -e:.ond ord rr p. rturr. -ti.:-,n count ributiorjn t.:. th ran.oij, non-

lirIc r -ca pr.:c-' r hr.:.u h C mr. :ii r of the s; j ,F.,etr of Ch.-

randLoi r : t ll: tlon : aout the mli a 15 th third t 3 tati tiC

mirLi..i'rt o.. :.. .-r.-.: FF,:rmnaii, 1 th'e ass'f r.pti..n of tajtion, rit'

and h.-rr cn.: i results in mn kir ng ti fiinctii ons fron -correla-

tion theor, -., cr alri rc trivtiarianre etc: *ep-.ndent

onl. -in rth diff-eren.:c in the temporal ani d -patiai p5ar Ji -ter

*.f its arguI i nt. t ithiugh th- a3s iumpt in..'n of i t tionjrit.

snd hoticr:.g-g netct. ar.- u .: d, the : ...piicit d7 p.nd c -ric f thec e'

:orrelatiri tunc:r t i .: cr n the temporal an.] ;rpa'rti l pjrar,,te r

t and ..ill b e r:ta in d in order that the relat ion=hip .tt t een

rjndJ, m ieai ire- and the ;pecir c ral dis tr ihir t ion function 1 iven

b. Eq. ..I ba.. e-m r.io dJ. Thi; d.ef'inriti r, f r the spec-

tral di trit ilt ion function involve: the. irac deiltia ji3 tr -

but .irn -3 i result of the a sijmpttion- of -t ti. onari[tv and

ho-i,.ogeneti .- The [irac d lta disl tr'i ution grer tl., ficiliit tes

the deri.'iation of certain results obtained in later section; .

The trivariancc function for both time, and Spi:e param-

etern n1 olv'es t.'o lai- 3n.] mn'la re repre'- enttid b'

l 13.1)
rI rl ri

If the riarnd,.JO e a r ali :rat on, ri(t, is a iisuir-d r.o b

cnrpose.i *-." *i lin.c r CaU :ian e tim t- plus a ncnl in r r, n.rn-

C-auSi ari ecorij or ler per turlb.at ion cc-mronr nt.r t Ihe r.aru.Lnm

pr-:,cc- ; mc ', be '. -:p.nJdJ nn a p,:--cr ser 1et i ten by

nlit, xi = Irl'it., t r, it .. i 13.: 1

There the or. R r o the pertr.rt -ti.:.n F r, i. r serir s pa r :-t : r

h'a; been inmplici r l.' abt.s rt..e itnrt- the random a.: f l U-.:tiurn in.J

the order -ft the pertOrturb-ati n para-er.er i- iric.jl at b.,- the

prec:.iJingc numeric il pr fix. ubh titut ing Eq. I 2) into q.

13.1) sand using chr linrir y of thee I a f t :pctt iur. ope rator. ,.

/ields tihc follo,.ing:

S t r 1 i l( ; .'1 ir, i tr1l..' i *1 .1 r i t, : T I

S i I: L T ; r* i T* T *r II

I ir, 1. :t i ,r, t t T i r i r, I t T r7 I l
E i. niict,1 r tt -l; r' tr .- .

E 1O l ;,-1 Ii-..I)

.here re I''r, i rearn: .. urd-. r r fi in the pe rturh a rconr para,-


AsXiuni ir trat the rlnd -riom mieaure ofh the linear ijur i an

estimate iS uniformly dJstrit ut..j herneen -1 ,nj th Fourier-

Stieltjis i nte ral for the lir.ear e- ctirate ma,' be replaced

by the dcn;it, cof the ranJom mzaiii ure to obtain the folloa-ing:

Ir 't, :.' = ."F.F I l *exp il...t-.* F.1 .1

= IF |F rri] > il .:r- k -.xp i t- l l J ad 13.11

where r i 1 is an indeperJeiit rani.lm rpha'.a an z;le un i rf rml,'

di ;tr ilr.,,teJ betr.-.e in 1- ., rsp .Jli i 6,9"] I deiiin -trates

tchi a randJ:r,' functi:r, .. ith this di jstribution tend; rapidl,

toward Gui si sijr rjnrJon. tuncti.an. Thi firir e .pectation

operator in E EqI. 13.I3 is theie ?t'f.r,, 3 third order istat sti-

cal miiom.nt c.f a i i.au3i 5i ar rand.i.n 'jriable a. J is *-f third

*,rd-r in the pErrtuli.ati.rn rarmeter. Appenji % i jerr.)nstratesi

that .ifft rent at iri. the r.r.menr, t ..ner aer.3tin function for a

jj.ss n rle andom riab r tie e ', iel js i :er.: pr-.iluct

f'or this te rm. The 6iraniSl hirng .f thi t. rT fi r 3 C3au; -ia

r3ard:.r ftunct r :nr ma' als.1 -,. s- -c n b:y ijubstiit ii n Eq (3.

into th. fin r-t term cf r.1. 17.3 ~ th the f.-llc.-in result;:

E t I r I t. T : i

*d.1- r,. j1 T"'*

*exp i l '*l, ,r *

* e .-:p iI.', rF- 1 r 1 .51

c l ::-.l : i ( "- 1 I (1d '1 .11, ir., ".i. ) i d dK.2dr.3 r 3 I

Since- the random phase angle l'I sre independent, thi

e.Xpect tic.n operator .3.,' bre .Cui posed into the pr.:duct of

three expect tions, i.e.,

rfe~p -i.,q. ., .I ,'

[ex,:p ii l 1 I I Ei:.:p i i 1 2 1 *rle I p i1',.

i c. i

e'ah oif i.ihih hs, :ero :xpcct at i n and is, thF:re f.:re. cQiica-

lent to the pro jhbili tl.: rc 'ui t J.eriti d in \ppn dr.l .1

Cther :'xa..implc of uni orfm l, 1 i rtrihiuted random ri at ~il are

given t., Bc.rgmarn [21 .

The e rebuilt dcrm rin tr at, in important cr Mch d .t.f rrim.i: r-

ing a random p r ces s h- i h i t ji.au j i in *.:rnl i in i t' fir rt.

order e-ti mate The i.':l.' r. *:,rder ri oricr ni 'l; ing t :itis; t' al

me ja.ure of the tri v ri nce Fiur,.:t ion g, %:n bt Eq. ( .. irf r a

random prl'' s s ]-i C:ir'ja n linc ir eS tiiri te i f fto.irtli,

or.ier in the perturbat ion piram ter ard 1i due '51c.:1, t'

e:(iu r:n e .of the -seci.:.nJ order r r, nllP ire. r cI: rrect i or. 't tcth: lil'c r

aussi n ist i, ate. lorc er this 1-ei t .:rd.er rno 'n iln hirng

mIcasurc i ] ; ~l a it r .con c T rde, r in th pertrt I ti[ :r. p'r, F' m

e ter as no otri Ililher pert rb t i:.ln ..rjicr mT.ia. fc.:rrm ti rd.

order 'tat i t tiCal mi r. n I: L LicIi ire .c:f fo. rth :-r icr in th.i

perturbatinon ip r a im,,t r. in *jrder tc interpret further tle

effect: of htm i-c*jn d o*rJdcr corr:c.riion ori the rricarran.:c

fuincl tio in e? p li .it forr, :.f this correction is required .nd

will be co, mput:d in the follo,,ing :ecto;n.

-I. Non 1Ine r i e ,E i tc r cti ns I E rcct
to Cccr.n J F rtiirbh it iion i''rjle r

An e.'cellrt 1 f, rim.ulati orn of the t,.undar, valu, c p.r 1:..ier

for ;urfaic gr i tt.. Jjrtic i' pi en .' hehiiiseri ( 1 pp. 44'-

J_' S] ir.j this f.-.rr.ul' ti-c.r, i s e~ .ua.,il val 1 for ran .Jor, -ui f ce

cravit',- e.-. -w nteJ in cit i.on 1, the r-:n 'iC r .r-n:. .*f

rinJd,:. r fir, t i-r .rs, jeri tie i nd i n r .li ral uced in the rirdi.jr.

bourj ar, 'a lu'. Fprhle ,m il.l l r: in r a .an sq ,ii.ar'. -n -add, tit rn ,

F':.urier- '.r.t 1 ltje integral -.h ich re uir re no a priori 3i unp-

tiori rccarding the e i; terce of a dens i.te rinction will be

utili:ed l f. Fharuj: h, -P .i: d .1[ .l

The flu' id is uiuriced t:, be iricvi .:z d nd irn, c .n re-; ib le.

ITh t'l id nr:t i:-, ri :ui :. ti.. b.: irr otat t final snd three -

Jd rer, r : ri al, The , I':.c'lrdrinit: a.e ; lie in a hori:ontal

plane It the m-ie n se le\e l, := I, i th the : a.(i5 pi ti E

upF rd- random riel-cit., p-rterit l, I, c' .cc u..h th a

u I ..:., ,:,t = I (J.11

F:.r rn:t ati nr al corf '3p tri- the h':. i iorn t l .*:,i p l re .' 11 he

der.r ce i-. ,

S : .', r4 ;'i

The gi:.' rninr parti-l diff rernt l fielJ equli' ior fcr the

ra3 dorIj i relocit po'.teCti; .1, i the: cquatio n f c-: continuity

for an irrotaticn al fluid. fl-..;

-1 = = 1 : Ix r.'=,-h : r ,( ,t ,t ,

wherT i the three -.dL ir'rl ln; ral tria ien t *.'p,'ratC r, I- ,1, 1 ien


S *I* C *-
I* I , ----, = I* -.-I

and l are the orthop r.i. ul -iit 'ecit,:r: in the thr-e*di:- :, i.-lu il

carcT i ijar ccL rdintrte t h- terp. The rini t p, dr pt fluid i:.rPirin

i sr.'surmed t. .e u.rb.:.ro d.i in th[e horl:co ntal 1.imeri-ri 'r s.:

that the rI,.:,.undar". :ondi i i .ns re.qulred at the h.:ori:. :nt l bt.o'-

diri:h lc.catel it infin ir re th.c : .hrIch preclude rp rti il

starn in g ,a rc ; and the Loind. ari,' c:rinditions recuirre. to quan-

tif .' th i ndn..r n co',efft' c.i ients :' r ei; en i l, i are pre'r- r b'l d

at the free rree rf .i arid h:.r on til b:trr : m t l,-unda3rie Ti~

fluid ;jis assumed tc: oc:cup?' a fini te Fpth h, in the 1. er

half pl nr.: and t o ir.ti the f:oll 1 ing nn fIl.: b,.,rtt:.ir. Ir.:,n-

dare c-ndJi tti.:n I' EE i ac r:c. th- h'.r i : l imp.: rmep abtle be,'tte r.-'

:"- : = : : = h I; ',t 1-. J

The fluid sur fac l(t,:i ,i rc n" t r .rain- d and 'j nin: a it :

f ee ijrft a ce b.oun.a r. conditic .r, I' F':B') i i- reqi re-. t.: pre-

scribe mathematic ,ll 1 the ,.:'.,ntinuitu'' c .t f luid m t,: r i cn at th,

interface and tc. enii iure pli:, ical th at i.: fiuidI p3rticl.'I

are conIccted acre :. [hi ifntertfac-: i .. ,

i c: r : r ,= rix, l,|. l ,t.) i 4.6

there tthe t. -dimen ton l h I 'rci:rntal grad e t oper tor,

, '" is given n ;,

.7 (", = e e I-.

