Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 416212 6 pageshttpdxdoiorg1011552013416212

Research ArticleReturn Period of a Sea Storm with at Least TwoWaves Higher than a Fixed Threshold

Felice Arena Giuseppe Barbaro and Alessandra Romolo

Natural Ocean Engineering Laboratory Department DICEAM ldquoMediterraneardquo University 89122 Reggio Calabria Italy

Correspondence should be addressed to Felice Arena arenaunircit

Received 12 August 2012 Revised 8 January 2013 Accepted 28 January 2013

Academic Editor P Liatsis

Copyright copy 2013 Felice Arena et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Practical applications in ocean engineering require the long-term analysis for prediction of extreme waves that identify designconditions If extreme individual waves are investigated we need to combine long-term statistical analysis of ocean waves withshort-term statistics The former considers the distribution of standard deviation of free surface displacement in the consideredlocation in a long-time span of order of 10 years or more The latter analyzes the distribution of individual wave heights in a seastate which is a Gaussian process in time domain Recent advanced approaches enable the combination of the two analyses In thepaper the analytical solution is obtained for the return period of a sea storm with at least two individual waves higher than a fixedlevel This solution is based on the application of the Equivalent Triangular Storm model for the representation of actual stormsOne of the corollaries of the solution gives the exact expression for the probability that at least two waves higher than fixed level areproduced during the lifetime of a structure The previous solution of return period and the relative probability of exceedance maybe effectively applied for the risk analysis of ocean structures

1 Introduction

The concept of Equivalent Triangular Storm (ETS) wasintroduced by Boccotti in the eighties [1 2] A completeanalysis was then given in Boccotti [3] where the ETS modelwas applied for solving in a closed form of some long-termproblems which are very useful for the design of coastal andoffshore structures

In other words the ETS model represents each actualstorm as a triangle where the triangle height is equal to themaximum significant wave height during the actual stormand the triangle base is such that themaximumexpectedwaveheight in the actual storm is equal to the maximum expectedwave height in the triangular stormThen the triangle heightis achieved from the significant wave height time series of thestorm while the triangle base is calculated by means of aniterative procedure

If the equivalent sea is considered which is given bysubstituting a sequence of triangles to the sequence of actualstorms some analytical solutions were found for the returnperiod of severe storms Boccotti [3] found the analytical

solution for the return period R(H) of a sea storm in whichthe highest wave has height exceeding a fixed threshold H

The solution for the return period R(H) was then givenwith a different formal derivation by Arena and Pavone[4] They considering the ETS approach combined the long-term and the second-order crest height distribution [5 6] toachieve the return period R(C) of a sea storm in which thehighest nonlinear crest height exceeds a fixed threshold C

Recently Arena and Pavone [7] proposed a generalizedapproach for the long-term modelling of extreme waves bygiving the general expression of the return period 119877

119873(H) of

a sea storm in which exactly N waves higher than H occurFollowing this approach the solution is also given for thereturn period119877

ge119873(H) of a sea storm inwhich at leastN waves

larger than a fixed threshold H occur with 119873 = 1 2 3 andso forth

Different approaches to combine short- and long-termstatistics were proposed by [8] who gave a generalization ofBorgmanrsquos results [9] and by Tromans and Vanderschuren[10] and Forristall [11] who introduced the analysis ofextreme waves during storms and considered the loads andresponse on structures for engineering applications

2 Mathematical Problems in Engineering

In this paper by following a different logic with respectto that one adopted by Arena and Pavone [7] a new solutionis found for the return period 1198771015840

ge2(119867) of a sea storm in which

at least two waves higher than a fixed thresholdH occur Thenew solution for 1198771015840

ge2(119867) is achieved from direct combination

of short-term and long-term wave statistics while the Arenaand Pavone solution gave the119877

ge2(119867) solution as a function of

both119877ge1(H) and119877

1(H) bymeans of the following expression

1

119877ge2 (119867)

=1

119877ge1 (119867)

minus1

1198771 (119867)

(1)

where119877ge1(H) and119877

1(H) are the return periods of a sea storm

inwhich respectively (i) at least a wave higher thanH occurs(ii) a single wave higher than H occurs

The final comparison between the two solutions which isproposed starting fromNOAA-NODC (USA) buoys data andfrom Italian buoys network (RON) from ISPRA Institutefor Environmental Protection and Research (Italy) showsa full agreement This result confirms that the EquivalentTriangular Storm (as well as the Equivalent Power Storm byFedele and Arena [12]) model is very powerful to determinethe long-term statistics of extreme waves during storms (seealso Arena et al [13])

The results are of interest for engineering applicationsbecause they enable to perform a complete analysis ofextreme waves that we may expect in the lifetime of a struc-ture which may be for example either an offshore structure[14ndash19] or on an upright breakwater (see eg Boccotti et al[14 15] and Romolo and Arena [20 21]) This may be donein terms of return value of the extreme individual crest-to-trough wave height as well as by achieving the second wave(the third and so on) in terms of height occurring in a fixedtime span on the considered structure

2 Return Period 1198771015840ge2(119867) of a Sea Storm in

Which at Least Two Waves Higher thana Fixed Threshold 119867 Occur

The return period 1198771015840ge2(119867) of a sea storm in which at least two

waves higher than a fixed threshold119867 occur may be definedin alternative as the return period of a sea storm in whichthe second wave height (if waves are ordered by decreasingcrest-to-trough wave heights) is larger than119867

If a large time span 120591 is considered we may define

1198771015840

ge2(119867) =

120591

119873119868119868 (119867 120591)

(2)

with 119873119868119868(119867 120591) the number of waves during 120591 which are (i)

higher thanH (ii) the secondwaves (if waves occurring in thestorm are ordered by decreasing height) in their own stormIn the following the solution is given for 119873

119868119868(119867 120591) Let us

consider the Triangular Sea which is given by the sequenceof equivalent triangular storms

The number of waves with height between 119909 and 119909 + d119909occurring in sea states with significant wave height 119867

119904in

(ℎ ℎ+dℎ) in the triangular storms with height between 119886 and119886 + d119886 and base between 119887 and 119887 + d119887 during 120591 is

[119901119860 (119886) d119886119873 (120591) 119901119861 (119887 | 119886) d119887] Δ119905 (ℎ dℎ 119886 119887)

sdot119901 (119909119867

119904= ℎ) d119909

119879 (ℎ)

(3)

where119873(120591) is the number of triangles during 120591 119875119860(119886) is the

probability density function of the triangle heights 119875119861(119887 | 119886)

is the probability density function of the triangle base withgiven height a and Δ119905(ℎ dℎ 119886 119887) is the time in which thesignificant wave height119867

119904is in the range (h h + dh) in those

triangles with height a and base b Finally 119901(119909119867119904= ℎ) is the

probability density function of the wave height in a sea statewith119867

119904= ℎ and 119879(ℎ) is Ricersquos mean period [22 23]

It follows that the number 119873119868119868(119909 ℎ 119886 119887) of waves with

height between 119909 and 119909 + d119909 occurring in sea states withsignificant wave height 119867

119904in (ℎ ℎ + dℎ) in the triangular

storms with height between 119886 and 119886 + d119886 and base between 119887and 119887 + d119887 during 120591 which are the second in order of heightin their own storm is given by

119873119868119868 (119909 ℎ 119886 119887)

= [119901119860 (119886) d119886119873 (120591) 119901119861 (119887 | 119886) d119887] Δ119905 (ℎ dℎ 119886 119887)

sdot119901 (119909119867

119904= ℎ) d119909

119879 (ℎ)

1198752 (119909 ℎ 119886 119887)

(4)

where in a triangular storm with height 119886 and base 119887 giventhat a wave with height between 119909 and 119909 + 119889119909 occurs in asea state with significant wave height between ℎ and ℎ + 119889ℎ1198752(119909 ℎ 119886 119887) is the probability that

