BASIC COASTAL ENGINEERING Part 6 doc

two ways: 1 by identifying individual waves in the record and statisticallyanalyzing the heights and periods of these individual waves and 2 by conductinga Fourier analysis of the wave r

two ways: (1) by identifying individual waves in the record and statisticallyanalyzing the heights and periods of these individual waves and (2) by conducting

a Fourier analysis of the wave record to develop the wave spectrum The formerwill be discussed in this section and the latter in the next two sections

Wave Height Distribution

Figure 6.3 shows a short segment of a typical wave record A question arises as

to which undulations of the water surface should be considered as waves andwhat are the individual heights and periods of these waves The analysis proced-ure must be statistically reasonable and consistent The most commonly usedanalysis procedure is the zero-upcrossing method (Pierson, 1954) A mean watersurface elevation is determined and each point where the water surface crossesthis mean elevation in the upward direction is noted (see Figure 6.3) The timeelapsed between consecutive points is a wave period and the maximum verticaldistance between crest and trough is a wave height Note that some small surfaceundulations are not counted as waves so that some higher frequency components

in the wave record are Wltered out This is not of major concern, since forengineering purposes our focus is primarily on the larger waves in the spectrum

A primary concern is the distribution of wave heights in the record If the waveheights are plotted as a height–frequency distribution the result would typically

be like Figure 6.4 where p(H) is the frequency or probability of occurrence of theheight H The shaded area in thisWgure is the upper third of the wave heights andthe related signiWcant wave height is shown

For engineering purposes it is desirable to have a model for the distribution ofwave heights generated by a storm Longuet-Higgins (1952) demonstrated thatthis distribution is best deWned by a Rayleigh probability distribution Use of thisdistribution requires that the wave spectrum has a single narrow band of fre-quencies and that the individual waves are randomly distributed Practically, thisrequires that the waves be from a single storm that preferably is some distanceaway so that frequency dispersion narrows the band of frequencies recorded.Comparisons of the Rayleigh distribution with measured wave heights by several

authors (e.g., Goodnight and Russell, 1963; Collins, 1967; Chakrabarti andCooley, 1971; Goda 1974; Earle, 1975) indicate that this distribution yieldsacceptable results for most storms.

The Rayleigh distribution can be written

p(H)¼ 2H(Hrms)2e

s

(6:2)

In Eq (6.2) Hiare the individual wave heights in a record containing N waves.Employing the Rayleigh distribution leads to the following useful relation-ships:

(6:4)For our purposes, we are more interested in the percentage of waves that have aheight greater than a given height, i.e.,

p(H)

Figure 6.4 Typical wave height–frequency distribution.

1
P(H) ¼ e
(H=H rms ) 2

(6:5)Since Hs¼ 1:416Hrms[Eq (6.3a)]

2.2 2.0 1.8 1.6

H / Hrms

1.4 1.42 1.2 1.0 0.5

Figure 6.5 Raleigh distribution for wave heights (U.S Army Coastal Engineering Research Center, 1984.)

From Eqs (6.3a) and (6.3b) we have

Hrms¼ 1:72

0:886¼ 1:94 mand

Hs¼ 1:416(1:94) ¼ 2:75 mFrom line b in Figure 6.5 H5=Hrms¼ 1:98 so

H5¼ 1:98(1:94) ¼ 3:84 mFrom line a in Figure 6.5 at

Maximum Wave Height

There is no upper limit to the wave heights deWned by the Rayleigh distribution

In a storm, however, the highest wave that might be expected will depend on thelength of the storm as well as its strength Longuet-Higgins (1952) demonstratedthat for a storm with a relatively large number of waves N, the expected value ofthe height of the highest wave Hmaxwould be

Wave Period Distribution

It was mentioned previously that the highest waves and the largest energyconcentration in a wind wave spectrum are typically found at periods aroundthe middle of the period range of the spectrum Consequently, for engineeringpurposes, we are not usually as concerned with the extreme wave periods as wewere with the higher wave heights

The joint wave height–period probability distribution is of some interest Thegeneral shape of this distribution is depicted in Figure 6.6 (see Ochi, 1982) ThisWgure shows the distribution of the wave height versus wave period for eachwave in a typical record, nondimensionalized by dividing each height and periodvalue by the average height and average period, respectively The contour linesare lines of equal probability of occurrence of a height–period combination.Note, in Figure 6.6, that there is a small range of wave periods for the higherwaves and these periods are around the average period of the spectrum of waves.For the lower waves (but not the lowest), there is a much wider distribution ofwave periods The signiWcant period Ts is considered to be more statisticallystable than the average period so it is preferred to use the signiWcant period torepresent a wave record If one is using a spectral approach to analyzing a waverecord (see the next section) the period of the peak of the spectrum known as thespectral peak period Tp would be used as a representative period From inves-tigations of numerous wave records the U.S Army Coastal Engineering Re-search Center (1984) recommends the relationship Ts¼ 0:95 Tp

2.0 1.0

0 1.0

of occurence

Figure 6.6 Typical dimensionless joint wave height–period distribution.