Aii addlit; :.rin l .j', .i free surfa'cet bo u i r,' c nr t rar int it

recilulre, t c ensure Uthat al.l tr'es' ; ac :rni .n)ilLon the free,

:urTf3'Ce i arc centLinuiou In the il-,s nc: of _urf-ic: tension

and for rin rtmc.rIp eric r- re :cI re i'. umei.J t:. bl e :ero, the j d ,i im .

fre:c urft :c t ouln .1. r. condi t i I i F.Ij) il., he br a ined re .

rthe erri li i e uat i ,' n n.e.

t .1rl l 1 Ir: r '|I3I E. .
* ,,- I I = i,.' t ,i ; : = n i' I ',|* = t .._. ( 4. 3 ,

The .tot l d ri i i e i .f tie Iernrrn ll equ'ir lr If. Fhbillips

[10)], p. 231 c:comb ine l i th the t.F :i ci e iminat,: the turi nC.-n

'c-,i ar free -surf:f e elc at i-n, t. I, ftrfa the r'rLstC b,..,und ar.

c.-ndit l :-r to: '.*le]d the f,-llcu I inc combine free surface t.eun-

Jar. c::n-Ji t ri iCF' .: 'I:

IC ; 3,- |1 |.-ititi = h f = L., t 1 .-,t .i

t-hi :h ri' be e:,:pnled t, the t',lL, .inr ftutrii.

t t ,- -3- = r. : = niy,t) ,|I2| :=, .
14. 10)

In :,rjer t.- av:.i d e -ialu tine the C'F PB it i th. t irU .nr..n fr-ee

zilrface i I: '.i t ion, rli'.cr anrd in order to utili :e a separable

coorJinnate s 's tem tr. .:rte the ch ter ing i n fie l equation

[Eq. 1. .3)], the CF'EBC; is e .pan.il in a 'i' cl3urin ser it

ih.,ut the til l iter li v.C l, -=', i.e.,

no tT :i*[T. E lj = :; = Ol:--

In orl' r t. jLbt l n 3appr. inF.ie olu ti n to Eq. (i.5i

dbl-ecti t.j, th- ro.rl1 near IF 'Eti, th rinl r.m jTriCi : jar a .- ur d

tc. r-ipo_. : :.3" 11 p.erturTh it ion: .n a mnr ir, i i i l 1ine r i- T pr.:. i,. -

t on Th.e mle a-ur.e :,f th:i p.: T ur.bj tin .r, i : S J um: ff.:d t tI: r ro-

p rc. ion.al to rh- 1 are slc.p.; .. d will b1 u tJ a. Le i3i n :ure

of the pe rturb.irn o r. c r iR ri : p.r i3m tt r. lIr. r, it i T rnor

ra prri. Cle r th at the i.:.rdte rin i:f [h- p rturt bai on r ti. a

pi.ram-ter i: i Ilid for tithi.r a nt. ir.,uou .: r .l : :rcr. s..:

trum of r nd.:n.m irfac: f r 'viit. :iei i ] lich [ 1'i' ha.!

Je\voted c.:n i d.r ble- r- search i ri pr: i- i hi riL ,alth t- at :ial

rigOjr t.j pro. lci l- r if pF. rturt-rin. ce r In r II l' Tcrr t c.ic: irLJ

his csrtal- l i heJ '.- ffi ici: nt tC:t o fpr'Ojf ctr.- ur. the

val ijdi ,' f the pc rct ir-i tio- n ,*: C r ,: .-- F c F ri,-dr i h. I : ( 'I) I .r, :: -

tinJiu d 1 1- i-'o-r- ini liate J i r \, Felli ,,%c F jra d ,:.:.r,[ril iIt.: .1 irtr..: r

to che cci pl. t ren-' s rif th,: pr.-,f? : The pr .). i hic h ir-

reIqui reJ t. es t aat. li t1 he i : t n.,: f p: r ult- tl n .-.rd.: r-

ing par a Ir.-- r C':.r a cont iiruuJ ,:,_u r ri 1r.r r, ilberL: -:a:e e re

too co C ,ple tx : b:e c:,: r-ed h: r:. Th nareri l 'i. n t elii :h

[1 i ] and Fri edrich- ( j and u u ..i-.r. :ri : .d l.' un t:rd r.id

Sch-w: r : [ .] eSt jili h th c li it.- of .3 pC r t urt ati on *.rd r-

ing p jr meter for c onr i ir.uluu s, pe ct r uir, Lid 11 e s 'iiire.l

Wulth.:.ut prToof in the folii: i i deve ii : .r. r c-nt.

The random m v-:loc: it- pr.terit ial, r ra jon ea i =ur fac, reaii -

zationr, and th: P rnoull con. tar t .ire aj unced to t-e c' :pinl-

able in the folloWiin poicr Scri- s wIth the pe rurb t in

ordcrirng pirarmeT r iMplpicitl'. ih,:rhJc irnt: the functi,:'n l

rnct j tLr.n

If .,:,t = 1' i ,:,tl 14.12jl

I i.' .t = r I .,tI 1 't.

t= 1

i.t-' ir.,jtu n Fqi. 1.1 2 ,t ,.: irnt.: the e.quations of the

t.-.i m.J r'' i ilu r.: ,t l r rd equatinr :quil .r er. f ct thel

f. ll:1 in, t lirn .ar t.' -ndar.' I e 1pr_'. r- i ita ir d:

S = 1: : = ; -h-: ,',l,| | 1 .a1'

.- .r, -iI A4.
r 1 = 1; : = | | t l4.13tl
1 r' = 11 1 1 } =J 1J -

I = o ; h.: , | 14 .14 )

: = ",| | .' .* E (4.14c

J = 't t: *'

: = 0,1\ l ., ,t:1 (4. 14d)

A ;iiut ioa n which iti; 1 it e e\.a.:t 1,' [q 4. 13 b I i

pi 'eri b, the foll.; .inc Fourier- tiltie, iitr c ral .

It.( :,t i = -1 i" _1 e. p 1-1 l, l l [r .t ]
1- co-hi | Ii| I
C -cvshr ij
(.. 15
whe re .il\l .,ti i the r in .:' i meci ur-: of rth vel i i t. po r. -

t i l. Equiaion (4i .1 I rc lllr i r .j r r r j a sumi ptil rn ri? ar -

inig ilther the e, litenc:. ..f i J nr it;. func tilri ir the teffmp rai

d.:pcn,-id nce f the: rarjn d3i v.- lo cit .' piter. iral .

Absorbin the temrtporal .J. r i t i e ,-f th f.,: rnculll I .:.or.

stant in Eq. ( .1 l ci int. the r rjnd.O measiur. dl [ r. t.

Tequirin thra- t h raiin .l.r prr ce :i- rii i tii i :eri Troit, .'*i j

the ft ll .:.ir. initial v iu, problem tfr theI urin: ,dep-nd.-rieC .t, f

rth rand-or, i me 3 s re

S p i .'[ C j tanhl li !l. ji i.l.

Tlhe ldete rminit; .t c ,lreen'- fiun rtlon which sIit sfl.:- this int-

ti'l valu: fri, leim as first de ri ed b.* Fin i t:t rn ( 15.

Frie dman, ard -hinbr. .t [ J1. ] h e:.. teniJ ed h r r ults t:*

gencrali:ed norill domaiinrs indj ha3t diScics c d the d r.e lLrrni -

tic olut iorn in exhaust S-e d, tail. The s lut in-.i to the

rando.ii initial viluc problems, h., ieicT, are less perspicuouI

3n.l involved ar, irre ersiibilit.* parade'. witth reg ardj to the

statistics of the initi l .:.riitions. ifn r.:rder t., a. id the

corplicatioins attendant i.ith thc'e initial talue :tatI3 ti.: ,

the rarindo pro.ccs :- 1il be assumeidi r..-, be ab.th t.ation r., and

ergcdic. These -asuriptiuns rcr 'er th:- solution -tatis ticall.,

LrInd .per.ddent .f jaii, init l c-jr-'li cri di cussi. l n o lf the

effect o1f Tr. e_: 3a iurnipt LI-rL uOr thi- s tocha3tic intro gral a ivfn

L.,' Eq. i 15 1 m .' be fc-und In inm n I' :: ,h. 81.

Eq.jl tlLc.ri Ic . 1:) ri,, :t ..l 'ced. b, the i pp 1 i lcat1 i .f t th

rP-ectrjl thec r' of randrcm rP r.erroi'rs ci f unrl frd jraI chn3rt:

I c.] cr Fr i:dmin r [ ,] i. 11-: i rte rand ri Fq. iJ.1 ,- ma', b-

writr r. in :'pctral oper3rtor nr:rtat i:n 1 bv

L[ r] l l I= I d.IIl

.hcre tIhe i rn 3 r IpeC tral operate or, L, Is gii' n t.:.,

L[,,t] = k--- glr t rnhl I hi 1i.13

Cquati -ln I .a"I si ati fied either be the trivial -.Ilution

dlAl ,t V E,t 1.191

ur h, the :er.':e o the -pectr' l operJat:or

S[ ,t] = 1:' J1.20 1

Eq ,jat r. iI. 1i re-iure- ha the p -ctrm o the t r te rar.jdnm

mei 'ae, .11A1[Ftl. a3ni h i enticli ci er.'.here e.xccptr 1lopg

the hY!perur fi ce given i :-' the zerEe: f.- : the -sp ctral P. ra-

tor. Ar. Inrei rae l wh ich i ti .: f, q I 1.i mi., be repre-

sented by t'h fcl lw in :

dlA[E,t] = e .p i l tild .i[', ] 1 .21

where the eigenvalues V are determined ," the crOe;:. of the

specctral .-pc r tor ; vi:,

- = glt tarnhil |h (J.2:

Equation (i l. 22) eitabii hc i relat ionir,'ip het.eren the to-

chaz tic: integral gi P 'e n 0. Eq.. 11. 151 and the h1idro.l,n.ini.;i

of the boundar. I al.uc proc .l-m. The h-'dJro:dyr, ic bourLj rv

coniditi on reqirei th it the ;p:.:t 1i rtre ernt t lti.n C. f tli-

randor mea'ur- \i ri 1-h ever,-.here in the 1r,.:e .iae e.,.cq.pt jl,:n.g

the cure 'i eri i bt, Eq. I 4.:: This re trictciori on the

spectrum of di.:;'k ,' ..~, be Lo'ii b the f':.llcwin f-'irac d lta

distribut i-.rn (cf. .rIrin [ '], Ch. .3'1.

,1l[kr.: i = d 1[k ,.- 3 i i' .

Altern ter neth.j-,j for computlinr thi 1 a 1 .r e zi river,

by Eq. (1.2:1 an th,' syrectrum .:.f the randro mireasure ieni cn b.

Eq. 44. i f:or g reneral larir.j3mi d i'ff rent ial equa tic-:r. arT..

given ',' Bo'yce- [2'r 'Icthc.-J h lch jr: :r. cfit'ic ll, ppl -

cable to ri.docru wa,,' ir, a di spers rz e iedji are CVen t.','

Frisch [51 For th.: random _urfiace C ra,'itv e',: proj.il:, n

foriiul .ated .abCi.. the f lu .i i.. r i I . ; J l' ..i to e .:.ri rand.:m

rid the e itenric lue couiiut d hb'' Fq i ) .ire dr t:rm rin 1 tc.

Addition a iretho:ds; for apri.'ing th.- C F Ci gi ,n b.' Eq .

(4.13.c) and (1.1 c 1 are Ci ren bv Ph ili p: (1001 O ,h .. h.,i

pointer, d oit the s imil aii ,' o: l th.. FP:' F t the equiatlion oft

an undamped hhsrmionic :' a tit:tor.