(i) just a wave higher than x will occur

(ii) all the other waves of the sea storm will have heightsmaller than 119909

More in general if an actual storm is considered 1198752is the

probability that a wave with height between x and 119909 + d119909occurring in a sea state with significant wave height betweenℎ and ℎ + dℎ is the second wave in height in its own storm Itis then given by

1198752=

1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+

119873

sum

119894=1

119875 (119909119867119904= ℎ119894)

1 minus 119875 (119909119867119904= ℎ119894)119873119894]

times

119873

prod

119894=1

[1 minus 119875 (119909119867119904= ℎ119894)]119873119894

(5)

Mathematical Problems in Engineering 3

01

1

10

100

0 5 10 15

Retu

rn p

erio

d (y

ears

)

Wave height (m)

1198771

119877ge1

119877998400ge2

Figure 1 Return periods 1198771015840ge2(119867) 119877

ge1(119867) and 119877

1(119867) calculated

from data of RON buoy of Crotone (Italy) with the parametersdefined in the Appendix

whereN is the number of the sea states in the storm Equation(5) in integral form may be rewritten as [9 24]

1198752=

1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+ int

119863

0

119875 [119909119867119904= ℎ (119905)]

1 minus 119875 [119909119867119904= ℎ (119905)]

1

119879 [ℎ (119905)]

d119905]

sdot exp int119863

0

ln [1 minus 119875 (119909119867119904= ℎ (119905))]

1

119879 [ℎ (119905)]

d119905

(6)

where119863 is the storm durationIf the triangular sea is considered for a sea storm with

height a and base b the probability 1198752(119909 ℎ 119886 119887) is given by

1198752 (119909 ℎ 119886 119887)

=1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+119887 (119886)

119886

times int

119886

0

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

1

119879 (ℎ10158401015840)

dℎ10158401015840]

exp [119887 (119886)

119886int

119886

0

1n [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840]

(7)

Table 1 Parameters of the Weibull significant wave height distribu-tion (A4) and of the base height regression (A5) for some locations(from [25])

BuoyLocation

1198621

(hour)1198622

(mminus1) 119906119908

(m)ℎ119897

(m)NDBC 46004510∘Nndash1360∘W 11025 643 sdot10

minus2 1484 2489 065

NDBC 44008405∘Nndash694∘W 756 557 sdot10

minus2 1100 1146 055

RON Crotone390∘Nndash172∘E 9184 348 sdot10

minus2 0956 0590 008

The number of waves during 120591 which are higher than Hand the second in their own storm is

119873119868119868 (119867 120591) = int

infin

119867

int

infin

0

int

infin

ℎ

int

infin

0

119873119868119868 (119909 ℎ 119886 119887) d119887 d119886 dℎ d119909

(8)

In conclusion combining (2) (4) (7) and (8) the expres-sion is given for the return period 1198771015840

ge2(119867) of a sea storm in

which at least two waves higher than a fixed threshold 119867

occur

1198771015840

ge2(119867)

= int

infin

119867

int

infin

0

1

119879 (ℎ)

119901 (119909119867119904= ℎ)

1

1 minus 119875 (119909119867119904= ℎ)

sdot int

infin

ℎ

minusd119901 (119867

119904= 119886)

d119886

times exp[119887 (119886)119886

int

119886

0

ln [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840 ]

sdot [119887 (119886)

119886int

119886

0

1

119879 (ℎ10158401015840)

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

dℎ10158401015840+

minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

] d119886 dℎ d119909minus1

(9)

where according to conclusions of Arena and Pavone [4] thebases of the triangular stormsmay be considered constant (119887)with respect to a because a and b are stochastically inde-pendent to each other alternatively the Dirac distributionmay be considered [3] because it is slightly conservative inthis case the function 119887(119886)may be represented by means of aregression as given in the Appendix

Figure 1 shows the return period 1198771015840ge2(119867) which has been

calculated by considering functions defined in the AppendixValues of parameters used for calculation are defined inTable 1

4 Mathematical Problems in Engineering

3 An Alternative Approach forCalculation of 119877

ge2(119867)

A different approach for calculation of the return period119877ge2(119867) was given by Arena and Pavone [7] by means of (1)

In that equation 119877ge1(119867) represents the return period of a

sea storm in which the maximum wave height exceeds thethreshold H and 119877

1(119867) represents the return period of a

sea storm during which one wave only with crest-to-troughheight larger than a fixed threshold H occurs

For the calculation of119877ge1(H) the solution given byArena

and Pavone [4] has been applied

119877ge1 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [1 minus exp(119887119886int

119886

0

1

119879 (ℎ1015840)

timesln [1 minus 119875 (119867119867119904= ℎ1015840)] dℎ1015840)]119889119886

minus1

(10)

The return period 1198771(119867) is calculated by considering the

following expression [7]

1198771 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [int

119886

0

119875 (119867119867119904= ℎ)

119879 (ℎ) [1 minus 119875 (119867119867119904 = ℎ)]

dℎ]

sdot exp[119887119886int

119886

0

ln [1 minus 119875 (119867119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840] d119886minus1

(11)

4 The Encounter Probability

In general the occurrences of severe storms are assumed toform a homogeneous Poisson process Then the probabilitythat in the given time interval L (which in the design of oceanstructures may be considered equal to the lifetime of thestructure) at least a sea storm with given properties occursmay be written as

119875 (119871 119877) = 1 minus exp(minus119871119877) (12)

where R is the return period of the considered stormThen if we assume that

(a) the occurrences of the sea storms in which the highestwave is larger than119867

(b) the occurrences of the sea storms in which just a wavehigher than119867 occurs

(c) the occurrences of the sea storms inwhich at least twowaves with height larger than119867 occur

will represent the Poisson processes we have for examplethat the probability that during L at least a storm will occurwith the maximum wave height larger than H is

119875119886[119871 119877ge1 (119867)] = 1 minus exp[minus 119871

119877ge1 (119867)

] (13)

Equation (13) gives the probability that the maximum waveheight in the lifetime (time) L will be greater than119867

More in general if we define

(i) Π119886(119871119873) as the probability that119873 occurrences of the

process (a) will occur in the time span 119871(ii) Π

119887(119871119873) as the probability that119873 occurrences of the

process (b) will occur in the time span 119871(iii) Π

119888(119871119873) as the probability that119873 occurrences of the

process (c) will occur in the time span 119871

(the processes a b and c being defined above) it follows that

119875119868= 1 minus Π

119886 (119871 0) minus Π119887 (119871 1)Π119888 (119871 0) (14)

represents the probability that the second wave in order ofheight during the time 119871 will be higher thanH

119875119868119868gt Π119888 (119871 1) Π119887 (119871 0) (15)

represents the probability that the second wave in order ofheight higher than H during the time 119871 will occur in thesame storm in which the highest wave happens

119875119868119868119868equiv119875119868119868

119875119868gt

Π119888 (119871 1)Π119887 (119871 0)

1 minus Π119886 (119871 0) minus Π119887 (119871 1) Π119888 (119871 0)

(16)

represents the probability that given that the second wavein order of height during 119871 is higher than 119867 it will belongto the same storm of the highest wave In other words 119875119868119868119868

(equiv 119875119868119868119875119868) is the probability that the two highest waves in the

time (lifetime) L will occur during the same sea stormNote that both 119875119868119868 and 119875119868119868119868 have been defined by consid-

ering the stochastic independence of the processes (b) and(c)

Finally if the Poisson processes are considered theprobabilities defined in this section may be calculated as

Π119886 (119871 0) = exp[minus 119871

119877ge1 (119867)

]

Π119887 (119871 0) = exp [minus 119871

1198771 (119867)

]

Π119888 (119871 0) = exp[minus 119871

119877ge2 (119867)

]

Π119887 (119871 1) =

119871

1198771 (119867)

exp [minus 119871

1198771 (119867)

]

Π119888 (119871 1) =

119871

119877ge2 (119867)

exp[minus 119871

119877ge2 (119867)

]

(17)

Mathematical Problems in Engineering 5

0

02

04

06

08

1

5 10 15 20 25

Prob

abili

ty

Wave height (m)