6.4 Wave Spectral Characteristics

An alternate approach to analyzing a wave record such as that shown in Figure 6.3

is by determining the resulting wave spectrum for that record A water surfaceelevation time history can be reconstructed by adding a large number of compon-ent sine waves that have diVerent periods, amplitudes, phase positions, andpropagation directions A directional wave spectrum is produced when the sum

of the energy density in these component waves at each wave frequency S( f ,u) isplotted versus wave frequency f and direction u Commonly, one-dimensionalwave spectra are developed when the energy for all directions at a particularfrequency S( f ) is plotted as a function of only wave frequency An alternateform to the above described frequency spectrum is the period spectrum wherethe wave energy density S( T ) is plotted versus the wave period

From the small-amplitude wave theory, the energy density in a wave isrgH2=8 Leaving out the product of the Xuid density and the acceleration ofgravity, as is commonly done, leads to the following expression for a directionalwave spectrum:

S( f ,u) df du¼X

f

Xuþdu u

The exact scale and shape of a wind wave spectrum will depend on thegenerating factors of wind speed, duration, fetch, etc as discussed above How-ever, a general form of a spectral model equation is

S( f )¼ A

where A and B adjust the shape and scale of the spectrum and can be writteneither as a function of the generating factors or as a function of a representativewave height and period (e.g., Hsand Ts)

Analysis of a wave record to produce the wave spectrum is a complex matterthat is beyond the scope of this text Software packages are available for this taskthat take a digitized record of the water surface and produce the spectralanalysis Wilson et al (1974) discuss spectral analysis procedures and give a list

of basic references on the subject

An important way to characterize a wave spectrum is by the moments of aspectrum The nth moment of a spectrum is deWned as

mn ¼

ð10

So, for example, the zeroth moment would just be the area under the spectralcurve Since a spectrum plot shows the energy density at each frequency versusthe range of frequencies, the area under the spectral curve is equal to the totalenergy density of the wave spectrum (divided by the product of theXuid densityand acceleration of gravity)

As with the analysis procedures discussed in Section 6.3, it would be useful tohave a representative wave height and period for the wave spectrum that can bederived from the spectrum The spectral peak period Tpis a representative period(or one can use its reciprocal, the spectral peak frequency) The spectral momentconcept is useful to deWne a representative wave height

From the small-amplitude wave theory, the total energy density is twice thepotential energy density of a wave Thus,

E¼ 2
Ep¼ 2

T

ðT0rgh h2

dt

where T is the length of wave record being analyzed and the overbar denotesenergy density This can be written

where the overbar denotes the average of the sum of Ndigitized water surfaceelevation values from a wave record of length T From our deWnition of Hrmsand Hsthe energy density can also be written

Hs¼ 4pffiffiffiffiffiffimo

(6:17)where the designation Hmo will be used for this deWnition of signiWcant height.Recall that Hsis based on a wave-by-wave analysis from the wave record, but

Hmois determined from the energy spectrum or, more basically, from digitizedvalues of the water surface elevation given by the wave record

Analysis of the same wave records by the wave-by-wave method and byspectral analysis indicates that Hs and Hmo are eVectively equal for waves indeep water that are not too steep For steeper waves and waves in intermediateand shallow water Hs will be increasingly larger than Hmo so the two termscannot be interchangeably used Figure 6.7, which was slightly modiWed fromThompson and Vincent (1985) and is based on Weld and laboratory waverecords, shows how Hs and Hmo compare for diVerent relative water depths

As wave records become more commonly analyzed by computer the second

deWnition of signiWcant height (Hmo) is more commonly being used

6.5 Wave Spectral Models

As the Rayleigh distribution is a useful model for the expected distribution of waveheights from a particular storm, it is also useful to have a model of the expectedwave spectrum generated by a storm Several one-dimensional wave spectra

models have been proposed They generally have the form of Eq (6.11) and arederived from empiricalWts to selected sets of wave measurements, supported bydimensional and theoretical reasoning Four of these spectral models–called theBretschneider, Pierson–Moskowitz, JONSWAP, and TMA spectra–will be pre-sented These models are of interest from an historic perspective and because oftheir common use in coastal engineering practice TheWrst three models were dev-eloped for deep water waves and the last is adjusted for the eVects of water depth.Bretschneider Spectrum (Bretschneider, 1959)

The basic form of this spectrum is

S(T )¼ ag2(2p)4T

H100and T100denote the average wave height and period The parameters F1and

F are a dimensionless wave height and dimensionless wave period, respectively

1.0 1.2 1.4

Figure 6.7 Comparison of H s and H mo versus relative depth (Thompson and Vincent, 1985.)