I'ubstituting Eq. '14. 1.1 int.:. the ,;rneral solution :-i .'er

by Eq. (4.1 1 and using Eq. (A.2!) give. the foll.-,ir randon

velocit potent ial

i (.( ,:,ti = .. c _L i l'h :). e:...p 1.3t-r*^ dl. [F't' 1]
c- c shcih I h i

"her-; the rc jll t of the rjndo.m V loc j Lt t potter nt ial rc:qvjir-.s

the f.l.llo in, co.n.ple.' con 'iu.r e relati.rns.hip:

*Jl.A.l[- I :1 = dI j 1 ) ][' )I (4. 25)

Tr. r .:ri se i"ur face real :at i.:n i det.erm irn e..i fr:iri Fq.

i1.l ; i, i.c .,

r,Inl ,t'i = i o ,Fp i 'f c .-0r .- I l j ir, l l 1" .2o1

The s'pectral rer. r.-rscnt tt i n O.f rthe r and.j m meas 3 ure f the

rand-jrm vel.:.ccit r. .:.tenri l m'.,' he rejc irie in term .: .',f the

spectral rcpreirentation ol the ran.Jm sea surface e lrvat on

b, the foll].:.. trn i:

d F [ ] = S*C E1J [5 l T I t 4. 2 .j

In terms: of the :pectral rcprescntar .-in of the random 'se

I trface neasure.lri i 'I th. random '. l'c l-cit;v p-.:tential and

radi'.i':,, "e- sjrftc:e reali:. ati.:..r beci.e the follotwirn e:

(, ,: ,~ -i } cashf ci fht : I i I
coCsil l r| h t
(4. : ,-,)
!n fY,t) = yp ii.'t-."i.)d1r i.)jl 4.2 t.i

The ftnctiorn l f.-r.m of the s.oliucion toi the second s.:t .,f

equatl c.rni vhich are correct t :. c :.c-nd ordere r in the perturbs-
tion piraiiuct.: r nrd which are given b'. Eqs. i4.14a,b,c,d1 is

assumed to be 3iiiil-ir t:. the first order solution ivecn by'

Eq. i i r'elnLt inrg the fr-quenic', ,, inrd t c ~ cp rijti.on

contc jntr., I'.. in a .n r* r l f:.rm hich .art t. be eiilj

thrciigh the boundar. c .rnditi:on., a -oluLtio.r ..hh iii ati tLe'

x jctl qE Il. 14a ,b i 1 gi n bn the ft l.:. nLrg rn.:ri :.m nIc -

I= .p il t -, I-d [ ,.,I]
5- Icoh l I I e

ij. ;'
As nr.tei previ. h1 [q. ( 1.1-Ic i equi.. ltent t[. th:

eqiiu tia rt r ft'or 3 f : Ice'-d, undJairpeJ 1h3 i t: n ir c *: c lll t T il

eqi1atc .on h.i be-n "tu.ic d e :ten-r ic i. I r. the thor, oi" f'ce -

tral ..p.ratC-.rs an d m- b. hritt-.n n tin r fol.llo: .inr; pec:tr-l

cp-ritcr f'crmr

-i e .p iI..t-.* I .:.., ,i ..... -d.s[ ,r...i ] =

1 .r:p i l ) -.-i, t- i .,.K.. ;

*dir[ 1 r[k, ( : I 1. ,1

,her: .,- ( I ] ii a lin.a r '.pcctral .:pe r tocr rr.J

.v[1 ,. .. ..1 i l -i t ] is the :pectril r .-.p. rat.:.r f.j.r the no-n-

line ,r Jde ri tl.;-* g~iv .n h. th- ri ht s ide i f Eq. 1. 1 l .

The general fincti r: fnf. : r., *:.f the separati.n cr:-t rnt,

* (I.J and frequent ,' . jii'. nr.:-i. b-c q aijrtifi ed b, require ne

equi .alence :f the te ppoi il jnd spat i l iful'ct ion.- in Eq.

i I 3-0 This e iqui alence require:- ch.in,. c.f ari bless in

the integral for the seccrnd order randr.m Ieloci .t, potential.

Hi ldehbrnd [o4] or r l.A.,lr. i'kff and Pedheffer [11 ] gi 'r the

r.-q uii rernen t s for the arpl ication of the fo llo' inr cr hanCe of
vari abl i equat io'n:

I. .J f f r ,vi l u j iJ .- ud' -1.31I

where IJii l is the at,-.lurIc alu.: of the J.aco-t.an of the

chanri :. f anr abl or l ir ',r art- ttl r' e iri- si- f nlteCgratioln.

Introduced the fo il:.ini cha3nc- -f varinablcc: in Eq. 1.3i.l:

S= 1 2 4.32 a

I .. ( 1 .3.':,

ihich h,'cI t h: f.1 llc. inc nrrora a tshin Jac:ti a n:

'i i : ,g (i.3:)
I F 1 I -1
1 a l ..

t.he r C iL the iroup ic: .ct 1 o :f the i h ciofi .r rt. The

IFSR'C n:'. bec-cLrr c

-i -i ix p ii :.''t rL ,i -
L 1 1 -

= *. e p iifi.. t.ic -r l ) "

1-1 ,o.. F,~ 1 d r 1[k 1 ]dr r ,1 1.51

which may be = :.l d for the sp.ctral r presner rtat ion of the

r'nd.d'm telocity potential correct t-:' -econd order to giive the

fo l 1 :. tn g :

[. ',. ;F ]dl 1r[ 1']. i- i i]
1 1 [j ; P-- .r -1d ,r '- i, i
L X *-1 1 1 1-


pro ided th r ne either thc Ja.:t 1 l n IJ i nor the line ar

aspect ral opc rator, i ], arc SingulA r. The jdi .i cn l te rim

.:.f the J ac.hi n arc ct l inarL n..nidpi pr-1i ri'c r nd the

J c.-.bi n, there f:r na, b.: seen fr..n Fi. 1 4.3 to e no:r,-

sin ir lar.

Phil l iF [101] and H is-'ise Finn 5I S hia.e .em. nt rat. d t.,'

separ it prou.fs t tha the linear Ji p : Crsi ni quatr..n -ien -t.

Eq. 4.22 7 i con'r e.. t .. ard the i' n the th ,n plan in..l,

tlhc fo:re nr. :er.:.e i nriya oC-ur in L [- I. Dnr f.:.rd and 'chr.art:

[ 3] FriedJm n [I) I and c\'.e hnil:c 11 l L '' p-1:.l'.'n mi I

e 'parr;ion3n for the ratio .: f -s pectral o.er itc.r e uivalen t t

thuo e given in Eq. II._..i ; hcncr.:r -. re.i -Occur in the i pcvc

rral o:'perator [ 1 In a ddii t i.n r,ince E .I.3 ') i an

inh o-rir... nec..u trc.urdi. r.' .-.:.nd tiun re.i uit inj:t fTror t. pertur-

hatii..n c..pan i.ns, the h.mcLene.,,u s cii it onr to Fq. i 0 1

are identitcai to thon e giie n h'. Eq. I1.1 Therefo re, inc

the spectral r .per7tor in Eq. i 1.25 i noun~i irig ar anrd inc.:

the homiij-;e ere.:oi : ti-on 3 i.hich cl:rrc.sp.jn tc. the :cr. c; in

L ). are identical t.: the linear -sclution n and liii ted b? the

,Dirac de 1 -i istr bit ln i g .i'-n in Eq. I1-. 2, nn "free '-Ive"

i:,iluti ons m~a',- e.,.i t t -e :ond ordJ r.

c.ub rititut inf the apFpr. pr 'int dir\vatci c into the t-.r.

Spe ctral operat.-.r in Eq. (1.35 ) vi l Js the fo.lloWIn l 'pec-

tra1 rc poncase function for the rmniJom c c scond order vel:ocit,


"l", :'' l

.L[-' h 1 .

lh-.inr c th chanLe c.f v.rzatle il'cn b..' i-. Ji' 2 ,. i with
the Ja.:obi r i lve. n tv' ti. lJ. 33i reC ul .i in the fi:.l oi.'LnTr

eq'atirtn tCar the ranrJldrn 5ec:orni order \el.ctit. potential:

co. t i I I i h :1
,;;,:,t| = -i f -
-= c.- l i: 1-. ,| h t

'. :p i ( I . t- 1 r *

[ 1 II x ',hrn,1
1.i .-'. '. I'.. 'li

Sut.ttitut on for th,- rarnd-.r, ipectral repre sentation,

J, l[ I. i ,1 ) I, e n r b',' Ei. I .35 1 and for lie t pectra!

rep: r."Or t f. n rl -rl .Len b, Fqi. I 3'. i ,ield the fo llco, ng
e ..Fpre' rin for the random sir ond c.rder vel'. it ." potentti l.

h:,:.sh' l c. ',' i 1 .I : i)
,.l.r ., : ,t 'l = i / -"
S :.s-hi i l 1 Ih

Se:p i[l.' I L -. 1 I .D[ ,,

*dl (.7 .r l ': i l :1 (4. 38i

where the ronlini r interaction kernel, [', ,, t 1 ll, 1 .7

is equivalc:nt rc t he s pe.ctral rc:pon1 : tfunctic.n givcn b.' 'q.
I 3 .

The seconrl order correction for the rrndoin sea surfiac

rejli nationn r', notn be \ aluated b.,' substituri n into Eq.

I'J Id the proper deri ir ies of Eqs I .i ,b ) arn Eq..

. 33. Fv ,luai tin a be fore the sec'_n.l orler GC rnouli1

constant. i, in orje r t.: maintain j :er,: ie in secondd ':rdc r

random process .'i lds I rh fT llohii ng ...pressilon for the =ec'-.n

order random sea r' 311: atior:

,i yt) = /- .' e" :p i f *.c 1 11. .

*d rl [ 1 I d,- ~. I: '1 :. i

liher. the nonlinear intera:ti r l.c-rine [ .- ; 1' ,., i

the spec ral res.pon se funct ir'n dict rm ,in: b.t the ei r1 i t L, ._

of the right sile i of Eq. 1(J.1 l and is p licn ri.,

HI'l ],. l ;, ('V l 'J 1,' ] = l". -. ,'

*a[ *1 .l 1 I
S 1-i7 1- 11

N'ot that th.: s .c' nd ord.r iarn..jo sea reali i ti on gri.ri b.

Fq. '. 39'1 s i inversel., rro portion l to the gravt'i ta n ii

c ons tst t, a .

The nonlinear Interactsion iernel g l rin b'. Eq. '..Ji 'l

e:.pre s iedl in a 'i'ir r i tr c forrm. Different formrs m'r .' e ot. -

tained i-hen the appro ri ate J rivati eS are sub st i L te. int.-

the inha-iog.encou forcing ter tr he right .side of E .

I'. lJd). Wi n.:r (132] has p.-inrteIi .out that .n.- kernel oper-

ating on the product orf ientical random firunct r-nr, hiding

.r'mm-:tric .1.:,,n irn, f rirti cra3tri.: m .jy t.c rerniered S.'Timctric:

b, re' T er;in; tre are I:Ir.:; in th L.rnrici .jding th t o

keCrrn l-1 'ar.' dr.l'lJinv bt. tl,-'o. Tll pFrc.ce.1..re his bteer uced

to *-tts- ir. th,: ','mrici rII c e r u :d in Eq 'l. l arid ( .

h- .tq.ar i.:i, .r. icr the n,:.nl rin r =,: a 'irfacc r a .ii :it i:cn

i:c.rrc; r:. se,,ccn ri : cri.er i: .bt a ri.-d b., ad. inre E: i. b

. nd i 1i, .e. ,

ri f .t i = cirl ,tr i ." ,t ) = r* e .:p i c-'t r- *.l':.ir [ l.' )*-

S 1 r li .. 1 1 1 -

.,p -i.' t l .. I- lr [ '1l I' 1 1


I1 '.:n mai. 'iti r .: .:l:. I r.: c ; r. 2e f i r ible n th. irnt.t :rn i

for rthe :ec..n.j cr ier ran.idcm i'r .i,

. = 3 2, ". .' :1

the prr cron i r the rarsndi rr-. a re ili: r iun tec.:.ns
t.i or. Cc r t h rIcnd,-, I I I a E: C_ :-, re

ni ,tl = ." le:q i -i, 1 -dr[ ri.13 .