RON Crotone buoy

119875119886[119871 119877ge1(119867)] 119875119868119868119868119897

119875119868119868119897

119875119868

(a)

0

02

04

06

08

1

15 20 25 30 35Pr

obab

ility

Wave height (m)

119875119886[119871 119877ge1(119867)]

119875119868119868119868119897

119875119868119868119897

119875119868

NOAA-NODC 46004 buoy

(b)

Figure 2 Probabilities of occurrence119875119886[119871 119877ge1(119867)] (11)119875119868 (12) and the lower bound of probabilities119875119868119868 (13) and119875119868119868119868 (14) which are indicated

as 119875119868119868119897and 119875119868119868119868

119897 respectively for L = 50 years (a) RON buoy of Crotone (Italy) (b) NOAA-NODC 46004 buoy

Figures 2 and 3 show the probabilities of occurrence119875119886[119871 119877ge1(119867)] 119875119868 119875119868119868

119897 and 119875119868119868119868

119897 which are calculated from

data of RON buoy of Crotone (Italy) and of NOAA-NODC46004 buoy forL=50 yearsNote that119875119868119868

119897and119875119868119868119868119897

are definedby right hand side of (15) and (16) respectively they representthe lower bound of probabilities 119875119868119868 and 119875119868119868119868 respectively

It is interesting to note that if for a fixed value of L a largevalue of the probability is considered (08-09 eg) the heightvalue achieved from 119875

119868 curve is 3-4 smaller than waveheight given by 119875

119886[119871 119877ge1(119867)] probability This difference

increases as smaller values are considered of probabilityFor example if data of Figure 2 are considered we have

that

(1) for a value of probability 119875 = 08 we find a waveheight equal to 253m from 119875

119886[119871 119877ge1(119867)] and to

244m from 119875119868 (the ratio is equal to 0964) in other

words for a lifetime L = 50 years and a probabilityequal to 08 we find awave height which ismaximumin its own storm equal to 253m the height of thewave that in 50 years will be the second one in heightfor 119875 = 08 will be equal to 244m

(2) for a value of probability 119875 = 02 we find a waveheight equal to 286m from 119875

119886[119871 119877ge1(119867)] and to

268m from 119875119868 (the ratio is equal to 0939)

(3) for a value of probability 119875 = 005 we find a waveheight equal to 309m from 119875

119886[119871 119877ge1(119867)] and to

284m from 119875119868 (the ratio is equal to 0919)

The probability 119875119868119868119868 in conditions (1)ndash(3) will be greaterthan 005 029 and 049 respectively

Appendix

The following functions have been used for calculation

119875 (119867119867119904= ℎ) = exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A1)

119901 (119867119867119904= ℎ) =

8

1 + 120595lowast

119867

ℎ2exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A2)

that are probability of exceedance (A1) and probabilitydensity function (A2) of the crest-to-trough wave height in asea state with a given significant wave height 119867

119904= ℎ where

120595lowast is the narrower bandedness parameter [2 3]

119879 (ℎ) = 104radicℎ

119892(A3)

is Rice mean period as a function of the significant waveheight in a sea state with a mean JONSWAP spectrum [26]with 119892 the acceleration due to gravity

119901 (119867119904= ℎ) =

119906

119908119906(ℎ minus ℎ

119897)119906minus1 exp [minus(

ℎ minus ℎ119897

119908)

119906

] (A4)

is probability density function of the significant wave heightin a fixed location represented by means of a lower bounderthree-parameterWeibull lawThe location is identified by the

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

In this paper by following a different logic with respectto that one adopted by Arena and Pavone [7] a new solutionis found for the return period 1198771015840

ge2(119867) of a sea storm in which

at least two waves higher than a fixed thresholdH occur Thenew solution for 1198771015840

ge2(119867) is achieved from direct combination

of short-term and long-term wave statistics while the Arenaand Pavone solution gave the119877

ge2(119867) solution as a function of

both119877ge1(H) and119877

1(H) bymeans of the following expression

1

119877ge2 (119867)

=1

119877ge1 (119867)

minus1

1198771 (119867)

(1)

where119877ge1(H) and119877

1(H) are the return periods of a sea storm

inwhich respectively (i) at least a wave higher thanH occurs(ii) a single wave higher than H occurs

The final comparison between the two solutions which isproposed starting fromNOAA-NODC (USA) buoys data andfrom Italian buoys network (RON) from ISPRA Institutefor Environmental Protection and Research (Italy) showsa full agreement This result confirms that the EquivalentTriangular Storm (as well as the Equivalent Power Storm byFedele and Arena [12]) model is very powerful to determinethe long-term statistics of extreme waves during storms (seealso Arena et al [13])

The results are of interest for engineering applicationsbecause they enable to perform a complete analysis ofextreme waves that we may expect in the lifetime of a struc-ture which may be for example either an offshore structure[14ndash19] or on an upright breakwater (see eg Boccotti et al[14 15] and Romolo and Arena [20 21]) This may be donein terms of return value of the extreme individual crest-to-trough wave height as well as by achieving the second wave(the third and so on) in terms of height occurring in a fixedtime span on the considered structure

2 Return Period 1198771015840ge2(119867) of a Sea Storm in

Which at Least Two Waves Higher thana Fixed Threshold 119867 Occur

The return period 1198771015840ge2(119867) of a sea storm in which at least two

waves higher than a fixed threshold119867 occur may be definedin alternative as the return period of a sea storm in whichthe second wave height (if waves are ordered by decreasingcrest-to-trough wave heights) is larger than119867

If a large time span 120591 is considered we may define

1198771015840

ge2(119867) =

120591

119873119868119868 (119867 120591)

(2)

with 119873119868119868(119867 120591) the number of waves during 120591 which are (i)

higher thanH (ii) the secondwaves (if waves occurring in thestorm are ordered by decreasing height) in their own stormIn the following the solution is given for 119873

119868119868(119867 120591) Let us

consider the Triangular Sea which is given by the sequenceof equivalent triangular storms

The number of waves with height between 119909 and 119909 + d119909occurring in sea states with significant wave height 119867

119904in

(ℎ ℎ+dℎ) in the triangular storms with height between 119886 and119886 + d119886 and base between 119887 and 119887 + d119887 during 120591 is

[119901119860 (119886) d119886119873 (120591) 119901119861 (119887 | 119886) d119887] Δ119905 (ℎ dℎ 119886 119887)

sdot119901 (119909119867

119904= ℎ) d119909

119879 (ℎ)

(3)

where119873(120591) is the number of triangles during 120591 119875119860(119886) is the

probability density function of the triangle heights 119875119861(119887 | 119886)

is the probability density function of the triangle base withgiven height a and Δ119905(ℎ dℎ 119886 119887) is the time in which thesignificant wave height119867

119904is in the range (h h + dh) in those

triangles with height a and base b Finally 119901(119909119867119904= ℎ) is the

probability density function of the wave height in a sea statewith119867

119904= ℎ and 119879(ℎ) is Ricersquos mean period [22 23]

It follows that the number 119873119868119868(119909 ℎ 119886 119887) of waves with

height between 119909 and 119909 + d119909 occurring in sea states withsignificant wave height 119867

119904in (ℎ ℎ + dℎ) in the triangular

storms with height between 119886 and 119886 + d119886 and base between 119887and 119887 + d119887 during 120591 which are the second in order of heightin their own storm is given by

119873119868119868 (119909 ℎ 119886 119887)

= [119901119860 (119886) d119886119873 (120591) 119901119861 (119887 | 119886) d119887] Δ119905 (ℎ dℎ 119886 119887)

sdot119901 (119909119867

119904= ℎ) d119909

119879 (ℎ)

1198752 (119909 ℎ 119886 119887)

(4)

where in a triangular storm with height 119886 and base 119887 giventhat a wave with height between 119909 and 119909 + 119889119909 occurs in asea state with significant wave height between ℎ and ℎ + 119889ℎ1198752(119909 ℎ 119886 119887) is the probability that

