BASIC COASTAL ENGINEERING Part 6 doc

two ways: (1) by identifying individual waves in the record and statistically analyzing 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 former will 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 and what are the individual heights and periods of these waves. The analysis proced- ure must be statistically reasonable and consistent. The most commonly used analysis procedure is the zero-upcrossing method (Pierson, 1954). A mean water surface elevation is determined and each point where the water surface crosses this mean elevation in the upward direction is noted (see Figure 6.3). The time elapsed between consecutive points is a wave period and the maximum vertical distance between crest and trough is a wave height. Note that some small surface undulations 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 for engineering 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 wave heights 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 the height H. The shaded area in this Wgure is the upper third of the wave heights and the related signiWcant wave height is shown. For engineering purposes it is desirable to have a model for the distribution of wave heights generated by a storm. Longuet-Higgins (1952) demonstrated that this distribution is best deWned by a Rayleigh probability distribution. Use of this distribution requires that the wave spectrum has a single narrow band of fre- quencies and that the individual waves are randomly distributed. Practically, this requires that the waves be from a single storm that preferably is some distance away so that frequency dispersion narrows the band of frequencies recorded. Comparisons of the Rayleigh distribution with measured wave heights by several Surface Elevation, η Mean surface elevation T H Time, t Figure 6.3. Typical water surface elevation versus time record. 162 / Basic Coastal Engineering authors (e.g., Goodnight and Russell, 1963; Collins, 1967; Chakrabarti and Cooley, 1971; Goda 1974; Earle, 1975) indicate that this distribution yields acceptable results for most storms. The Rayleigh distribution can be written p(H) ¼ 2H (H rms ) 2 e À(H=H rms ) 2 (6:1) where the root mean square height H rms is given by H rms ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi X H 2 i N s (6:2) In Eq. (6.2) H i are the individual wave heights in a record containing N waves. Employing the Rayleigh distribution leads to the following useful relation- ships: H s ¼ 1:416 H rms (6:3a) H 100 ¼ 0:886 H rms (6:3b) The cumulative probability distribution P(H ) (i.e., the percentage of waves having a height that is equal to or less than H)is P(H) ¼ ð H o p(H)dH ¼ 1 À e À(H=H rms ) 2 (6:4) For our purposes, we are more interested in the percentage of waves that have a height greater than a given height, i.e., p(H) H H s Figure 6.4. Typical wave height–frequency distribution. Wind-Generated Waves / 163 1 À P(H) ¼ e À(H=H rms ) 2 (6:5) Since H s ¼ 1:416H rms [Eq. (6.3a)] 1 À P(H s ) ¼ e À(1:416)2 ¼ 0:135 so 13.5% of the waves in a storm wave record might have heights that are greater than the signiWcant height. Figure 6.5, which is adapted from the U.S. Army Coastal Engineering Re- search Center (1984), is a plot that is useful when applying the Rayleigh distri- bution. Line a in the Wgure gives the probability P that any wave height will exceed the height (H=H rms ) and line b gives the average height of the n highest fraction of the waves. Example 6.3–1 A wave record taken during a storm is analyzed by the zero-upcrossing method and contains 205 waves. The average wave height in the record is 1.72 m. Estimate H s , H 5 , and the number of waves in the record that would exceed 2.5 m height. 2.62.42.22.01.81.6 H / H rms 1.4 1.42 1.21.00.5 1.0 0.5 0.1 0.05 P or n 0.01 0.005 0.001 a b Figure 6.5. Raleigh distribution for wave heights. (U.S. Army Coastal Engineering Research Center, 1984.) 164 / Basic Coastal Engineering Solution: From Eqs. (6.3a) and (6.3b) we have H rms ¼ 1:72 0:886 ¼ 1:94 m and H s ¼ 1:416(1:94) ¼ 2:75 m From line b in Figure 6.5 H 5 =H rms ¼ 1:98 so H 5 ¼ 1:98(1:94) ¼ 3:84 m From line a in Figure 6.5 at H H rms ¼ 2:5 1:94 ¼ 1:29 we have P ¼ 0:19. So the estimated number of waves exceeding 2.5 m in height is 205(0:19) ¼ 38:95, or approximately 39 waves. When a spectrum of waves reaches the shore, wave breaking causes the wave height distribution to be truncated at the higher end. Some authors have mod- iWed the Rayleigh distribution to account for nearshore depth-induced wave breaking (see Collins, 1970; Ibrageemov, 1973; Kuo and Kuo, 1974; Goda, 1975; Hughes and Borgman, 1987). 