As will be shown in the next section, Bretschneider empirically related F1and

F2 to the wind speed, the fetch, and the wind duration to develop a forecastingrelationship for the average wave height and period and, using Eq (6.18), thewave period spectrum

Inserting a, F1, and F2into Eq (6.18) leads to

Pierson–Moskowitz Spectrum (Pierson and Moskowitz, 1964)

The authors analyzed wave and wind records from British weather ships ating in the north Atlantic They selected records representing essentially fullydeveloped seas for wind speeds between 20 and 40 knots to produce the followingspectrum:

oper-S( f )¼ ag2

In Eq (6.20) the wind speed W is measured at an elevation of 19.5 m whichyields a speed that is typically 5% to 10% higher than the speed measured at thestandard elevation of 10 m The coeYcient a has a value of 8:1  10
3 Note thatthe fetch and wind duration are not included since this spectrum assumes a fullydeveloped sea At much higher wind speeds than the 20 to 40 knot range it is lesslikely for a fully developed sea to occur

The following relationships can be developed from the Pierson–Moskowitzspectrum formulation (see Ochi, 1982):

JONSWAP (Hasselmann et al., 1973)

This spectrum results from a Joint North Sea Wave Project operated by tories from four countries Wave and wind measurements were taken with

labora-suYcient wind durations to produce a deep water fetch limited model spectrum

If we eliminate the wind speed from Eq (6.20) by incorporating Eq (6.22) thePierson–Moskowitz spectrum can be written

The coeYcient a and the peak frequency fp for the JONSWAP spectrum aregiven by

TMA Spectrum (Bouws et al., 1985)

The previous three models were developed for deep water conditions As windwaves propagate into intermediate and shallow depths there is a period-depen-dent change in the shape of the spectrum versus that for deep water The TMAspectrum is a wave spectrum based on the generation of waves in deep water thatthen propagate without refracting into intermediate/shallow water depths Thespectral form is a JONSWAP spectrum modiWed by a depth and frequencydependent factor F( f , d ) Thus, S( f )TMA¼ S( f )JF( f , d ) where F( f , d ) is arelatively complex function deWned graphically in Figure 6.9

Hughes (1984) further proposed that a and g in the JONSWAP spectralformulation be modiWed to

for the TMA spectrum In Eqs (6.28) and (6.29) Lpis the wave length

Directional Wave Spectra

The components that make up a wave spectrum at a particular location willtypically be propagating in a range of directions A point measurement of thewater surface elevation time history will not detect this directional variability, so

an analysis of this time history yields a one-dimensional spectrum But, duringrecent years, wave gages that can detect the full directionality of the waveWeld at

a given location have come into more common use Consequently, directionalspectral data are becoming available and signiWcant development of directionalspectral models has taken place

The directional spread of wave energy in a wind wave Weld is frequencydependent Generally, the short period components of the wave spectrum have

a wider range of directions, while the wave energy is more focused on thedominant direction for the frequencies near the spectral peak Models for direc-tional wave spectra commonly are one-dimensional spectra corrected by a factorthat depends on the wave frequency and direction, i.e.,

S( f , u)¼ S( f )G( f , u) (6:30)

2.0 1.5

1.0 0.5

where G( f , u) is a dimensionless directional spreading function Since modifying

a one-dimensional spectrum to a directional spectrum does not change the totalenergy density, we have

A much more complex directional spreading function (Mitsuyasu et al., 1975),which is based on extensive measurements of directional wave spectra, is

In the above equations G is the gamma function of the term in parentheses, which

is tabulated in some mathematical handbooks The parameter s was originallygiven as a function of wave frequency, wave peak frequency, and wind speed.Higher values of s give a more widely spread directional spectrum Goda andSuzuki (1975) and Goda (1985) give a simpler deWnition of s that is useful forengineering applications, i.e.,

s¼ Smax( f=fp)5when f < fp

¼ S ( f=f )
2:5when f > f (6:35)

For design purposes, Goda (1985) recommends

Smax¼ 10 Wind waves

Smax¼ 25 Swell with short decay distance

Smax¼ 75 Swell with long decay distance

As a wind wave spectrum approaches the shore, wave shoaling and refractiontend to reduce the spread of wave directions which would cause some increase in