(i) just a wave higher than x will occur

(ii) all the other waves of the sea storm will have heightsmaller than 119909

More in general if an actual storm is considered 1198752is the

probability that a wave with height between x and 119909 + d119909occurring in a sea state with significant wave height betweenℎ and ℎ + dℎ is the second wave in height in its own storm Itis then given by

1198752=

1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+

119873

sum

119894=1

119875 (119909119867119904= ℎ119894)

1 minus 119875 (119909119867119904= ℎ119894)119873119894]

times

119873

prod

119894=1

[1 minus 119875 (119909119867119904= ℎ119894)]119873119894

(5)

Mathematical Problems in Engineering 3

01

1

10

100

0 5 10 15

Retu

rn p

erio

d (y

ears

)

Wave height (m)

1198771

119877ge1

119877998400ge2

Figure 1 Return periods 1198771015840ge2(119867) 119877

ge1(119867) and 119877

1(119867) calculated

from data of RON buoy of Crotone (Italy) with the parametersdefined in the Appendix

whereN is the number of the sea states in the storm Equation(5) in integral form may be rewritten as [9 24]

1198752=

1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+ int

119863

0

119875 [119909119867119904= ℎ (119905)]

1 minus 119875 [119909119867119904= ℎ (119905)]

1

119879 [ℎ (119905)]

d119905]

sdot exp int119863

0

ln [1 minus 119875 (119909119867119904= ℎ (119905))]

1

119879 [ℎ (119905)]

d119905

(6)

where119863 is the storm durationIf the triangular sea is considered for a sea storm with

height a and base b the probability 1198752(119909 ℎ 119886 119887) is given by

1198752 (119909 ℎ 119886 119887)

=1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+119887 (119886)

119886

times int

119886

0

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

1

119879 (ℎ10158401015840)

dℎ10158401015840]

exp [119887 (119886)

119886int

119886

0

1n [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840]

(7)

Table 1 Parameters of the Weibull significant wave height distribu-tion (A4) and of the base height regression (A5) for some locations(from [25])

BuoyLocation

1198621

(hour)1198622

(mminus1) 119906119908

(m)ℎ119897

(m)NDBC 46004510∘Nndash1360∘W 11025 643 sdot10

minus2 1484 2489 065

NDBC 44008405∘Nndash694∘W 756 557 sdot10

minus2 1100 1146 055

RON Crotone390∘Nndash172∘E 9184 348 sdot10

minus2 0956 0590 008

The number of waves during 120591 which are higher than Hand the second in their own storm is

119873119868119868 (119867 120591) = int

infin

119867

int

infin

0

int

infin

ℎ

int

infin

0

119873119868119868 (119909 ℎ 119886 119887) d119887 d119886 dℎ d119909

(8)

In conclusion combining (2) (4) (7) and (8) the expres-sion is given for the return period 1198771015840

ge2(119867) of a sea storm in

which at least two waves higher than a fixed threshold 119867

occur

1198771015840

ge2(119867)

= int

infin

119867

int

infin

0

1

119879 (ℎ)

119901 (119909119867119904= ℎ)

1

1 minus 119875 (119909119867119904= ℎ)

sdot int

infin

ℎ

minusd119901 (119867

119904= 119886)

d119886

times exp[119887 (119886)119886

int

119886

0

ln [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840 ]

sdot [119887 (119886)

119886int

119886

0

1

119879 (ℎ10158401015840)

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

dℎ10158401015840+

minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

] d119886 dℎ d119909minus1

(9)

where according to conclusions of Arena and Pavone [4] thebases of the triangular stormsmay be considered constant (119887)with respect to a because a and b are stochastically inde-pendent to each other alternatively the Dirac distributionmay be considered [3] because it is slightly conservative inthis case the function 119887(119886)may be represented by means of aregression as given in the Appendix

Figure 1 shows the return period 1198771015840ge2(119867) which has been

calculated by considering functions defined in the AppendixValues of parameters used for calculation are defined inTable 1

4 Mathematical Problems in Engineering

3 An Alternative Approach forCalculation of 119877

ge2(119867)

A different approach for calculation of the return period119877ge2(119867) was given by Arena and Pavone [7] by means of (1)

In that equation 119877ge1(119867) represents the return period of a

sea storm in which the maximum wave height exceeds thethreshold H and 119877

1(119867) represents the return period of a

sea storm during which one wave only with crest-to-troughheight larger than a fixed threshold H occurs

For the calculation of119877ge1(H) the solution given byArena

and Pavone [4] has been applied

119877ge1 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [1 minus exp(119887119886int

119886

0

1

119879 (ℎ1015840)

timesln [1 minus 119875 (119867119867119904= ℎ1015840)] dℎ1015840)]119889119886

minus1

(10)

The return period 1198771(119867) is calculated by considering the

following expression [7]

1198771 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [int

119886

0

119875 (119867119867119904= ℎ)

119879 (ℎ) [1 minus 119875 (119867119867119904 = ℎ)]

dℎ]

sdot exp[119887119886int

119886

0

ln [1 minus 119875 (119867119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840] d119886minus1

(11)

4 The Encounter Probability

In general the occurrences of severe storms are assumed toform a homogeneous Poisson process Then the probabilitythat in the given time interval L (which in the design of oceanstructures may be considered equal to the lifetime of thestructure) at least a sea storm with given properties occursmay be written as

119875 (119871 119877) = 1 minus exp(minus119871119877) (12)

where R is the return period of the considered stormThen if we assume that

(a) the occurrences of the sea storms in which the highestwave is larger than119867

(b) the occurrences of the sea storms in which just a wavehigher than119867 occurs

(c) the occurrences of the sea storms inwhich at least twowaves with height larger than119867 occur

will represent the Poisson processes we have for examplethat the probability that during L at least a storm will occurwith the maximum wave height larger than H is

119875119886[119871 119877ge1 (119867)] = 1 minus exp[minus 119871

119877ge1 (119867)

] (13)

Equation (13) gives the probability that the maximum waveheight in the lifetime (time) L will be greater than119867

More in general if we define

(i) Π119886(119871119873) as the probability that119873 occurrences of the

process (a) will occur in the time span 119871(ii) Π

119887(119871119873) as the probability that119873 occurrences of the

process (b) will occur in the time span 119871(iii) Π

119888(119871119873) as the probability that119873 occurrences of the

process (c) will occur in the time span 119871

(the processes a b and c being defined above) it follows that

119875119868= 1 minus Π

119886 (119871 0) minus Π119887 (119871 1)Π119888 (119871 0) (14)

represents the probability that the second wave in order ofheight during the time 119871 will be higher thanH

119875119868119868gt Π119888 (119871 1) Π119887 (119871 0) (15)

represents the probability that the second wave in order ofheight higher than H during the time 119871 will occur in thesame storm in which the highest wave happens

119875119868119868119868equiv119875119868119868

119875119868gt

Π119888 (119871 1)Π119887 (119871 0)

1 minus Π119886 (119871 0) minus Π119887 (119871 1) Π119888 (119871 0)

(16)

represents the probability that given that the second wavein order of height during 119871 is higher than 119867 it will belongto the same storm of the highest wave In other words 119875119868119868119868

(equiv 119875119868119868119875119868) is the probability that the two highest waves in the

time (lifetime) L will occur during the same sea stormNote that both 119875119868119868 and 119875119868119868119868 have been defined by consid-

ering the stochastic independence of the processes (b) and(c)

Finally if the Poisson processes are considered theprobabilities defined in this section may be calculated as

Π119886 (119871 0) = exp[minus 119871

119877ge1 (119867)

]

Π119887 (119871 0) = exp [minus 119871

1198771 (119867)

]

Π119888 (119871 0) = exp[minus 119871

119877ge2 (119867)

]

Π119887 (119871 1) =

119871

1198771 (119867)

exp [minus 119871

1198771 (119867)

]

Π119888 (119871 1) =

119871

119877ge2 (119867)

exp[minus 119871

119877ge2 (119867)