E t
i 1 .i: I':1 ',> ', ,

*exp i 1 1-1[ i ,l ) .1 ,, t r 1 .1 Fl ,l ,i r[ i',. :, i i -

*e.p tit) 14.40

C'oirparink a Eq. 1 1. 31 lith Eq. i 1.1), the random non-

l rie'ir sea ,urf' ce correct. tc second order i; s.e ri to F

n, .. ,t = t e .rp i .t .d; :{ 1 i'] J.. i l

"he re

J. |! .'' ] = e p i f- .F *d [ r ..- ]

-;- I ^ I 1 I 1 '1'

e :.:p i il l i 1 1 I l ] "i.- [[ .L. i ] i 1 I

The i np.-' rr fit poi. t natc ab,:,ut thhie fin l *:* pr.: ixon

for the rand-: er, : surf3 ac r e li:-ti:n ir that fo.r fi tl..c:.]

et :f hori rentall spatial cc.,.-rdinat rhe quadrat r.:.r ri-

butitr-n : :ccitur Lat the siae tre^ucrcir- a th:-"e *jf the l ine ar

Gau'si arn real i -at in Thi- ;i s a kcVc. pciner ir, t-he jprl i cIt : n

*:. rh.: fast i .,ric cr trans, i ri nr. th,- rc su! r ng s i pl fi i :at i.n

cf the ci-rputa 31 n o i.: n :.nl 1 n t r rando-..i rcall:ations. I]JCc

3al .:. th.t the cr,:.:.n or c r *,:,ntritut. i :.ri. tr: the r rnd:.i, iie.,-

sure, I. I I ,1 ]i which arc L i i.:n ti thle i r.tc. ral t rii i

Eq. I' 1 a re ftiiri', n :-f t h.: um. anrd d, i F f: r r.r::. iF

,ave r.,r.mbers, '. i 'nC 'quc t 'it ., 3 1 : the i 1-ri:rjd *'r.Jer

c.-,ntriL hutiL .:n i. ich rc i ii t frtri [h is c: r'.l i i .:. int :rir l itr

nr t phase locked .o.r niariispersi .'e .. ith r :epect t. the li r.ar

Gau' ianr est imate at that :3iT-: f r..qcri..'. hc E 5 irple ari ple

for deep r.arter i.jai s ;icnc in -ect i:.n 2 f i'Th apr.er 1 iil .-'-

trated the mannrir in i hich the-c sec.n:.n order crnt rrb utrcorn

are nondi sp r i i;.irh rer.pect t.:. th linear Gau:s iar, pcctrii

and the resulting linear pr.-.-ile.

S. The I.ispectruii

The biLi sr.. t ru is th Fourier transfori of r the triv L 'ri-

ancr function rin. r-pr esnts thl spectral JErin i .' -,f the on-

tr lbur i i'r to trhe '~1 r C cl f tbi. r -idr. i -.:.r re 1j : i; a LLon,

r i.,tl I a ri uct of t hreT Fourier densities hose resultant

freqiic-nr *-qjjl- :er Ic: li f. HFli, sci jrn et ai. [ I. This

Fourier t ran.i- f .:rn pm r i-i .; .I-nl.rT in S action 3 trj y.iel..l a iw: j

zure che eco':, J o':rde r *::onrtr ltut io tc. t the rai djom cr r. l i -

:.ti l:n hi ch i. cl: e -i j [,, the.: se.icon crder in rerturb ation

p arT l ter. v In:ier miich ii, ful in'i f rrjtir n on r -ir lng sitatli ne r,.',

hiemi.-.c:n.:.n -us ran i-.m proce s. es r ti: *i.r tairj~ fiMro t i; thic trns-

f,,ri, pair, 3 hnri f di.3 cui i-n of t;r~,. bispcctrunm is de -elopeJ

in thr i i sc. t ion.

Th. lo'I ct orri r n.:rniini h iTrg third order ;t ti tiical

i.uirenril fo 3r a l1.n ar C s in -- ist iri t: as colrputed in Eq.

3. 1 r.d ,I m no- bc Ie,::,rpc;%-J e plicl c 1i into t first aIri

c -:crn. or r._r p- rturbation r contr ibuti i.)n using the r .sults of

.ct ion 4:

C, :rt r, 1 I
I T 1 YI I

ri 1 ,4 1T I k^*J~ e..F 1.

1 ] .p Il rl 3*r 2

E *'1r[ l, 1. l j r[ ,l i J r. dF [k il dir k l' 10 )])

0 (en 1f

Thii is a fc.:urth order nomcnrt o i the -3au':i.,an randun,

ineea:Iure, JFr( I'? ], ihich mri, t.e factored as shown in .'ppcndi...

A into. the ;um of three product pairs .)f all p siiible pirmii

tation : of the foir rf il i ji v- ariate i f the procc' ; is.

3asum-d to be spatial 1'. rimo.:.eneous th. ipec'tral distribut i:.

funij ctiu:,r ir, a', be c.rtre ;ed t., Eq. 12. 1 i ri h hichi tirh .rguii ':rit

of the Dirac delt' S d ti trilut ion. arc 1ll pr- 'it le pe rmutationi

of wae runbh r;. ithich recilt from tihe prjoductI of the dco:im-

porsition o,.'f fourth order Causian ranJd m me ria re. Ihr fre-

*quencyi dependence *:.f the i. i. number T 'ier b.' the irin ar

disper i.:.n 'eq-u ti.:.rn n Eq (4.221 als .' re -ult; in a teimp: ra i

1v stationary. process whir t the teL rt at iE.nr t'er the Diira-

dc It a di.= t rib tiurs .,r' pe r fP 'rme.d. hi tac t r ha alre ad.

b:en iusj d t,:' eli rinate the frcqiinr deprc rcn cc fr.': the r'n-

d.m spectral measure in Secti r 1. it in be -h.:., n, fin'll, ,

that the t:erm of io' .:.t ordel r in the porturh.atti'.n paraieter

Wh c Lh is s n:.r n i'ri ir. il in the t ritari nr.:e furi ct in for hi.:..i -

ecneoulJ st at on r',' random surface era'vit.' ia t reduces to.

*rIi ,T:r 1 r, u = 1 C i- 1 r -
rI rl i i

"-rl" 1 :T lr- ,r r , ]dr dJr is. '

where the non near inter. .:t i:r n k.e rnmel .1 ,.:1 .,r 1 :

1 ,2,1 include; nine permurtation'- of the prcvioau, Ir Jdeied

nonlinear interaction Icrn l. .1 ,l' ,.3 .2 and tte ci.mpl; :

unit ector; of the tine .and space laI 3 and T

Formj all., th-e dJou .le Fourier tr an, s orm of the tril'arL-

an.ce functi.rn i the bisp.i ctrru ird m ., be e ..prec F sed t.

r6ri j l_ j-, T, ,
rl[ .1 I -r { . L] . iT TI ; -[, .] -

c 1 T I T T r* I Jr..r 1 E j I 5.

lia :e Irrearn r t r, [ :-] h ie g n.r, the rnmetr p, rcoper-

tie:- of rhe F,.Tr Ler tr ar, s .rm pair f,*,r the tri'artan:e funr -

r ion arid bi:spe,:trtir. r.i h ir ilt s ti ate'd th: ti. pectririr of

Fress.ir- re.:.rdj tfroim 'urfi :e gr \'it, 'ave- to determi ne the

Jirecti .: r l :-rr-p ad .f the i. i sre-tra. Er ilint e r and

R.'r -en tl att ,[ ,7:1 g .e -mc.. .t hing funct on. and .:i the r useful

re atic.:.n:" r these ner n l hi a her .tierd r prol.'rpectri.

Plac:Dcn ld 3 '.:], Hin.ch and 1a', [r. j rnd Haubrich [62)

derive usefu, l ..pre's i ns i.nf r Jde : riri in n b ispe:tr Il csti-

mjat s l'1 tih Fi.t FounTei Tranir f.T rmi al i in th-rm. .Al th-I1Jii

much :etful intfurmatr .:.n ab-n ut the rrndc.m a rc:, rce s appear

to. be : dli crnib h l ir from bi pcctral e timrtec cnl, the c.iarse

mrIe :-. uf thi- trc '..riai nc funri ct .c r for tim e 3nd -pace 13

Shich are identicill -ero 1. :., the sl:c :r.n ss I ill be

empl,-'.:d In thi- stud .

At .r'.pt: t. t:btain a convolutic.n rel ation:hip from Eq.

5S.') for the bi .pectral estiimates grcen in Eq. (5. 3) whichh

ioulld bi e similar o thos'c girc-n b. Ti.: [I: .] for the second

order spe:tra did north prove icciesiful.

f6. F r.d. P- rturbaticn ,'rJd r r-': ctra

Alth:cugh the autc, s o jariance funct icn for rrndi:*o, -

surface reail;aticu n is not c l.:.-ed t thI s.cond pertiirtj-

tion c rd r, it 1i otf inir .re lt t: c,.mput th- sp.ctral c:nrirl-

bution .hi.:n results frem the sec-n rd p rturt action crdj.: r :*:.r-

rectinr tc. the rtn.idiom lirnar =' i r- li 3r.tion.

Th- autoco\ ar' an':' uncTi f-:or hmc:ge-rn.:us, rtariorn 3r.

rand-.-i i Ft c,-rr c:t tc. s c.',n.J p. rt. rbur ati :rn ,:rd-.J-r i i n ien .:

S T ,r) = 1 [ i l' i t I, 1 [ l t r ; i ;

P1 D
r r, l t;' I* l ir..r. '' n t, I lr: it* r; *r I

c r c rnt t r .h r rs is ti r d rd r i 7r in -nrt oIf : -

pe*nden rijdo irc. ari d l ( idertic l qul *o

since the e:.pectation ,f the product ,sf the first i.nJ sec,:,rid

ord.-r contributions is a third ,ordcr C us.i ian momentt ,.f irnd.:-

pendent random, variables and ident ic ill, 1 qual to :.r,:,.


U inri th.; i.jJ nt t t" -.et n .r r .an, .1 .c me' ure I NJ r.ta h :pec-

tral Ji :t rr t.uctiCr gi ir- t. E i 5 an tr del'c.-..r..po itl on cf.

a C.aa u in rfcurth rl. r iTi:.r.r:nt gi en ir 'ppcr.Jix ., th-'

f-.ll. 'irn sp-ctTral r:pr-s'ent t i..rn of tr.e autc.:.''arian.: ftun.:-

tr Oi, f.r i rIn.-i .n. -- .: rr-ct t: .r'.: onJ :.rdc r 1 -c rciin-d.

t l7 1 = '.f ."/ e ..*.p 1 l i .l ., 1 1 i 1 l'l -, -. *.^. *

nl ,. -]* I- '-I

il,. r' i.' 1 t r r r .

1 1 -1 J .
I .. iI. *dL .Tk I II I T. '
..? 1.1,'1 T I I_.. '

Int r ir a T nl i ,,, in l.- th- .h IL it rng pr- :, r rt. t : I-. t c ,, ir

J-elT .i tr il uc ic.r, th-: f .- .IrI ni: xpr.:' ; ri f':r thr auto-

Co -arian i . 'ir a n d:

r, l rr e ..p I -. 1

.,* ,'., c ',p r-.7. -...' r K- ont[- ]i u- ,- 1
I crt inrlu d

S .[. 1 1 1 .* 1 jc .' [ l 1 I 1

[ i ; , ] [-. ,I, 1 :;- ,-. i ]-
1 2 1 2 1
*cp 1 j(., I-T I I I rI

5 1.1k..1 1c ,
3 ,jI ,' j Ido F ( '. ; i

Ihe 'wetric properties .f b:.tl the spe.:rril di trribir i:on

funct ion, [ ,: *,], inrd the rN n 1 in 3r in era3ct ion L.e rrel ,

I[:3 : . ,L I. reduce a Eq 1 '1 to. the foll,:I ~ n :

(T,r) = ."." LI -.T r. [ ,.i3 l d.- r /.'. '.