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 the length of the storm as well as its strength. Longuet-Higgins (1952) demonstrated that for a storm with a relatively large number of waves N, the expected value of the height of the highest wave H max would be H max ¼ 0:707 H s ffiffiffiffiffiffiffiffiffi ln N p (6:6) For example, a storm having a 6 hour duration of high waves having an average period of 8 s would about 2700 waves and H max would be 1:99H s . In the nearshore zone, the highest wave might be limited by wave breaking, provided the storm can generate suYciently high waves for this limit to apply in the water depth of concern. However, in deeper water beyond the depth where waves would break, the value given by Eq. (6.6) should be appropriate. Wind-Generated Waves / 165 Wave Period Distribution It was mentioned previously that the highest waves and the largest energy concentration in a wind wave spectrum are typically found at periods around the middle of the period range of the spectrum. Consequently, for engineering purposes, we are not usually as concerned with the extreme wave periods as we were with the higher wave heights. The joint wave height–period probability distribution is of some interest. The general shape of this distribution is depicted in Figure 6.6 (see Ochi, 1982). This Wgure shows the distribution of the wave height versus wave period for each wave in a typical record, nondimensionalized by dividing each height and period value by the average height and average period, respectively. The contour lines are 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 higher waves 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 of wave periods. The signiWcant period T s is considered to be more statistically stable than the average period so it is preferred to use the signiWcant period to represent a wave record. If one is using a spectral approach to analyzing a wave record (see the next section) the period of the peak of the spectrum known as the spectral peak period T p 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 T s ¼ 0:95 T p . 2.0 1.0 0 1.0 2.0 H H 100 T/T 100 Increasing probability of occurence Figure 6.6. Typical dimensionless joint wave height–period distribution. 166 / Basic Coastal Engineering 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 surface elevation time history can be reconstructed by adding a large number of compon- ent sine waves that have diVerent periods, amplitudes, phase positions, and propagation 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)is plotted versus wave frequency f and direction u. Commonly, one-dimensional wave spectra are developed when the energy for all directions at a particular frequency S( f ) is plotted as a function of only wave frequency. An alternate form to the above described frequency spectrum is the period spectrum where the wave energy density S( T ) is plotted versus the wave period. From the small-amplitude wave theory, the energy density in a wave is rgH 2 =8. Leaving out the product of the Xuid density and the acceleration of gravity, as is commonly done, leads to the following expression for a directional wave spectrum: S( f ,u) df du ¼ X f þdf f X uþdu u H 2 8 (6:7) where H is the height of the component waves making up the spectrum. This simpliWes to S( f )df ¼ X f þdf f H 2 8 (6:8) for a one-dimensional frequency spectrum. For a one-dimensional period spec- trum we have S( T )dT ¼ X TþdT T H 2 8 (6:9) It can be shown (see Sorensen, 1993) that the following relationship holds: S( f ) ¼ S(T )T 2 (6:10) Equation (6.8) indicates that the dimensions for S(f) would be length squared times time (e.g., m 2 s) and from Eq. (6.9) the dimensions of S(T) would be length squared divided by time (e.g., m 2 =S). This is consistent with Eq. (6.10). Wind-Generated Waves / 167 The exact scale and shape of a wind wave spectrum will depend on the generating 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 f 5 e ÀB=f 4 (6:11) where A and B adjust the shape and scale of the spectrum and can be written either as a function of the generating factors or as a function of a representative wave height and period (e.g., H s and T s ). Analysis of a wave record to produce the wave spectrum is a complex matter that is beyond the scope of this text. Software packages are available for this task that take a digitized record of the water surface and produce the spectral analysis. 