]

(17)

Mathematical Problems in Engineering 5

0

02

04

06

08

1

5 10 15 20 25

Prob

abili

ty

Wave height (m)

RON Crotone buoy

119875119886[119871 119877ge1(119867)] 119875119868119868119868119897

119875119868119868119897

119875119868

(a)

0

02

04

06

08

1

15 20 25 30 35Pr

obab

ility

Wave height (m)

119875119886[119871 119877ge1(119867)]

119875119868119868119868119897

119875119868119868119897

119875119868

NOAA-NODC 46004 buoy

(b)

Figure 2 Probabilities of occurrence119875119886[119871 119877ge1(119867)] (11)119875119868 (12) and the lower bound of probabilities119875119868119868 (13) and119875119868119868119868 (14) which are indicated

as 119875119868119868119897and 119875119868119868119868

119897 respectively for L = 50 years (a) RON buoy of Crotone (Italy) (b) NOAA-NODC 46004 buoy

Figures 2 and 3 show the probabilities of occurrence119875119886[119871 119877ge1(119867)] 119875119868 119875119868119868

119897 and 119875119868119868119868

119897 which are calculated from

data of RON buoy of Crotone (Italy) and of NOAA-NODC46004 buoy forL=50 yearsNote that119875119868119868

119897and119875119868119868119868119897

are definedby right hand side of (15) and (16) respectively they representthe lower bound of probabilities 119875119868119868 and 119875119868119868119868 respectively

It is interesting to note that if for a fixed value of L a largevalue of the probability is considered (08-09 eg) the heightvalue achieved from 119875

119868 curve is 3-4 smaller than waveheight given by 119875

119886[119871 119877ge1(119867)] probability This difference

increases as smaller values are considered of probabilityFor example if data of Figure 2 are considered we have

that

(1) for a value of probability 119875 = 08 we find a waveheight equal to 253m from 119875

119886[119871 119877ge1(119867)] and to

244m from 119875119868 (the ratio is equal to 0964) in other

words for a lifetime L = 50 years and a probabilityequal to 08 we find awave height which ismaximumin its own storm equal to 253m the height of thewave that in 50 years will be the second one in heightfor 119875 = 08 will be equal to 244m

(2) for a value of probability 119875 = 02 we find a waveheight equal to 286m from 119875

119886[119871 119877ge1(119867)] and to

268m from 119875119868 (the ratio is equal to 0939)

(3) for a value of probability 119875 = 005 we find a waveheight equal to 309m from 119875

119886[119871 119877ge1(119867)] and to

284m from 119875119868 (the ratio is equal to 0919)

The probability 119875119868119868119868 in conditions (1)ndash(3) will be greaterthan 005 029 and 049 respectively

Appendix

The following functions have been used for calculation

119875 (119867119867119904= ℎ) = exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A1)

119901 (119867119867119904= ℎ) =

8

1 + 120595lowast

119867

ℎ2exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A2)

that are probability of exceedance (A1) and probabilitydensity function (A2) of the crest-to-trough wave height in asea state with a given significant wave height 119867

119904= ℎ where

120595lowast is the narrower bandedness parameter [2 3]

119879 (ℎ) = 104radicℎ

119892(A3)

is Rice mean period as a function of the significant waveheight in a sea state with a mean JONSWAP spectrum [26]with 119892 the acceleration due to gravity

119901 (119867119904= ℎ) =

119906

119908119906(ℎ minus ℎ

119897)119906minus1 exp [minus(

ℎ minus ℎ119897

119908)

119906

] (A4)

is probability density function of the significant wave heightin a fixed location represented by means of a lower bounderthree-parameterWeibull lawThe location is identified by the

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

01

1

10

100

0 5 10 15

Retu

rn p

erio

d (y

ears

)

Wave height (m)

1198771

119877ge1

119877998400ge2

Figure 1 Return periods 1198771015840ge2(119867) 119877

ge1(119867) and 119877

1(119867) calculated

from data of RON buoy of Crotone (Italy) with the parametersdefined in the Appendix

whereN is the number of the sea states in the storm Equation(5) in integral form may be rewritten as [9 24]

1198752=

1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+ int

119863

0

119875 [119909119867119904= ℎ (119905)]

1 minus 119875 [119909119867119904= ℎ (119905)]

1

119879 [ℎ (119905)]

d119905]

sdot exp int119863

0

ln [1 minus 119875 (119909119867119904= ℎ (119905))]

1

119879 [ℎ (119905)]

d119905

(6)

where119863 is the storm durationIf the triangular sea is considered for a sea storm with

height a and base b the probability 1198752(119909 ℎ 119886 119887) is given by

1198752 (119909 ℎ 119886 119887)

=1

1 minus 119875 (119909119867119904= ℎ)

times [minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

+119887 (119886)

119886

times int

119886

0

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

1

119879 (ℎ10158401015840)

dℎ10158401015840]

exp [119887 (119886)

119886int

119886

0

1n [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840]

(7)

Table 1 Parameters of the Weibull significant wave height distribu-tion (A4) and of the base height regression (A5) for some locations(from [25])

BuoyLocation

1198621

(hour)1198622

(mminus1) 119906119908

(m)ℎ119897

(m)NDBC 46004510∘Nndash1360∘W 11025 643 sdot10

minus2 1484 2489 065

NDBC 44008405∘Nndash694∘W 756 557 sdot10

minus2 1100 1146 055

RON Crotone390∘Nndash172∘E 9184 348 sdot10

minus2 0956 0590 008

The number of waves during 120591 which are higher than Hand the second in their own storm is

119873119868119868 (119867 120591) = int

infin

119867

int

infin

0

int

infin

ℎ

int

infin

0

119873119868119868 (119909 ℎ 119886 119887) d119887 d119886 dℎ d119909

(8)

In conclusion combining (2) (4) (7) and (8) the expres-sion is given for the return period 1198771015840

ge2(119867) of a sea storm in

which at least two waves higher than a fixed threshold 119867

occur

1198771015840

ge2(119867)

= int

infin

119867

int

infin

0

1

119879 (ℎ)

119901 (119909119867119904= ℎ)

1

1 minus 119875 (119909119867119904= ℎ)

sdot int

infin

ℎ

minusd119901 (119867

119904= 119886)

d119886

times exp[119887 (119886)119886

int

119886

0

ln [1 minus 119875 (119909119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840 ]

sdot [119887 (119886)

119886int

119886

0

1

119879 (ℎ10158401015840)

119875 (119909119867119904= ℎ10158401015840)

1 minus 119875 (119909119867119904= ℎ10158401015840)

dℎ10158401015840+

minus119875 (119909119867

119904= ℎ)

1 minus 119875 (119909119867119904= ℎ)

] d119886 dℎ d119909minus1

(9)

where according to conclusions of Arena and Pavone [4] thebases of the triangular stormsmay be considered constant (119887)with respect to a because a and b are stochastically inde-pendent to each other alternatively the Dirac distributionmay be considered [3] because it is slightly conservative inthis case the function 119887(119886)may be represented by means of aregression as given in the Appendix

Figure 1 shows the return period 1198771015840ge2(119867) which has been

calculated by considering functions defined in the AppendixValues of parameters used for calculation are defined inTable 1

4 Mathematical Problems in Engineering

3 An Alternative Approach forCalculation of 119877

ge2(119867)

A different approach for calculation of the return period119877ge2(119867) was given by Arena and Pavone [7] by means of (1)

In that equation 119877ge1(119867) represents the return period of a

sea storm in which the maximum wave height exceeds thethreshold H and 119877

1(119867) represents the return period of a

sea storm during which one wave only with crest-to-troughheight larger than a fixed threshold H occurs

For the calculation of119877ge1(H) the solution given byArena

and Pavone [4] has been applied

119877ge1 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [1 minus exp(119887119886int

119886

0

1

119879 (ℎ1015840)

timesln [1 minus 119875 (119867119867119904= ℎ1015840)] dℎ1015840)]119889119886

minus1

(10)