Smax Mitsuyasu et al (1975) employed Eq (6.34) with a JONSWAP dimensional spectrum

one-Refraction and DiVraction of Directional Spectra

It is common in much of coastal engineering design to select a representativewave height, period and direction (e.g Hmoand Tphaving the oVshore directionthat is dominant in the wave spectrum) This wave is then treated as a mono-chromatic wave which is shoaled, refracted and diVracted (if necessary) to thepoint of interest in the nearshore area, employing the methods presented inChapters 2, 3 and 4 However, shoaling eVects depend on the wave period, andrefraction and diVraction eVects depend on both the wave period and direction.Thus, a more complete analysis of the changes that take place as a directionalwave spectrum propagates from oVshore to the coast will be dependent on thefrequency and direction distribution in the oVshore wave spectrum Each fre-quency and direction component in the spectrum will shoal, refract and diVract

diVerently The result will depend on the combination of these components at thepoint of interest in the coastal zone

For a directional wave spectrum the combined shoaling/refraction coeYcient(Kr)sis given by

(Kr)s¼ 1

(m0)s

X1 0

Xp

pS(f , u)Ks2Kr2Df Du

In order to apply this equation, the directional spectrum would be broken intodirectional and frequency segments (Du and Df) Then, a representative value of fand u from each frequency/direction segment would be used to shoal and refract

a monochromatic wave to the coast, yielding the Ks and Kr values for thatsegment The results would then be recombined using the above equation to yield

a value of (Kr)sfor the directional spectrum Then,

(Hmo)c¼ (Kr)s(Hmo)oWhere the subscripts c and o refer to the signiWcant wave height at the coastalpoint of interest and oVshore

This approach is obviously very onerous to apply in practice, especially if asigniWcant number of direction/frequency components are to be used And, itwould have to be repeated in toto for a diVerent nearshore hydrography or adiVerent oVshore directional wave spectrum

Goda (1985) employed this procedure to develop results that give someindication of the diVerence in results for a classic monochromatic wave shoal-ing/refraction analysis versus a directional spectrum shoaling/refraction analysis

He considered a coastal area with straight shore-parallel bottom contours Thisgreatly simpliWed the shoaling/refraction analysis because the refraction coeY-cient for each frequency/direction component could more easily be calculatedfrom Equations 4.2 and 4.3 Goda employed a modiWed Bretschneider spectrumwith a directional spreading function given by Equation 6.34 and Smax¼ 10, 25and 75 Dominant oVshore directions for the directional spectrum included 08,

208, and 408 The nearshore coastal point selected for analysis was the pointwhere d=Lo ¼ 0:05, where Lo is calculated using the peak period for the spec-trum

Goda’s results for (Kr)sas a function of the oVshore direction and Smax, andthe comparative result for KrKsfor a monochromatic wave are:

5 percent Considering how well other factors such as the eYcacy of the shoaling/refraction analysis and how well the design wave conditions are known, this

diVerence is not exceptional As expected, the diVerence between the matic and spectral results diminishes as Smax increases (i.e as the waves moreclosely resemble a monochromatic wave)

monochro-An eVective diVraction coeYcient for a directional spectrum (Kd)sis given by

(Kd)s¼ 1

(m0)s

X1 0

Xp

pS( f , u)Kd2Df Du

and Kdis the diVraction coeYcient for each frequency/direction component.Goda (1985) also compared diVraction analyses for a semi-inWnite breakwaterand a breakwater gap using monochromatic waves versus a directional spectrumusing spectral and directional spreading conditions employed in the shoaling/refraction analysis comparison The above equations were used for the direc-tional spectrum analysis.

At a particular point in the lee of the breakwater there was a shift in thespectral peak frequency away from the incident spectral peak frequency Formonochromatic waves there is no change in the wave frequency as waves diVract

to the lee of a breakwater A shift in the spectral peak frequency should beexpected because at a particular point in the lee of the breakwater the value of r/

L and thus Kdwould be diVerent for the diVerent frequencies in the spectrum Sothe recombined components then yield a diVerent peak frequency

Also, at a particular point in the lee of the breakwater, the monochromaticand spectral diVraction coeYcients were diVerent In many cases these diVer-ences were quite signiWcant, with the monochromatic analysis often yielding amuch lower wave height than the spectral analysis at a particular point As might

be expected, comparison of calculated results with some availableWeld data on

diVracted wave conditions behind a breakwater at a coastal port indicated thatthe spectral approach gave better results

6.6 Wave Prediction—Early Methods

Early methods for wave prediction were simple empirical formulations relatingthe wave height and period to some representative wind speed, fetch, and laterduration

Selection of Wind Conditions

Prediction of wind generated waves by the simple empirical methods or by theuse of spectral models requires selection of representative values of wind speed,fetch, and duration Winds from more than one approach direction may generatewaves that must be considered for design analysis at a given coastal site Thefetch may be limited by land boundaries and it may be suYciently short so thatone can assume fetch-limited conditions to make the wave predictions

The best wind data source would be local speed/direction measurements over asuYcient length of time to do a return period analysis and select a design wind