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 a spectrum. The nth moment of a spectrum is deWned as m n ¼ ð 1 0 S( f ) f n df (6:12) So, for example, the zeroth moment would just be the area under the spectral curve. Since a spectrum plot shows the energy density at each frequency versus the range of frequencies, the area under the spectral curve is equal to the total energy density of the wave spectrum (divided by the product of the Xuid density and acceleration of gravity). As with the analysis procedures discussed in Section 6.3, it would be useful to have a representative wave height and period for the wave spectrum that can be derived from the spectrum. The spectral peak period T p is a representative period (or one can use its reciprocal, the spectral peak frequency). The spectral moment concept is useful to deWne a representative wave height. From the small-amplitude wave theory, the total energy density is twice the potential energy density of a wave. Thus,  EE ¼ 2  EE p ¼ 2 T Ã ð T Ã 0 rgh h 2  dt where T Ã is the length of wave record being analyzed and the overbar denotes energy density. This can be written  EE ¼ rghh 2 ¼ rg P h 2 N Ã (6:13) 168 / Basic Coastal Engineering where the overbar denotes the average of the sum of N Ã digitized water surface elevation values from a wave record of length T Ã . From our deWnition of H rms and H s the energy density can also be written  EE ¼ rgH 2 rms 8 ¼ rgH 2 s 16 (6:14) If the zeroth moment of the spectrum equals the energy density divided by rg, with Eqs. (6.13) and (6.14) we have  EE ¼ rgm o ¼ rg P h 2 N Ã (6:15) and  EE ¼ rgm o ¼ rgH 2 s 16 (6:16) Equation (6.15) provides a useful way to determine the energy density and the zeroth moment of a wave spectrum from digitized water surface elevation values. Equation (6.16) leads to a signiWcant height deWnition from the wave spectrum energy density or zeroth moment: H s ¼ 4 ffiffiffiffiffiffi m o p (6:17) where the designation H mo will be used for this deWnition of signiWcant height. Recall that H s is based on a wave-by-wave analysis from the wave record, but H mo is determined from the energy spectrum or, more basically, from digitized values of the water surface elevation given by the wave record. Analysis of the same wave records by the wave-by-wave method and by spectral analysis indicates that H s and H mo are eVectively equal for waves in deep water that are not too steep. For steeper waves and waves in intermediate and shallow water H s will be increasingly larger than H mo so the two terms cannot be interchangeably used. Figure 6.7, which was slightly modiWed from Thompson and Vincent (1985) and is based on Weld and laboratory wave records, shows how H s and H mo compare for diVerent relative water depths. As wave records become more commonly analyzed by computer the second deWnition of signiWcant height (H mo ) is more commonly being used. 6.5 Wave Spectral Models As the Rayleigh distribution is a useful model for the expected distribution of wave heights from a particular storm, it is also useful to have a model of the expected wave spectrum generated by a storm. Several one-dimensional wave spectra Wind-Generated Waves / 169 models have been proposed. They generally have the form of Eq. (6.11) and are derived from empirical Wts to selected sets of wave measurements, supported by dimensional and theoretical reasoning. Four of these spectral models–called the Bretschneider, Pierson–Moskowitz, JONSWAP, and TMA spectra–will be pre- sented. These models are of interest from an historic perspective and because of their common use in coastal engineering practice. The Wrst 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 ag 2 (2p) 4 T 3 e À0:675(gT=2pWF 2 ) 4 (6:18) where W is the wind speed at the 10 m elevation and a ¼ 3:44 F 2 1 F 2 2 F 1 ¼ gH 100 W 2 F 2 ¼ gT 100 2pW H 100 and T 100 denote the average wave height and period. The parameters F 1 and F 2 are a dimensionless wave height and dimensionless wave period, respectively. 0.001 0.01 1.0 1.2 1.4 1.6 Maximum Average H s H mo Intermediate Deep 0.1 d/g T 2 p Figure 6.7. Comparison of H s and H mo versus relative depth. (Thompson and Vincent, 1985.) 170 / Basic Coastal Engineering As will be shown in the next section, Bretschneider empirically related F 1 and F 2 to the wind speed, the fetch, and the wind duration to develop a forecasting relationship for the average wave height and period and, using Eq. (6.18), the wave period spectrum. Inserting a, F 1 , and F 2 into Eq. (6.18) leads to S(T ) ¼ 3:44T 3 (H 100 ) 2 (T 100 ) 4 e À0:675(T=T 100 ) 4 (6:19) Employing Eq. (6.10), Eq. (6.19) can be converted to a frequency spectrum that would have the general form of Eq. (6.11). Ochi (1982) recommends the rela- tionship T 100 ¼ 0:77T p from empirical data and the average wave height can be related to the signiWcant height through the Rayleigh distribution. Thus, given one of the common representative wave heights and periods the