The return period 1198771(119867) is calculated by considering the

following expression [7]

1198771 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [int

119886

0

119875 (119867119867119904= ℎ)

119879 (ℎ) [1 minus 119875 (119867119867119904 = ℎ)]

dℎ]

sdot exp[119887119886int

119886

0

ln [1 minus 119875 (119867119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840] d119886minus1

(11)

4 The Encounter Probability

In general the occurrences of severe storms are assumed toform a homogeneous Poisson process Then the probabilitythat in the given time interval L (which in the design of oceanstructures may be considered equal to the lifetime of thestructure) at least a sea storm with given properties occursmay be written as

119875 (119871 119877) = 1 minus exp(minus119871119877) (12)

where R is the return period of the considered stormThen if we assume that

(a) the occurrences of the sea storms in which the highestwave is larger than119867

(b) the occurrences of the sea storms in which just a wavehigher than119867 occurs

(c) the occurrences of the sea storms inwhich at least twowaves with height larger than119867 occur

will represent the Poisson processes we have for examplethat the probability that during L at least a storm will occurwith the maximum wave height larger than H is

119875119886[119871 119877ge1 (119867)] = 1 minus exp[minus 119871

119877ge1 (119867)

] (13)

Equation (13) gives the probability that the maximum waveheight in the lifetime (time) L will be greater than119867

More in general if we define

(i) Π119886(119871119873) as the probability that119873 occurrences of the

process (a) will occur in the time span 119871(ii) Π

119887(119871119873) as the probability that119873 occurrences of the

process (b) will occur in the time span 119871(iii) Π

119888(119871119873) as the probability that119873 occurrences of the

process (c) will occur in the time span 119871

(the processes a b and c being defined above) it follows that

119875119868= 1 minus Π

119886 (119871 0) minus Π119887 (119871 1)Π119888 (119871 0) (14)

represents the probability that the second wave in order ofheight during the time 119871 will be higher thanH

119875119868119868gt Π119888 (119871 1) Π119887 (119871 0) (15)

represents the probability that the second wave in order ofheight higher than H during the time 119871 will occur in thesame storm in which the highest wave happens

119875119868119868119868equiv119875119868119868

119875119868gt

Π119888 (119871 1)Π119887 (119871 0)

1 minus Π119886 (119871 0) minus Π119887 (119871 1) Π119888 (119871 0)

(16)

represents the probability that given that the second wavein order of height during 119871 is higher than 119867 it will belongto the same storm of the highest wave In other words 119875119868119868119868

(equiv 119875119868119868119875119868) is the probability that the two highest waves in the

time (lifetime) L will occur during the same sea stormNote that both 119875119868119868 and 119875119868119868119868 have been defined by consid-

ering the stochastic independence of the processes (b) and(c)

Finally if the Poisson processes are considered theprobabilities defined in this section may be calculated as

Π119886 (119871 0) = exp[minus 119871

119877ge1 (119867)

]

Π119887 (119871 0) = exp [minus 119871

1198771 (119867)

]

Π119888 (119871 0) = exp[minus 119871

119877ge2 (119867)

]

Π119887 (119871 1) =

119871

1198771 (119867)

exp [minus 119871

1198771 (119867)

]

Π119888 (119871 1) =

119871

119877ge2 (119867)

exp[minus 119871

119877ge2 (119867)

]

(17)

Mathematical Problems in Engineering 5

0

02

04

06

08

1

5 10 15 20 25

Prob

abili

ty

Wave height (m)

RON Crotone buoy

119875119886[119871 119877ge1(119867)] 119875119868119868119868119897

119875119868119868119897

119875119868

(a)

0

02

04

06

08

1

15 20 25 30 35Pr

obab

ility

Wave height (m)

119875119886[119871 119877ge1(119867)]

119875119868119868119868119897

119875119868119868119897

119875119868

NOAA-NODC 46004 buoy

(b)

Figure 2 Probabilities of occurrence119875119886[119871 119877ge1(119867)] (11)119875119868 (12) and the lower bound of probabilities119875119868119868 (13) and119875119868119868119868 (14) which are indicated

as 119875119868119868119897and 119875119868119868119868

119897 respectively for L = 50 years (a) RON buoy of Crotone (Italy) (b) NOAA-NODC 46004 buoy

Figures 2 and 3 show the probabilities of occurrence119875119886[119871 119877ge1(119867)] 119875119868 119875119868119868

119897 and 119875119868119868119868

119897 which are calculated from

data of RON buoy of Crotone (Italy) and of NOAA-NODC46004 buoy forL=50 yearsNote that119875119868119868

119897and119875119868119868119868119897

are definedby right hand side of (15) and (16) respectively they representthe lower bound of probabilities 119875119868119868 and 119875119868119868119868 respectively

It is interesting to note that if for a fixed value of L a largevalue of the probability is considered (08-09 eg) the heightvalue achieved from 119875

119868 curve is 3-4 smaller than waveheight given by 119875

119886[119871 119877ge1(119867)] probability This difference

increases as smaller values are considered of probabilityFor example if data of Figure 2 are considered we have

that

(1) for a value of probability 119875 = 08 we find a waveheight equal to 253m from 119875

119886[119871 119877ge1(119867)] and to

244m from 119875119868 (the ratio is equal to 0964) in other

words for a lifetime L = 50 years and a probabilityequal to 08 we find awave height which ismaximumin its own storm equal to 253m the height of thewave that in 50 years will be the second one in heightfor 119875 = 08 will be equal to 244m

(2) for a value of probability 119875 = 02 we find a waveheight equal to 286m from 119875

119886[119871 119877ge1(119867)] and to

268m from 119875119868 (the ratio is equal to 0939)

(3) for a value of probability 119875 = 005 we find a waveheight equal to 309m from 119875

119886[119871 119877ge1(119867)] and to

284m from 119875119868 (the ratio is equal to 0919)

The probability 119875119868119868119868 in conditions (1)ndash(3) will be greaterthan 005 029 and 049 respectively

Appendix

The following functions have been used for calculation

119875 (119867119867119904= ℎ) = exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A1)

119901 (119867119867119904= ℎ) =

8

1 + 120595lowast

119867

ℎ2exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A2)

that are probability of exceedance (A1) and probabilitydensity function (A2) of the crest-to-trough wave height in asea state with a given significant wave height 119867

119904= ℎ where

120595lowast is the narrower bandedness parameter [2 3]

119879 (ℎ) = 104radicℎ

119892(A3)

is Rice mean period as a function of the significant waveheight in a sea state with a mean JONSWAP spectrum [26]with 119892 the acceleration due to gravity

119901 (119867119904= ℎ) =

119906

119908119906(ℎ minus ℎ

119897)119906minus1 exp [minus(

ℎ minus ℎ119897

119908)

119906

] (A4)

is probability density function of the significant wave heightin a fixed location represented by means of a lower bounderthree-parameterWeibull lawThe location is identified by the

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

3 An Alternative Approach forCalculation of 119877

ge2(119867)

A different approach for calculation of the return period119877ge2(119867) was given by Arena and Pavone [7] by means of (1)

In that equation 119877ge1(119867) represents the return period of a

sea storm in which the maximum wave height exceeds thethreshold H and 119877

1(119867) represents the return period of a

sea storm during which one wave only with crest-to-troughheight larger than a fixed threshold H occurs

For the calculation of119877ge1(H) the solution given byArena

and Pavone [4] has been applied

119877ge1 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [1 minus exp(119887119886int

119886

0

1

119879 (ℎ1015840)

timesln [1 minus 119875 (119867119867119904= ℎ1015840)] dℎ1015840)]119889119886

minus1

(10)

The return period 1198771(119867) is calculated by considering the

following expression [7]

1198771 (119867)

= minusint

infin

0

d119901 (119867119904= 119886)

d119886119886

119887

times [int

119886

0

119875 (119867119867119904= ℎ)