-[k ,i e n( 1 -1 1 T x 1-., -r i 1i 1

e [ d i .. J

ForT mal1,'1 tIh F uirier tr r nstorr, ':f the Ilt c- rTa rin.:.C f'unrcti.:n,

'.'leldi the pec.tr ri dj r' t, u furct', n. Thush iiultr.pl :in t.':.th

sides of Eq.. C'I.J h,' e. p I -wTr*- rl .dnd intc gr ing o.: r

time and space v'iCld

Si ( T ,ri p i -w T *r d dr =

r= fl,' s[ ,o ]*c-.p 11-lo- I '~-',I T I d)Lir.jidrdr'

Hi,,-, .- ] c,-,- p i -.Ir -r dr dl l.d d1 .i dT-j


',**: .. 1 1 i. 1i. ,1

1 ,,p r -., 1- d il l, tr 5'
.p it -i. ... .... il . i- .i d1 .1 .. r (*. .5i

in rtin: rh-e *.rJer :.f in rtcgr]t .ri n *'- er the i. ne r iL' tr ari

frequerc,:. diffe nr i al: -ti. The t .n anrd -ra:ce di f.:-r.:rintial

re ult; in int.-:c r i : ,uil'-ri t c*:* the iras .: lt3 .li t ril.u-

ric.n ci r, ':., E '. I:.13 f1 r the sr i al integri r i:n and a

Si, 1 l r in.d:nritr i f-.r the tI enl. :.r il n er. igrT r. Subst tr ting

theS e ir eriCrti. f.:.r [the i E ral : Jelri di trl i..uti..nr, trh

f.:ll1:, ing p r:. 1 i:n i: iobtline

i' ,i nr e r, ii._v.Z"rldTdr =
1T e r1 r I d T d r

1 [fr / .I 1 1 I l" i I 1' I
I .

1': r i-' .,; ., 1[r ,.'.7[i- ..p i .-1,-, 1 -i'i ].

i: ; r ,-i. ', j i I t'i. id 'l

S I -.. I I

DiviJdin; r th :t i i: lc I a '. inrd c., prinr, tlhe e Tre s ion fcr

the ein er r ;i ;ed -r...ujri er tr r f.:.r pai r if..r their .pe c r, l den-

sit. fu .ti.:.ns \'i,- l s r[he f li,; i n,:

S[-,,,7] = --- --v /."' ,l 1-e'i'e', l1-. *< '*J2 d dr
rl r I -i 1 ~ i, i r

The fi r: c intEgral inr Eq .. I'.r. ) give-:


.-hi h is the spcctral dIeristv for the linear -Caus'ijnr e ti-

mate. The second Integral % r-"hcs identically becau-e the

argument of the D[irac delta Ja sribution is not in the dojain

of the integral. Integr tinr: the third" intee r l and using

the shiftinE prnr.2rtv of the [irac delta Jistr iutt in civ.:s

,s [; ,,.,] = ".r.i \[ l,. l .- t 1. .....1
f rI -
H [o :L1 .. 1.1 1 1 )

Eqluati:rn r, i L the ijiiportant rcsilt first given bi', ic1

S1'3] for c:.n tinuou s pc-.trum~ i of dicp water isir s nJ 1 water

bv Longuet-Hi.ginrs ['1] f:r a discrete spectrium of deep waterr

waves that the ;ce.ond nr3cr spectrum which results r.:nl from

first and ;scor, d pj ertur .bation o:rd,:r :conirtrTbuitoi ns i 3 .i:.n-

volution o'f products of all first order spectral d:nsiti.: _

whosi umii an-.J ifferencer s in fr.qienc;es contribute to the

same frequency i:f the second o.rdj.r spectral r denlr itv. Equat inr

( '1 is 1 a generall exprr.? e on f:.r the -ec:ond ord-r -pectral

densit., for both tine and spa-.e lac e. The hv'drod: 'narii: rela-

tionship which restriict the spectrumi .:. the rand-.:i measure

to these value; in the .,i plane .givn b., the line, r Jdi per-

sion equation has not heen inrered.

The simulation of a tim.- series for a randio ;sea requires

that th,: sp ct ral denrsit be given in the frequency domain

vice the wave runt-cr do-ain. h-e spectral densit., ha~ beer

given in the wave ruaiier d-omain a? a re_.ult of ;IolinL the

sp.ti a houndar-, value problem in Section 4. Th t rars forma -

tion front, waie number r-pace to freqitern-',' space will he con-

Side r.-d nc.,t.


PPL [IC T IuN il 'm(-F J luJL %.E.P PF'.rJDOilI Si1iLiULA i ltJ

S cti r:: n 1 tr ra for', th. .direc.: r i al raindrj.ni, i il re

Expre sed in i a e ri rir.er :r'a.ce fcr the linear 3au:i S 3n rarnd.j:im

ie i int the s:pec tral der, j i ,' c \xprc d in frcquicnc- spacc.

'eccti:ri 2 d' ;s:u- es trh; .. t' F uuri r transr fcrirl iilcorlthr.i rind

Sit appli: at ri t: I.irh-r i ri d ra'nr m J c re l i:,itioni.r

Sec t -l.: 3. il tr.:.duic s tie i'r. t :hri ide r Spctr.Irl r .aliuat: the

tic. ciLon tant pirr etieteT: rjlird rt.- repre.'-r t mc sure.J hllrn -

-:3ari.=: c er r-j iai.e irp ct ri, arnd dii.:ils ie the applicat ion r

the Phill ip equi 1 it. ri m Spectr iu in the imul at on ,:,t non-

lnr: ar rand.m n .as. '-ct n I briefi'l'. d.'scr -: iL thi mea ur d

hirriciiIe generated t n ,rj p re Sure force r ali3 at i.rn

rcc.:,rded t. ',* 1o.' Frc.j)c t I drurinc Hurricanr, Carla in the

;ul f c.F e i, :.. Secti:. n i c r. pares the Jii ribut i a..n ,rid

rp ctra c.: f m : a ured ;sa i JcurfaC:C r.: i :at i:ons flr.:Fi four hau rrin -

cane pcrnrated record- %,ith the iimulnatcd lnrear and rin onlineir

rand..-i .,: a surfacee rcal 1 :at ions. Section . dl nriivez. th e qua-

tir'ns. requir-d in the applicat ior of the di it ra linear fitter

t[crthniqu-: tr pricdict r ,ave pifresSr-' foi.rce= from r rdom i s a

surf ic:e reaii: atir.n S. ta'ion co:,mp rimpa s rie district uti.ns

and v.pe :tr a o f rh i mc asi rcd and predicted \%.-.e preI'-ure forceS'

at the 55.3 feet d,narll:'mneter elevation o:n a vertical piling

loc-tcd irn '9 feet of uat.-r.

1. Fourier seriess .pprT o' itri ton-
nt* the I'.an m Ilea ulre

4Altholich the nr'nlin. ear interarcti:n crnle e]. fr water

of finite depth cr.riputel tL% Tick [ 2-'5 and by h-laise li.n [:.j

have been a ai l.sbi ftr :c'.cral ,'ears, r no :u.:cer s ful atterirts

tc. eripl',' trhe e-e erne 1 t' c.-.ni ntrT ict seconil 'ortjer n.ornl irneJ r

sea3 imulat i,:,n: hate been publ i1 r cd. Ti: [125] attribut.t j

this tr. the conp.i ratin al dil fficiiltes inv'",ls'ed rn a.pl', ini

the theory .

The 3vallabtil i tr c.f the Fja t Fourier Transf,:.riT IFFTI

alg,'nrithii [3"] nro. initle the s.iruliati'n f n'rniinc.ar rand-rin

;eas correct to secr.nd order in the perturLbatii:n para3Teter

relatri'eli' simple. Hr.in- h-,' th. rerenticn .:f ci:mple> phaic

angles b:, the FFi algorithim ijii, i it p.i-s iblc to cinmipiute the

spectral .Jenn ities c'.rrect r.-- sec;:,,nd rlr entirely in the

frequency, dcmaiin jn. to mai.e one inversion t,: the t iiT j. iia in

for the final sea -urfjce real: at1 ',n. lth u.:.ch no e iriultane-

ciua temporal slm latI1 ons *f:., the sea surface at car''ingn h-:.ri-

:ontal ;pat al locat ions- cre made: in thi: itui .,' the c ten-

sir-n of the techniques Jeveloped in rthis chapter 1i tri\'iil si

mill become bi'tousL later. These sir.ltaneouJ siriul]atins at

varying h':.ri cntal spjt ial ]n- ti..ns are required for the

dynamic analysis of multileg-ced pile-supported platf1 rmii .

SiFulat ior L.' digital computer of nonlinear randuir, -cua

correct tr. s-cCond :.r. Jer ising Eq. 14.431), Chapter 2, Sect i.:n 1 ,

requires thjt the e\act Fourier-Stie lt ies ntegral. bw ppro'l-

mated b' Fourier series. The pro.:ces of t ran f,:rmTing the

Fcurier-'tieitie in tegr-il requ ires firt Ethjt the p Eccrjl

di rrslut :r f.n trict n 1 h, jc Jcris I t,

.JF[ ,-, = F l [ j)].d- (1.1)

iThne ecitr ji, :f tlhe rand:rn mea.- ure, F[ | I. ll, L in .ave nrrimter

pajc.: ai a r- ult 11 :f s l. i .ltng spati al r..:jndar.' \value [.robic m

in .'ijapter 2 rand t ransfr ar rnjati:.n tr.. freo quencv. space ii

re.quircd in aorde r tr imrul ji a time series. The trans. or ir -

ti'.n fr,;m rectjng ul 3r cjrtesi3ri co:,or..:lir res [o: p-.: 13r c :rd i-

n Lrtc i g icen b.,

, = II r in I' .'b l

Sirh th-, ft.l:l]. i chanrc. r.: f i j r i bles ij.,:ojbi in:

F I I :I 1 d In U

SiFl = l11.31

i.:.tiitutin. rrhi chilnge :F 'jar i les .iclds the t'oii.:c'ing

discrete polar spectral reprcsentati.;n:

Fli'(-, ]-.*iF = Fll:,I H -dl.. (1.1)

F nillF the t rinsrformi rti:or from the scalar W'ai' nurihc-r sp :e,

I, to ia.-e frequency : space, 3, 1 i5 rajd: v\ a the linear dis-

persior:.n equjti i:n to o.Itain the f:.ll 11 in.:

F I ..-dl. -de = F .., j]-If I Jd., d (1.59

where tih Jachobln of thr: change of iriables is the f.llow-


!.ii | = l -1 = i [i (1 lh ] -1 (l..)
L inhi | |h i

The determinitiron of the direc-ti rnal d ependenric if j

general three -dimrnnsicinal random process is not trivial een

for a linear ,Cau a13n cstimatc. A gooj d dicscription *: f the

thc-oretical derivjtio:n ir gi.en b',' Long r uct-Hitggin;i S .h ich

s based on the th-eories of PaIleich [1 ,,] and [iiL. (1I9)

Additional irsighr im-nt, he dirrctiorn l _pc ctr.r wi pr.,l-l-.m th

some pec ific a3plicati.n: gi a.e'n I *.' in manr [ ; 'h. ii.

Applic i tic.nr to:. direction ril 1 e 3 rra's are gi'en Por r:ma r

(1[ ] and cnvder and im it h [ 121. 'n de r and Smith [121] out-

line an cxtenri ion of the line r direction3l spectrum t :.

nonlinear .iirecticn.dl sir ct r il f.r th. .:ca e :t aIn arrj,' .:.f

I'aj recorders hasscl1mann ct al. ['5 ] aie ric.nlinear asrpli-

cati,:.n for the d*.ijeteriisn t n crf th' Ji rect io r l sr read frmc- a

since pre-.iure realizstia ..n. These referfenc:es r flect the

complex xitie.s in ole'.d in describing the d i rcct irnal depernJ dnc'.-

of the spectral rcepresentition o.f a random sea.

For the simulation -*7f a tnlD scri: s of a random sea, the

hori:ontal spatial co: rdinate mi.a, be considered as fied and

the directional deperid n-e:.' ma..: be e.-.pres-ed implicit, in the

random measure bv the follo srin inrtgr3l.