119879 (ℎ) [1 minus 119875 (119867119867119904 = ℎ)]

dℎ]

sdot exp[119887119886int

119886

0

ln [1 minus 119875 (119867119867119904= ℎ1015840)]

119879 (ℎ1015840)

dℎ1015840] d119886minus1

(11)

4 The Encounter Probability

In general the occurrences of severe storms are assumed toform a homogeneous Poisson process Then the probabilitythat in the given time interval L (which in the design of oceanstructures may be considered equal to the lifetime of thestructure) at least a sea storm with given properties occursmay be written as

119875 (119871 119877) = 1 minus exp(minus119871119877) (12)

where R is the return period of the considered stormThen if we assume that

(a) the occurrences of the sea storms in which the highestwave is larger than119867

(b) the occurrences of the sea storms in which just a wavehigher than119867 occurs

(c) the occurrences of the sea storms inwhich at least twowaves with height larger than119867 occur

will represent the Poisson processes we have for examplethat the probability that during L at least a storm will occurwith the maximum wave height larger than H is

119875119886[119871 119877ge1 (119867)] = 1 minus exp[minus 119871

119877ge1 (119867)

] (13)

Equation (13) gives the probability that the maximum waveheight in the lifetime (time) L will be greater than119867

More in general if we define

(i) Π119886(119871119873) as the probability that119873 occurrences of the

process (a) will occur in the time span 119871(ii) Π

119887(119871119873) as the probability that119873 occurrences of the

process (b) will occur in the time span 119871(iii) Π

119888(119871119873) as the probability that119873 occurrences of the

process (c) will occur in the time span 119871

(the processes a b and c being defined above) it follows that

119875119868= 1 minus Π

119886 (119871 0) minus Π119887 (119871 1)Π119888 (119871 0) (14)

represents the probability that the second wave in order ofheight during the time 119871 will be higher thanH

119875119868119868gt Π119888 (119871 1) Π119887 (119871 0) (15)

represents the probability that the second wave in order ofheight higher than H during the time 119871 will occur in thesame storm in which the highest wave happens

119875119868119868119868equiv119875119868119868

119875119868gt

Π119888 (119871 1)Π119887 (119871 0)

1 minus Π119886 (119871 0) minus Π119887 (119871 1) Π119888 (119871 0)

(16)

represents the probability that given that the second wavein order of height during 119871 is higher than 119867 it will belongto the same storm of the highest wave In other words 119875119868119868119868

(equiv 119875119868119868119875119868) is the probability that the two highest waves in the

time (lifetime) L will occur during the same sea stormNote that both 119875119868119868 and 119875119868119868119868 have been defined by consid-

ering the stochastic independence of the processes (b) and(c)

Finally if the Poisson processes are considered theprobabilities defined in this section may be calculated as

Π119886 (119871 0) = exp[minus 119871

119877ge1 (119867)

]

Π119887 (119871 0) = exp [minus 119871

1198771 (119867)

]

Π119888 (119871 0) = exp[minus 119871

119877ge2 (119867)

]

Π119887 (119871 1) =

119871

1198771 (119867)

exp [minus 119871

1198771 (119867)

]

Π119888 (119871 1) =

119871

119877ge2 (119867)

exp[minus 119871

119877ge2 (119867)

]

(17)

Mathematical Problems in Engineering 5

0

02

04

06

08

1

5 10 15 20 25

Prob

abili

ty

Wave height (m)

RON Crotone buoy

119875119886[119871 119877ge1(119867)] 119875119868119868119868119897

119875119868119868119897

119875119868

(a)

0

02

04

06

08

1

15 20 25 30 35Pr

obab

ility

Wave height (m)

119875119886[119871 119877ge1(119867)]

119875119868119868119868119897

119875119868119868119897

119875119868

NOAA-NODC 46004 buoy

(b)

Figure 2 Probabilities of occurrence119875119886[119871 119877ge1(119867)] (11)119875119868 (12) and the lower bound of probabilities119875119868119868 (13) and119875119868119868119868 (14) which are indicated

as 119875119868119868119897and 119875119868119868119868

119897 respectively for L = 50 years (a) RON buoy of Crotone (Italy) (b) NOAA-NODC 46004 buoy

Figures 2 and 3 show the probabilities of occurrence119875119886[119871 119877ge1(119867)] 119875119868 119875119868119868

119897 and 119875119868119868119868

119897 which are calculated from

data of RON buoy of Crotone (Italy) and of NOAA-NODC46004 buoy forL=50 yearsNote that119875119868119868

119897and119875119868119868119868119897

are definedby right hand side of (15) and (16) respectively they representthe lower bound of probabilities 119875119868119868 and 119875119868119868119868 respectively

It is interesting to note that if for a fixed value of L a largevalue of the probability is considered (08-09 eg) the heightvalue achieved from 119875

119868 curve is 3-4 smaller than waveheight given by 119875

119886[119871 119877ge1(119867)] probability This difference

increases as smaller values are considered of probabilityFor example if data of Figure 2 are considered we have

that

(1) for a value of probability 119875 = 08 we find a waveheight equal to 253m from 119875

119886[119871 119877ge1(119867)] and to

244m from 119875119868 (the ratio is equal to 0964) in other

words for a lifetime L = 50 years and a probabilityequal to 08 we find awave height which ismaximumin its own storm equal to 253m the height of thewave that in 50 years will be the second one in heightfor 119875 = 08 will be equal to 244m

(2) for a value of probability 119875 = 02 we find a waveheight equal to 286m from 119875

119886[119871 119877ge1(119867)] and to

268m from 119875119868 (the ratio is equal to 0939)

(3) for a value of probability 119875 = 005 we find a waveheight equal to 309m from 119875

119886[119871 119877ge1(119867)] and to

284m from 119875119868 (the ratio is equal to 0919)

The probability 119875119868119868119868 in conditions (1)ndash(3) will be greaterthan 005 029 and 049 respectively

Appendix

The following functions have been used for calculation

119875 (119867119867119904= ℎ) = exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A1)

119901 (119867119867119904= ℎ) =

8

1 + 120595lowast

119867

ℎ2exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A2)

that are probability of exceedance (A1) and probabilitydensity function (A2) of the crest-to-trough wave height in asea state with a given significant wave height 119867

119904= ℎ where

120595lowast is the narrower bandedness parameter [2 3]

119879 (ℎ) = 104radicℎ

119892(A3)

is Rice mean period as a function of the significant waveheight in a sea state with a mean JONSWAP spectrum [26]with 119892 the acceleration due to gravity

119901 (119867119904= ℎ) =

119906

119908119906(ℎ minus ℎ

119897)119906minus1 exp [minus(

ℎ minus ℎ119897

119908)

119906

] (A4)

is probability density function of the significant wave heightin a fixed location represented by means of a lower bounderthree-parameterWeibull lawThe location is identified by the

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0

02

04

06

08

1

5 10 15 20 25

Prob

abili

ty

Wave height (m)

RON Crotone buoy

119875119886[119871 119877ge1(119867)] 119875119868119868119868119897

119875119868119868119897

119875119868

(a)

0

02

04

06

08

1

15 20 25 30 35Pr

obab

ility

Wave height (m)

119875119886[119871 119877ge1(119867)]

119875119868119868119868119897

119875119868119868119897

119875119868

NOAA-NODC 46004 buoy

(b)

Figure 2 Probabilities of occurrence119875119886[119871 119877ge1(119867)] (11)119875119868 (12) and the lower bound of probabilities119875119868119868 (13) and119875119868119868119868 (14) which are indicated

as 119875119868119868119897and 119875119868119868119868

119897 respectively for L = 50 years (a) RON buoy of Crotone (Italy) (b) NOAA-NODC 46004 buoy

Figures 2 and 3 show the probabilities of occurrence119875119886[119871 119877ge1(119867)] 119875119868 119875119868119868

119897 and 119875119868119868119868

119897 which are calculated from

data of RON buoy of Crotone (Italy) and of NOAA-NODC46004 buoy forL=50 yearsNote that119875119868119868

119897and119875119868119868119868119897

are definedby right hand side of (15) and (16) respectively they representthe lower bound of probabilities 119875119868119868 and 119875119868119868119868 respectively

It is interesting to note that if for a fixed value of L a largevalue of the probability is considered (08-09 eg) the heightvalue achieved from 119875

119868 curve is 3-4 smaller than waveheight given by 119875

119886[119871 119877ge1(119867)] probability This difference

increases as smaller values are considered of probabilityFor example if data of Figure 2 are considered we have

that

(1) for a value of probability 119875 = 08 we find a waveheight equal to 253m from 119875

119886[119871 119877ge1(119867)] and to

244m from 119875119868 (the ratio is equal to 0964) in other

words for a lifetime L = 50 years and a probabilityequal to 08 we find awave height which ismaximumin its own storm equal to 253m the height of thewave that in 50 years will be the second one in heightfor 119875 = 08 will be equal to 244m

(2) for a value of probability 119875 = 02 we find a waveheight equal to 286m from 119875

119886[119871 119877ge1(119867)] and to

268m from 119875119868 (the ratio is equal to 0939)

(3) for a value of probability 119875 = 005 we find a waveheight equal to 309m from 119875

119886[119871 119877ge1(119867)] and to

284m from 119875119868 (the ratio is equal to 0919)

The probability 119875119868119868119868 in conditions (1)ndash(3) will be greaterthan 005 029 and 049 respectively

Appendix

The following functions have been used for calculation

119875 (119867119867119904= ℎ) = exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A1)

119901 (119867119867119904= ℎ) =

8

1 + 120595lowast

119867

ℎ2exp [minus 4

1 + 120595lowast(119867

ℎ)

2

] (A2)

that are probability of exceedance (A1) and probabilitydensity function (A2) of the crest-to-trough wave height in asea state with a given significant wave height 119867

119904= ℎ where

120595lowast is the narrower bandedness parameter [2 3]

119879 (ℎ) = 104radicℎ

119892(A3)

is Rice mean period as a function of the significant waveheight in a sea state with a mean JONSWAP spectrum [26]with 119892 the acceleration due to gravity

119901 (119867119904= ℎ) =

119906

119908119906(ℎ minus ℎ

119897)119906minus1 exp [minus(

ℎ minus ℎ119897

119908)

119906

] (A4)

is probability density function of the significant wave heightin a fixed location represented by means of a lower bounderthree-parameterWeibull lawThe location is identified by the

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

02

04

06

08

1

10 15 20 25 30 35

Prob

abili

ty

Wave height (m)

NOAA-NODC 44008 buoy

119875119886[119871 119877ge1(119867)]119875119868119868119868119897

119875119868119868119897

119875119868

Figure 3 See caption of Figure 2 Data of NOAA-NODC 44008buoy

parameters 119906 119908 and ℎ119897(see Arena [25] for some values of

the parameters)

119887 (119886) = 1198621 exp (minus1198622119886) (A5)

is mean value of the base of the triangular storms with heighta which is calculated by means of an exponential regression[3 25]

References

[1] P Boccotti ldquoSome new results on statistical properties of windwavesrdquo Applied Ocean Research vol 5 no 3 pp 134ndash140 1983

[2] P Boccotti ldquoOn coastal and offshore structure risk analysisrdquoExcerpta of the Italian Contribution t the Field of HydraulicEngng vol 1 pp 19ndash36 1986

[3] P Boccotti Wave Mechanics for Ocean Engineering ElsevierScience Oxford UK 2000

[4] F Arena and D Pavone ldquoReturn period of nonlinear high wavecrestsrdquo Journal of Geophysical Research C vol 111 no 8 ArticleID C08004 10 pages 2006

[5] G Z Forristall ldquoWave crest distributions observations andsecond-order theoryrdquo Journal of Physical Oceanography vol 30no 8 pp 1931ndash1943 2000

[6] F Fedele and F Arena ldquoWeakly nonlinear statistics of highrandom wavesrdquo Physics of Fluids vol 17 no 2 pp 1ndash10 2005

[7] F Arena and D Pavone ldquoA generalized approach for long-termmodelling of extreme crest-to-trough wave heightsrdquo OceanModelling vol 26 no 3-4 pp 217ndash225 2009

[8] H Krogstad ldquoHeight and period distributions of extremewavesrdquo Applied Ocean Research vol 7 no 3 pp 158ndash165 1985

[9] L E Borgman ldquoProbabilities for highest wave in hurricanerdquoJournal of the Waterways Harbors and Coastal EngineeringDivision vol 99 no 2 pp 185ndash207 1973

[10] P S Tromans and L Vanderschuren ldquoResponse based designconditions in the North Sea application of a new methodrdquoin Proceedings of Offshore Technology Conference pp 1ndash15Houston Tex USA 1995 paper OTC 7683

[11] G Z Forristall ldquoHow should we combine long and shortterm wave height distributionsrdquo in Proceedings of the 27thInternational Conference on Offshore Mechanics and ArcticEngineering (OMAE rsquo08) pp 987ndash994 June 2008

[12] F Fedele and F Arena ldquoLong-term statistics and extreme wavesof sea stormsrdquo Journal of Physical Oceanography vol 40 no 5pp 1106ndash1117 2010

[13] F Arena V Laface G Barbaro and A Romolo ldquoEffects ofsampling between data of significant wave height for intensityand duration of severe sea stormsrdquo International Journal ofGeosciences vol 4 pp 240ndash248 2013

[14] P Boccotti F Arena V Fiamma and G Barbaro ldquoFieldexperiment on random-wave forces on vertical cylindersrdquoProbabilistic Engineering Mechanics vol 28 pp 39ndash51 2012

[15] P Boccotti F Arena V Fiamma A Romolo and G BarbaroldquoA small scale field experiment on wave forces on uprightbreakwatersrdquo Journal of Waterway Port Coastal and OceanEngineering vol 138 no 2 pp 97ndash114 2012

[16] P Boccotti F Arena V Fiamma and A Romolo ldquoTwosmall-scale field experiments on the effectiveness of Morisonrsquosequationrdquo Ocean Engineering vol 57 no 1 pp 141ndash149 2013

[17] A Romolo G Malara G Barbaro and F Arena ldquoAn analyticalapproach for the calculation of random wave forces on sub-merged tunnelsrdquo Applied Ocean Research vol 31 no 1 pp 31ndash36 2009

[18] F Arena ldquoInteraction between long-crested random waves anda submerged horizontal cylinderrdquo Physics of Fluids vol 18 no7 Article ID 076602 2006

[19] F Arena and V Nava ldquoOn linearization of Morison forcegiven by high three-dimensional sea wave groupsrdquo ProbabilisticEngineering Mechanics vol 23 no 2-3 pp 104ndash113 2008

[20] A Romolo and F Arena ldquoMechanics of nonlinear randomwavegroups interacting with a vertical wallrdquo Physics of Fluids vol 20no 3 Article ID 036604 2008

[21] A Romolo and F Arena ldquoNonlinear wave pressures given byextreme waves on an upright breakwater theory and exper-imental validationrdquo in Proceedings of the 33rd InternationalConference on Coastal Engineering (ICCE rsquo12) pp 1ndash15 ASCESantander Spain July 2012 paper waves33

[22] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[23] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 24 pp 46ndash156 1945

[24] L Borgman ldquoMaximumwave height probabilities for a randomnumber of random intensity stormsrdquo in Proceedings of the12thConference on Coastal Engineering pp 53ndash64 1970

[25] F Arena ldquoOn the prediction of extreme sea wavesrdquo Environ-mental Sciences and Environmental Computing vol 2 pp 1ndash502004

[26] K Hasselmann T P Barnett E Bouws et al ldquoMeasurementsof wind-wave growth and swell decay during the joint NorthSea wave project (JONSWAP)rdquo Ergnzungsheft zur DeutschenHydrographischen Zeitschrift Reihe vol A8 pp 1ndash95 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of