? -I I
F [(.] = j' -1 ]- . $ il- ico .''H in l l. 1 .,'l d i1.l i

i her e ''"I ~' r.: fi.\C 3 val,:;1 of th: hori~ : nt l i.p ti.jl c:..oordi-

The aFpr:. ~ i atiion *:.f the -t :,cha ic irntgral repr rn clr -

tiocn 0 f 3 r:Tind,. : e re lL :tlt ior I: .* F.:urier set : r .: r.:q'jire

trh t th,: diiscr te toch :ttic : rplitiude .:" the ajpprc:.'1rm tin

erri; repre e nt the eqiint al'crt .:n.:rg. -nInt: r i th t n-a n

tir.l in the e ct inter al r pr.-: rtit i.ri -..er a di fferer-

ti l inter l, J i .e.,

IF l nr l" = I F rn:. ', -. 11 .-

Including tih? Ji r.: ct i.nal 1: -pender c imr. i, citl thru.:.ij h ti[,e

c.nr.l I pit .; .7. t th t e stO'ch i ti i pl itul .- Fl r I the to-

chrastcl integrall y iv n -,. Fq. I'1.. 11 i n Ch3pt.:r 2, *ect ijn 1,

ma.. h.: 3Fpro.lj. te.i L- a.' th l.:. 11:., j-,; iCrie :

ri lI = F Inle pi' I n t I
= -

,* F ri 'r 1 r in n II f1 ]

S 'p iltni- tl (1.9p

Tth iimul ati -.r by digital computer -of the r inl.im proce-

repr sert d i-., r h. ;tocha s tic series in Eq. '1.9.1 requJire.: the

cv i luation of i tui t j-.l e cprc:i i : t o fL r Cie : : r chi -: t r n.JOim

function, Fllr i arw nj 15lc. required c that thi function repre-

sent ai rTur rin r.ndjni pr .ae;;. De-criptions. c.f various

Sti'c.-hasti c anpl tude- which mria be used to ciniulitre the rinJom-

linear cs a are civen b', Pirk.ho:ff and Kotik: [1l St. E nis

[11 ,IIS Le Icehjur.: ["11 KLnsman i: ], int-r alios.

Technique; for simula ing by digital -omr.pte r a ari .c m

Gausi 'ianr really ation utili; inc r.hcz: st:.ch sr t i c amplitud.s

or deterrr ini 1tic amplit u e- are glir, b', tlihram (i[S and

E .rgmian I 3 ].

The technique s li fte. tfor simui a ine Iinear Gsii-.in

random e 3a for the purpose *,if this stiAJ. s1 the following:

(1) :C nerate i sequence of random r riu berii r L .'I ch

are unif i.rmIl, d i stributcd be tweern ( ,1 .

i2) rlultiply., the urii t'orml di ; trihute.3 random number

sequence v .' .-' in nrdcr to t'btain a rindoi, pha-.:

angle which h is uri L forml. distritute, d het cen


(: i orm ccomFlrpi : rnLuber :.f unit amplit ude vi

Euler'5 eqJuajtion ijing the ur i f.:.rmi.: distr t.utcd

random p .l f c: anrl i the ar rguint.

(II Ilultiplv. the complex. nu ber computer in I'' b,

the square root f the- di fer cntai nei rg. con-

tent ci en by the Pretschneidcr pe-truiri.

Following thi slmu i ijlat in proc'edire, the tocha tic complex .

amplit udc in Fq. i1.9 1 may be e.-.pre-sed y r th f ll:.'ing ti:,-

sided FOur ie r -pe c rum:

F1 l ) = nI [ ..p il- in) I1.

.here Siiif i( the val' of r..e ion t: iej Erretchrne.Jdr ;pcc-

rru-J at frequ.:-nc.' *-r IF, Tp is the l.eneth tf si. i ul ted

record, mnd v'rfr.i i th rrandon. ph S.;e 7 rl-erated in pr -

:edu re (I i ::.,

T'hrl pr.r : dure ar,,' h. :how-'n to : uiji l ri to filt rr-

inc i' -u : r i hit rn' :s D'*. t.ai n be f.:.ird i Ei ndat 1 .

[ l 'rnpo.rt arnj i,..ot l51, Ijrung in.l Battiri I ], rnc.r 3Z :; .

E refetr encin th. h r c.i:.nrtal p.ti al i c:rcirdnir, e tr. a

ftr .i.d l.:* ;.C nn, ( \= ., the c.: irl.- ., l tochi:tic iamplitude

fur the rin.jir' -,ea ; inr l .ti. irn i r., bet. c.'.rr, utcd L-.' Ini iali ing

th.- .:,,: pl.:... .-.'*e? f f i 1:1:nl ; .: f the f r. C.Ju r ie r t r an;, f.t'orfr i ith

th. l ine ir S t c'f a t ca: ilu L i .'n i F.. i1 1 [1 plu.t the

:ec.o:nd c.rd.:r conr,r i r ut i'.-s :. i eri bL. the d.:.ibl-l suriF.atr 1i n ir

Fq. 11. '1. Ii .:. rl r r..t urde rstnJ in th i rr th.: b ttc r, 3 brie f

Jes-.cr i option .f m he ta t c, i:ri.,: r tr'n r'-r, a :.r ti-hr. i rined

and I.11l t1 t pr.:ieri tc next.

2. Fa t f i c i ier T i nir' forn (FFT

Th fl_[ ilnu' ir tr ,i, f.:.i lr r l;.r thrf iicd t.o 1iniul ate

the i n. -, n.:nl in-ar i-ta urfai e p .i o ilc is the C'ubr:o tine

NILOGNi .rirtern b. Pobbinson [111 ]. Since most FTFTKiII c*opipler-

do rot. ijmit :e t: e or ne i tg i i e sut';i cripts for arra.'s th- n ot. -

tic.n u'ied t'i F. ch inirson [1I10] and i.v P. orh riarn [ 11 has bsenr

m.:..Jifii,: t,:. ,c.:, r -e p.:.[rd Co rhe sub.scripts .'aliue. ic ept ab. Ie

to F'.P.T A_.'i crin il ers. The iubr routine e n1LO3'Il app rc..i.imatee theo

Fourier integral t rarnsfrm pair by the f.llo.ing discrete

F. rllr i r Serle appr.:..: inat lion


(','lI = .e.f in l *e .p il ri m- 1 i (n- 1il L.t 2. 1lb )

"Imi = *t r l rl.'e ., i -.:nn,-l 'ri-], L') (.2.1b)

where the t Lie seq.iue ce, rlnl, i: e itr, t L Jii crc te i aiue

of tine whichh are p3rt tt1-.r- e into' ejai l LrterValr ,tf .e

second" jnJ the FcirTer Iprectrjl c e:jrC1nce, i;iT, i: iU ln at

L'' Jis:r te \laluc iof ireq ,iin-' ..hich are pa3rt ti rneJ trn tI :

equal intervals a* : f Hert:. in t rder for the in-c re trni -

formr to be e.:'act. the f :. ii ih. n, relati.:.nrhl tie. L een th- tirle

and frequent: Interrval; must rold:

I ,t * .f = ] 'L I' .21

and Lv must b.e --au l t:. ra si-cd t: an intieg r poher.

The ccirmple.: Fourier :r ec ral .;e ur cr r.: r-.pre er, ted b, Ea.

2.]b i r:r re ser. t ri .n-sidedJ .pc:tr run ilth or, e-half *tf th-

s pectral s:'Hii.ri. : r cointainc, In the *co.rple: arra',l e leiefit

2 r,. L 21' I ,.I

and the .:.iple :cri:njugate pect ral zequen:- ceonairnij in the

ComptIe arra ele.nrt;

(LvY 21 2 -: L X-.-

The mean alu t: of the time sequence i repree;ieried I., the

first arr .. clemi-ent tm=1 The ti .: sequence. riln), i- real

and, ther-: .-re the foll .-',nirg ci:ople:: cnnjuC te rel ti..rirhip

fi-r the C:iplr x i F-p ctrj1 ':iequenJ e : ds:

VI L' l = Y lm 2 51

T i eq 'a 1 ic, ia,' e jel i .m.n: tr it aed t- liub titut i IT =

I' L -:-m inrtoc E.q. I:.lt i to *h b ain

v fL: ** n = c : r, (n I er I-_- it. ,- 1- in-lj i. I i2.r.)

The e r.ponri rn i t l jri menrcr reduc.:-- to the' f ll. ing:

AF C = il i i -r, I r - L -1 l2 ri 1 I I' .

,h ih give' Eq. : .51 a a re ult cf the fo.llrciing idrntit i '.

exp l-2 l I 1 n= ,3. ( :.8

Th' r.:.rmiri co I_: t ant, LY, is a; .:. :la teJ i th the time

'cquer.- e i hn Ch u ;u .-.Iutlrne '1LiL:H, lcf. kobinrnn r [ill pp.

0'-6 1I v i.e ii th ,rh freque- n e c.' squncrL c : tabli:hed in rhe

: ar,ndar i :e nor tatiron pri ierl edj in Chapc[r ;. -on'r :- ntl.,

the *ompi'. :- 'uri .:r Co ffi.C:int ; for the linear ranJd'm se

given b: Eq. 11. 1l ) ere irnir jli:ed b, the following coriple .

expre~' i ri .

.fm i = F mi i -i. i

= -. .i .' .'i( .i) n ex' r rxp i -l- lm I2".91

where +(mli i the random phmae single centrated by the process

descril-,e in -ecti n.r 1 and (m .') i.: the value ,.f the

Brrtschn- idc r spect ral compcnrent a)t frcquenc) ~n I .

The IPIM uhbroutr ine k.VJlDI .a.; ii.,:d t.:. geni rate the randjo.i

number equiience required for the- I':.'rputat irn .,f" the r3radloiri

ph-as ani:le. u if orml dis tri.ut>eJ in the inter !i I' ,2 1.

A descrip:i.:.n -f the sim ulati:rn uf a l ine r I:wiis'sian sci iju in

unifo rrmi, Ji rrs triblteid phase: az nclIe- : iS cri. n y rE.cr. n [ .:U,

an d h, B T[ er v ] .

3. The Pre'l tshneider crectr rm anri
the Phil] ipS. F[u l ll, : ri r, '-pec:t rui

Br t;.tchn -ije r [ i51 h.:i di .c 1i-p: the foll:c"ing s ectrrll

repr es.ntation for a linrasr -: :

-in- efi t.'p 5r, -[ ) I'.lJ
BF. i' T

here and 7 -re pl:rametrric con st ar t Th is I.rC -sided srrpec-

SruiTm i c mple.tel. defined t th t, ..c p5ar jit'ter inr trer

to compir. r i q. 13.11 ith pei ctra from rarj:r.d e3as *r btain.d

frTom mireasured hurricane re.r: irds i' s r?:ces:.- ary to- tran tc'crr,

thb-s t-,:, constant a parn m-t r ir, prmarmete rs Lmt'icih are

me as~irab ie fr'cn the hurri ane records. This mi a be a:c-cnm-

pl ih'ed b. I'1). rteq iTrine that the: Dr: .sc:hnl i er specEtrumu i nd

the measured spectrum have equal variance and (') that the

frequcn.:.' of the t.eli-defined peal of the bretsc-hreir ier spe.:-

trur, Tre.: in t.est least- quare Senr ,e iith the. frequi;ncv

of the often ill-d fined "peak" : f the mea':ured -pectrar. \5

measured spc'tra frequently di'ff.-r in the number and the mng-

nitudes of "peaks" present dep-ndin., inter ilia, on the

degree of mco..thin cg wi ed, a. resorr t a be' t 1r aSt quarre

jgreem-enc h.s tien the fr:quivcnciev of che me i;ured and

br.e r .:brie ek r "pc.a[ is r:.iii red.

Thc V 'T r ii :C t i n"c-r i urd pJ t.- rum mavi tre .zonlpuiF c

either from rthe: me n 'quare ,f the re corjde rerali tii:n n,

E ., (t I dt 1 .2
f I '

.here TI i: the length of t' e ime '-urei rec.: rj or, equiva-

lentli from the inrittcr.] m:e" ths one- c- cd i p: c .-:z ral Jd rniit.'

fir ,- t i or,

E f. ii. i d. ~. .J
0 r, r

Th- frl.e inr:,. jt I h. :i- te t pe of the rer'c chnei.r spectrum

occurs ni3 e d .jrt:rmrined equat J iti n ri :: ro the dcr '.'tr itc

ii h r rpi: :t t o thi: ianrul' -r frejquenc,- of thi spec.trsi den' it .

furi. ton i jnr 3 luj 'tin r th re ul r in: ei el.br ai' equi. t oL n 'it

triAe frequer.': it hich thi: r3 of rhi : ctr mn occur'; 1. ,

-- t = 1 ; = 1i-.4J

Equj ai'tons .31 nd I .Jl iAclJ t.L. qu e 1- cioni hhic-. illo the

tio c:. o:S r s3 n p F r a me r- Lien b, i' P'retshnceidC r to tbe trins-

fir.rTed into two meri urible c. o st t i n p3jr-iir.terw i. ,

(l ,Tl I .,. l (3.5

Int.:-rratirn the- brCrtE chr,1e .i r spectrum is m[ost eja iiv dc-ne tb'

fir' t rmakine tihe fol .l,: rn ch nee. of vari ihle:

. :T f-]. :. b

flaking the c charngi2 o*f i' ri able" and ntei r aring. the tfolow-

irng express iun for the ar i rce of crie ipecrrim i; cbtanned:

S= r d. c.

Differentiat ing Er 13.1 1 ith respect to the arngular fre-

qucr..:', a, cquatinQ th.: r:. ult to zero, arid .v lu 3tinc tihe

rc ~ulting e. prts iorl 3t tht' rt quer-,n ot the peak in the

spectrti yields the follo'-ir e:

i. S( *' i

Equations I ;.- and 2. t) finally. jetc rmin the f':llo~',i tiJu

coni tant pa ri' tet: r

= f .z) z3

,? = (,. ( 3.9b l

Substituting the& e t o paaraic tc rs rF it:o F-. (3.11 result; in

the foll.jwin f form far the Prer-r:hri i.ler spectrrum:

.t 1-
-.(.*-I = I-' --.pt-l. t --2 ) (13.1u
t E. fu '

The test lea:.t-- quari, t ilE mJi e ft or the "p-, i: frcq.:- .. ,

S Fma'. : .:-j.itp Jt-d I. the ap location of the linear Ta.'ior

differertial correcr l i'r t .:hr. iQ'.i' Ilrar.qu, r.Jt ([ ] or

icC al la [S71 In thi tE:chriquc, a .re3n r i q. 3re error

b.: t-.-n the if.',j r:d jndi predict d -peCtrT at tre ;1. ji.crete

fre iu nc i- Samr-pled is f:irmeI as ft1 11-.2 :

mI .

*: m=l r,1 t (3.111

rhere fl. ccorrespcon-r, ti: the :.,t o ff frequ. ri: bove\ which the

iriT:'i-uri iT spectral eS: timat are nr,;gl ibIe.

Thie breti chn.:dc r spe-t ral d i tr il ti- u ln funr:ti' n i1

ex\p nd-e.J in 3 Ta.li--,r .eri.es iaoh t ti e be.t ei tim it- of the

p -a. fr .cii.:r, c. ir t, rr,- ot r'hr linear di T ferent al cc.rrcc:ri 3n

to the pe i fre-j-,ern -', i.e.,

r.ic ( mm i -
i I I I ii - (-.:.- l i i i 3. :
C inil= 1 nri, BB -

PFet ining .:nlv thIh linear corrections the ,eara square error

i di ff.:-r:nti ated with respect to: the di fferenti al correction

to the peF a frequenc..', -.. and the result equaij ted tJ :eru:,


I7. f l ) I' l
d F__E E
= i a

= 0 1 .1 2

which Iy e h ? f.: ll c-tin .i l:el r eq, 'at ':.i n for oa :

B [l1(uT)-Mni> ----
S pp
r" l I rl E p
< 1 p.

mi = 1 0

Thi- equation Ima, b e iTeraTe I rf:r the 11 coTre ctin O:.

the i j est aite for -f : follow .
1 S (mi

m=l nrlr BE I'
.( I'jl 1 1 = --1 ,.151

lM 1 1I!li

The i t ratin'r i terniriated hen iuccesil c correct iLon are

ic~eptabl.,1 small Table 3.1 dcmin tr tej the re'iii lt' :

ft tine fur iTe -a rTed pCctri fTr-'i llurricanric arl.a ith a

bi t le ast squ res r ti animate f r the rpe k: freqier c,'. The fIit

and number of tteratiCrn_ [r:. :orive rgencrre are the pocret i'or

the very n,arro- :p ct.rum rerrc:ertred by .ec,:rd Nl o. C'0,. 6 2

Thei number ':f srnec:trl et icmte, L. uied frum the measured

'p~cira E as6 equal t: -.0 and c.-rrcsp-..nded t:. a cuE-ct C fre r

qu'enc. equal .. 2.. '4 rad sec or 0.3" Hert:. fnr Fecocrd iJo'.

0S51ES/1, 0bS S,,' A and and to ': radi Sce r 0'. I H: fo:r

Record t.o. 06 5S'' 1.

2. -~


3 ,
l- :

l/ .

m 1^
ex -9





rl ~

i r`
iC i
i C

~. I- -~ ;1


? ~

E j




1 n~


A n.-tc .i" ciutin i in i:rJ er Li ith rc ard t[ the -ppli-

cati.n ro ic i in C r the Prct'c hnreidcr spectrum e"prcci s a3

2 fu.nct i n cf th t -c .o parameter; .' f tart. i 'c E-, anJ r.e l.

frcquenc,', Prc t :chn. ei r i2", ) ct.rrel's t.:J his S p ctru

with the twoi- p'lr 'met.:r; o ind -eed arLd f.- cii l c r th nrd

there is ro s 3at rtfactCc.r.' ,iar ant: : :C f :r.tentini I- 1th it re -

juenc, in it-jt ic'- in.dj.:ated bt br- i irnc i re: in the

Si rit lat .n 'f1 r lm r3 in n rlrin-: r 'c -a r all 3t i:r.5- ti e, ttie

total ener ',' contend and [e. k 1 fre.,ucnci : of the spect. rum ir.

used. In the b.sern e J tc ile-d s't bil i t an l .- i th.-

equiil: riuti r Fi ctruinm -derel o ed r i. Phill i [I.' 9l trcrm a imcr -

5 i:- n ''l -qi l.i.'- n a e nmpl r e..J to l ii t ht h- e *.:rerv -rntert in

the hich frequ.ncv region. Alth.-.uch the til.Jr it, c'f the

eolui l br iLurr. pe.t rum hai b-ecr ques tior: [ i J ',f.',ri 2 I, i,',i9 ,

Its 1aliJit',' f.r aipplic l j n r': ernLinr-.rin r Jic : i n appi: ars

to b appropri ate.

Fhill ips ic [ lii Lh. -. "pp. i 9-11' re a one.

by similarity c r.dLr tior n that th .: pcc:t ral c:rrpor.innt': in

frequency 'Fac f.:. r frequc e rc greater rth i tt i pheai t'fre-

quLenc- siho-uld de.-Ca j acc .:rdin to

5( ) = c .' : : ..'. (3.1 )

PF'h llip annJ c-thers ['4 ,fl'u, 1I ] have fo-und thi constant i tE o

have 3 t3ilue -f C' .011" -1 10', for c ci pretscd Ir. ri dii ai per

aecn.r J. If the frLquencv i. ex\pretsed in Hert:, the require -

ment that the cnc-rg c cc ntcnt .-f quJi vir I:nt di ff;rc-nt i l a spec-

tral dJns ities remain -cnr tant implies that

Srn 7 J = Jl. T l.d(i Fl (3. 17

,hiich :ii. T 1 T

r'i 2' i, TF ?'- -* I 1 T I

_-i" .*I1 T ti [m *il 1T ]

- r.
e' = II. 1 l 3 19

Thi z co.r.z rt for r the re. i liL.riui r.-:ctrVIri i C i iF li t:

di ffe r f i :. th [ I iver b.' Phillirp [li'' : p. li ,rhich,

s earr -to t I -n ,: i t .: J Th- ncnlin.: ri at r.n r'r'-.c: .dlr c

incl d.d i tEA f t' r high tfr.' u nr ,- ajcturatlon usiing the e.ui -

lit.rium i p.:c rum o Fh illips. Ii'rne *: the sp' --tra frcir the

rnoi 11rir : iti li jn.r 1 .F [ht e t E ie r IT j .. irr i cane -p c C 1i

?xc: d d i the -: qj I ri .l !im r : c r um.r.

I. Hurrican, karia Data i'ipt. enb.r r 10, 1 61 1

l.'a v F.:rc rri.~1 iI c I: ii ili r.; .:l-c, ,je r.i- i, ; ri;,t;r, -tr.-.-

s a f.)ur fae i- C jl a i on I Jan -. 3V' pr c'uri fr ce' -in a ve r tica

p ine hih su Jppoirtei an initr i..menr ed dri lling platform it a

l'catL i r in the rFilf c f 'lk :ic: i,'th a, t. r -e, i t -,p of 3pprr. r i-

mate '.. 11w') fiee The djt- ri ec rd-d during 'F II hae bcc n

m.idJ .' a il bl.-i t thec public thicugh thei uatir r l iice.,r.n raphic

Data Center. Thra-shcr -i d 'gJUjrd [122l provi.de d-etilS nf

th-e torn T r:c.-rdr d jrd anthe r inrformat ir. i ril at iv to the

d at. Hurricane Cill a was recorded during the p. rioa

September t.- 1 1 9' 1. THi. continuous; records '.hich are avail-

abl, frcn this period of rPII contain o.som- of the highest and

ro.s;t force ful avcs r :corde d during .eiL One wav% in

pirticu1 :r, hij an :i\eraje tric.uh to crtst I.ave heicht cf

3lmoi s t f fortr cr Trhe e ij r; 'c*:.rt irn f.oir re :c.r s fromll

Hurricane Ciarla with :.:.ntinucajs s a surface elevatiorn ajnd

pres rc force recr dr.ii. c4 c e c c ,'erin, apprT:'' iatI ici le. gth

of tin e frcim eleven rto fo'urte~n minutes. The datt.a .re d iI-

ti zed at rne'rer intervals of tie rariFirin arpr'o imati1 from

0.12 r 0.24 s,-conds.. Cal ibr at i:n. informiat i:n :ird equati n:r,

for transforming g th data into r, en ineering unitr, are given

bhv laril. [lr.). Table contains thV charac.:t erl t ics :rf tih

fcuir hurricar je records jsed in th.- c .nr aris.in with simulat-ed

re al i tc ion-. The data .-ere re igiti zd fcr thi :c:C n -.i 3ri -:on

anal. is b' line r interpol nation afrer jappli'ir th.: c::iitbra-

ricn equations civren b'.' lvni n. [l.] in orderr ro obt l n record

whichh wouldd be an.-l,:ed by the Fast Fo:urT r Transform IFFI

algorithm. N"lo time shi ft a' s used to effect :co:ir.:i..l: nce

bherten rhe ea iiurface re al :atipon recCi-rded b., the wa .'

staff and the pressure forces rez:rded on the irnstruiented

pile because the separation di-tanr:e between the itie st 'ff

and center-line of the pile a-s o.nl]. arpprcximatc-l. 55 nch:-.

Usina linear :3ave thec. r', a Itavc iiath a pe:r- I o f ten recconds

traveling in i n o,:: an *:rf U-Ifc rin depth .of orne hundred fcct

woulJ trr\el rhe straight-lInc distance bE tcen tth center of

the waie staff and thce center *:.f the pile in approxriatels

0.10 seconds.. save- in,:idcent froM'n directions other than al,.n

Ta.bl :. .

Chirrcteri_- tic-, of Hurric an CariT ecor-Ji

E.e-CrT.J ]J.:. llte rime ,in I I I:t

r.S I.'1 I.ept. I 'r.1 ll, 1 !. J,9r. .t :1

I'r,~ c,. pt. 10, r. 1 1'100 15.01 4 1. 20

0) ,. 5.. 2 .p t. 1ll, 19 ,1 .i..i .'.,. F? -09, 20

,'^,? '.'1 -.ept. 1", li'" i ,:1:.1,', 11.ii,) 1,9.i I". lr,

thi ; r ai lht l e .i .:.uld iqluire I tr1r ?r.Tl cirte In rl Tre.

f're, the [tie hl lf t l. rc le tc d. The ,fflct ,S, ci r-

in 1li I prc. i e ,u f':.r:e- .: ;lec tir,; th.: p-ati l : r ti r

t.:t E -n J le 't tf ari the CF t l -t li tf ir t i i = i r ,nt

ve rt ic I l p l rI ii d. ..r. rtite.I in rptr'Ji the i.pp ii: -

tic ri .*. f I inr i ti '- lio l r c r 1' th r:. in t h-:' l r i .r- :r.

eq ai t i:.ri .

Th- pr-E: i re fiir tor.- to ac in 1i\iiu 1 .J' r, a :,.-.-t r :'le -

v3)t.:.. ir ci r. in ort cori: 1 c irrp.r-.r,rrr- r hic.:h .are p r....t-

niF l' ,' li or.: Ij ith the pl [ f.l l 1 iC f. El rn 1 c] f.:.r

dirvenr i i r. I, ri .:in th'i l di c E 1 1t ; I. Trei pri.3 uir f.rc- :

r c r: : r-Jc J t, I r i.t irn.:.n. et r it the S .-. f :'.t lei i t .:.r :r.:

ir'sed ir. the c.'.mr :3 ir--' n n i; taniC t h '- : t ti r :. r -

t inu us. i. ib :c ro: d l uri ng t h, p .-S ;, i '.; r' -g .i. -e rin. l,-C 'a;t

rnr 'r n .l h to t th fi ree ,j i :f t :. i, r ::r. d the lelc c r

f" the in l l r hF,.,. r. i-c i h '. E in f ui n.- i.: ,

Sixth Jd t h. Thie j -, r a r c: rd.: d r h- "r ..1" T.t e ,:. r c r. -:,r,

w -ul .j h : bc. n Fr rl t.o. e t.:r th,, I P.r .J, ri*L ': tc rr- a r

rthrD le -si it r hut th- 1.:. r- '.: ution .1 the i c .' r.:-C c irrp ,:, r.: rt

rei lte d 1in irme -lure.i d lalu i c n.; r.. :.1 I-.,' in crrii nt=

-'ppr,:.i .% na tec .1 iF' i f t nrJ -i' f e t t .':, h- Ie-' icuratt.

tl -ir the dit:i 3at th .,. foo.:.t le .ati .:.n. Th: .rnp 1 :.r.

b_-t-er. i, ei uri d a nd i' l'1u atc .l press -sJr rt:c, i n J L tre

re ultanrt f.r :,S iu f ii? r. Lc- tr r, i .-r -riri (1 11. T-he

w c ierc assumE j t.: tr: c':ii ir. ir i Ld t t pr r.ia ir in

direction =i6 2 r lit t tc t-he i pla t or .:.vi I ::...rd.tin t ; r.ei.

The resulint fl.rcc it ti's= net r t-turned b:' the foll- ing

F I n' = Jri ':Ii I n i ~ i 1n .1 j

.-here F. in and 1. Ini ire r h ..:.rth.- ronal pre T sur fr rce ccrm-

p n,:nri t re at e t, th ci plj tfr-.rm i:. J iiti :-J at time n-t

ar, the ; j [uric i n .: isr d: t rnin.: t,. the foil lo inr

for nI Iln I .
qgr l_. ir l = n, 14. 1 '1
f :.r I ,in l-- ." "

i.he re

S In'
-in) C= I l i 1 R)

The effect the jppr, c liii. t ni:ul ir nr l 1ig ient i-.th

r.Fes : to the r pl at'.ri r c.ordinate a.es of the d'.'r aj ..r: Cter

at th.: 3.33 focr :levato i wo R n: c iniiule-J in the determl-

n ,t ion of the djire,:tion f rth r-.uitant f:.r.e

:. C r:ipa rison ,f 5.: : urf ia e P.: li: tia is.

The r.ea.ured s pectra ifrom four Hurricane Caria records

Le re uiJ d ci: ir:iuil- t e re j i:.,tl :cr o:f rand:di rr h rrTTcanc -

gen.r:rr.:d u ae *: c-:rrect to s.ec: nd perturb itiln or --r r for teo

Jdifferent input Fpectra. The s mi.-:-.tihed "mesurjed Jpectra of

these fto tr r :cnr. ar.e h e,..n in Fics. .1, 1 .., ... nd 1 3. .

Al.n .h s :.n on these fEtures are the ret 4cihnc id.:r spectra

with equal arian.ce and best eastr squares fit tr. the peak

frequenc., s JIeternrineJ b., the proc:..Jure di as:u- =e in cc.:tion o .

I I I -

.- 0

-, ,-

o -,'

N 0 ,' 1 '"



I 7
I 1( 1)
i L 0 -

0 0 0



o 0



0 0 0
~, po r

,II .^


0 -

--i c

( g /

-o 1- i
r. "r

0 "
o 2:

- O

S 0

5 6'. D


:' :





C I-





L :

L: 1

Ir i

: I'
. Ir ' I
- -

:I '' r

- ;j=







c Lr,
N i'

I n V -..
k. I ,

N "


I /
i- o Er
**- Lu,

0 0 0 0 0

3as po. t''

zl I (


The me i s ire d-.'-e.t rT.i .., :e ~mr-.: the b,' t.bl.ck av' r3 irng :ver

an inr erir al of nine sEpc.tral ailuies. This t.p, ,if ioothinr

filter a'3s d.termiried b' Errorian ['1) to be approxiri atel.,

equal t,-- th; '.laus' ian simoothinr which hi used t. irnv'e ti ite

the chi- quarel cr':nfiJ.nc:e interval hcf rh-.'e 3mre rec:.? rds

'.Ef. il :o Fend t and Pi.crol j i Jine s.pectral co~nmp..nent

,.re blick ave raced vi-ce the eight coiponr ent- ui:.nd b.',' Eor-jTan

in --*rd-r that the aver5gine i.oL-ld be sy'mnetrici al. Th- bl:c

averaging as .-bhtrin-ned fro.:m the f.:.ll-.ing:

5(ml = 5'I(m-n) '9 5. 1
nnr n=- I rlr

here ih1. i h s iTIT, c thcd o...i-a iir d : rp : t ral estimr.ate at fr -

quenc. mif and i' is the raj.' pepectral estimate Th. di .:rr et

smootched 3pectral e-ti ,mits have been conncctej bi c :'n tlin.-

:ou: curve t) a 1id in r-9ading tre figures.

Alth.:T.Algh the cur've which i c cnn r i ct the im'rfitl'ed iea i.re.d

spectral t -timates appears t.:- tE rea na bl,' .1 mo:.,.:.th, the

erratic nature cf the raw., spectral e rtimate- cau; .d b. liar);

differ rnces inr the nic gnitudc of adjacent c timit-s. and b.,

:ut ller estimates i 'till idenr ih.: rai. spectral ..st -

mates o*rscillae rapidly and these ojsci lat i.'-.n are esrpc:i 'aii

prevalent in the ra,- 'pectra near the best le ct-squiars peal

estimates h here .several large outli,.r estirmjtes were c.-:i.puted

fror, the itT coefficients.

The variance ofc the four measured recordi varied from

22.3-25. ft" and the best least-squarci estimated of the peak

freque-nc%\' -a ri- J fr.--,m 0I I I 8 I 5-11 1 r3' J. ..ec. The roi,:t-r: arn-

Lquare errcTO:- s tre comrputed frc'm the diiscrete s-moothJ

measure i sp ctrsal tIm iter anrd the di.-i rete Erets .hne der

:p.:.tral I al i .e .- the followal g:

r 1 r-o-7---S---S

i= l r, r, L

wh ere :I i the ti:otal number c:,f zpectral .estima:ite u-sd.

Borgrman [21] dlete rTiired that the miicrit, del osf the raw rp .:-

tral e ;timat i: :arr, r~cjm n l : ib cl after three: hundred and firVe

31 ,.i talu-j s and. a value of r H -1r ti.a s. .: cted for this .:om-

pari ':.n anal',;i;. The r1' errors. ari-d I- et'ct n .:- 1 2

ft 'ir3,1 t.:) I ith the l TrcIest error ocL.currinr for the rel'-

tiv.l, nirrc-. ppectral ret re. entationr f'or Record l o. ii6o j 2.

All of the mejis.'red s-pectri d emion ;rated iiev'ral "pea l." and,

e\cept if r i.c:ord f..:. kOr' 2, there arr to, "peal:S" "f n-earl.

equal aIsmplitrudc- vcr,' near the best le st- t -qu re e 'timmate Cof

ra3 % run pe, l .

Frmii each of the measured jnd br.:t-chneidi r ;pect ra

repr r,--- c n irnle orie menasi d r, c-r rcor], i r ali :at ior, ere ire -1 u-

lated. -Three linear anrt three nonlinear re.all:]tions fTir both

the ;mn-.:thed mFeasured and the Bret; chnrider ;pectria there

simllated iJ nir the IBlM scientific Subroutine Frocram F.LDJPU

to gen.e rate the rand-om numb' -.r rc:quir,-d tc. c.,mput,:- the random

phiae aneles. The same thiee "seed" nm.ihe i required to

initialize the k~JDIJ randor, number geen erator -r re: iused for

each of the 'four recorJd. The-se three seeds -e re ch-osen

since theiv g Eri. rally. ,- ercrin treated the follu. inr three differ-

ent liriear mT. as:lre:" ..-f s h lneer : l n1 e tiie, I p.o i i ii .

an d I .1 l a r : .'er' r.: The r,- l : [ .-i n fruii the line r silmu-

lation whichh **: re gene ratei d iu irn1 this meth.-,j .'uIuld have

licemont r' t r J :r,. si Cn I: me r j ure had t he rsandrirr ni im-rr r rie r-

atc.r re al : J pu.re l,' a i .: i n p.rr.c' : c z.

The cm umiul at i ti pr b il. -I. i i .' di tribe iui r. r:i ei :h of

the f'-'ur r'c':rd5 ii hcr, ir, al i ... is, 1 3 tu .1 re al z] i.:.,.ns in

3ddi tio.in t.:, the i'ersilre.d r:3 a1 : ati on arc sIltr Ln in FILc 3.5,

3.0, ti ."', r,,r ." P a 1l1:ati-- n fro,,il the un iu...i...th-:J n,-a: -rc-d

rspectrum ic. u]i j h ',e .mion3- rat d e.. ctl. the iiae di : tr I-ili it i

-a the iimei ,re-d J'. i repre :enreJ i.., he [J rp- n circle i f the

identical nreasure. pr has.e an l: es J hid iE:n u j as a result :-f

the c 3act inv:eri.: rclat .:.sl ip in the FFT aig :r rhi The

variT tiJn.s bet' C n t:Ei .Ji strib h. livrt fr ir thic i. i.ircd spe-ct ril

demnonn tr a e te impc rt ance i:f thee phase nrg irle ir. d*eter-

Sn ing t he i t r i ut in .-f an. r' 1 i: a r. i r. The stra igh

line reprres irnts 3 rn.:rr-3 i. tribuir. 1.:. h l.i the i-lenricu

mrie ian and st indard Js ii ni n 3s the re l i : T :. .

11 r j li: r i-rns. .ieiir,;n ra3te c!: I e 3pre renr t rtL a

3aui. ian pruoce. tbet'e n Er n :.rmali -: I 'tandjard Ide ia131ti :nr .

The ordinate ialuje ha'te t~er, nurn. il, 1 L.J b.' the iC iai dard J'- -

ation of the simulated r.:. mea-uredT record; i.e.,

ntli = 1 =1,2,...,31 ,5.3j
N -



0 -

2 1I

Vl C

* L.

a 0

oc 0
* a a

q ii N

- 0

01 L -

(P -D

) ,-;

a oi
-? s-




S0l -







5" 0


ha- N

Hp C

- 0 ~ 'J ~ ~

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