Bretschenider spectrum can be plotted using Eq. (6.19). Pierson–Moskowitz Spectrum (Pierson and Moskowitz, 1964) The authors analyzed wave and wind records from British weather ships oper- ating in the north Atlantic. They selected records representing essentially fully developed seas for wind speeds between 20 and 40 knots to produce the following spectrum: S( f ) ¼ ag 2 (2p) 4 f 5 e À0:74(g=2pWf ) 4 (6:20) In Eq. (6.20) the wind speed W is measured at an elevation of 19.5 m which yields a speed that is typically 5% to 10% higher than the speed measured at the standard elevation of 10 m. The coeYcient a has a value of 8:1 Â 10 À3 . Note that the fetch and wind duration are not included since this spectrum assumes a fully developed sea. At much higher wind speeds than the 20 to 40 knot range it is less likely for a fully developed sea to occur. The following relationships can be developed from the Pierson–Moskowitz spectrum formulation (see Ochi, 1982): H mo ¼ 0:21W 2 g (6:21) f p ¼ 0:87g 2pW (6:22) The simple form of the Pierson–Moskowitz spectrum results in it being used in some situations where the sea is not fully developed. Wind-Generated Waves / 171 [...]... Eqs (6. 40) and (6. 41) Wind-Generated Waves / 185 Example 6. 6-2 For the same condition given in Example 6. 6-1, calculate the signiWcant height and peak period using the SPM-JONSWAP procedure Solution: The adjusted wind speed is WA ¼ 0:71(30)1:23 ¼ 46: 6 m=s Then, Eqs (6. 40) to (6. 42) yield  Hmo 9:81(20,000) ¼ 0:00 16 ( 46: 6)2 0:5 ( 46: 6)2 ¼ 3:4 m 9:81     9:81(20,000) 0:33 46: 6 ¼ 6: 1 s Tp ¼ 0:2 86 9:81... Tp ¼ 0:2 86 9:81 ( 46: 6)2     9:81(20,000) 0 :66 46: 6 td ¼ 68 :8 ¼ 65 80 s(1:82 hours) 9:81 ( 46: 6)2 Since the actual duration is greater than the calculated duration, the wave generation process is fetch limited and Hmo ¼ 3:4 m, Tp ¼ 6: 1 s These values are close to the values Hs ¼ 3:1 m and Ts ¼ 6: 4 s given by the SMB method and the process is fetch limited as indicated by that method 6. 8 Numerical Wave... (6: 39) where WA and W are both in meters per second Then the signiWcant height and peak period can be calculated from   gHmo gF 0:5 ¼ 0:00 16 2 2 WA WA (6: 40)   gTp gF 0:33 ¼ 0:2 86 2 WA WA (6: 41) The values determined from Eqs (6. 40) and (6. 41) use only the wind speed and fetch and are thus only for the fetch-limited condition A limiting wind duration would be calculated from   gtd gF 0 :66 ¼ 68 :8... largest height and m ¼ N for the smallest height Thus, from Eq (6. 46) , if data were collected every day for a period of one year, r ¼ 1= 365 ¼ 0:00274 years and N ¼ 365 Then, for the largest wave height in the tabulation of heights, m ¼ 1 so P(H) ¼ 1 À 1 ¼ 0:9973 365 þ 1 190 / Basic Coastal Engineering and   1 365 þ 1 ¼ 1:0027 years Tr ¼ 365 1 A useful concept, related to return period, is the encounter... deWned graphically in Figure 6. 9 Hughes (1984) further proposed that a and g in the JONSWAP spectral formulation be modiWed to   0:49 2pW 2 a ¼ 0:0078 gLp (6: 28) 174 / Basic Coastal Engineering 1.0 Φ 0.5 0 0.5 1.0 2πf Figure 6. 9 1.5 2.0 (d/g)1/2 Correction factor for TMA spectrum for the TMA spectrum In Eqs (6. 28) and (6. 29) Lp is the wave length   0:39 2pW 2 g ¼ 2:47 gLp (6: 29) for the spectral peak... wind speed, Eq (6. 23) yields Hm o ¼ 23:3 m and Eq (6. 24) yields Tp ¼ 24:3 s Thus, the condition in Example 6. 6–1 is much less than the fully developed sea condition.] 184 / Basic Coastal Engineering The JONSWAP spectrum is based on fetch-limited conditions Given the wind speed and fetch length, the wave spectrum peak frequency generated by this wind condition can be calculated from Eq (6. 26) and the spectrum... highest signiWcant height measured each week is tabulated below (wave heights are in meters): 1.05 1.70 2.94 1. 46 1.92 3.72 2.05 2.07 3.00 1.39 2.37 2. 26 2.04 2.54 2.27 1.39 3.11 1.39 4.48 1.33 1.47 3.50 2.39 2.51 1.53 2.37 2.53 1.37 0. 96 2.54 1. 36 1.15 3.05 1.80 0.82 1. 86 2.07 3.07 1 .66 3.10 1. 16 2. 56 2.84 4.53 3.31 1.58 3.35 Plot the above data [P(H) versus H] on a Gumbel plot and estimate the 10- and 50-year... Equation (6. 36) has been presented in the form of empirical equations and dimensional plots (see U.S Army coastal Engineering Research Center, 1977) Figure 6. 10 presents the relationship as a dimensionless plot as given by 1 g Hs / W2 , g Ts / 2πW gTs/2πW 10-1 gHs/W2 10-2 10-3 10 102 103 g F/W2 (solid), g td/W (dashed) Figure 6. 10 SMB wave prediction curves 104 Wind-Generated Waves / 181 Eq (6. 36) The... W2 (30)2 Figure 6. 10 then yields gHs ¼ 0:034 W2 gTs ¼ 0:33 2pW or Hs ¼ 0:034(30)2 ¼ 3:1 m 9:81 Ts ¼ 0:33(2)p(30) ¼ 6: 4 s 9:81 For the given duration and wind speed gtd 9:81(2)( 360 0) ¼ ¼ 2354 W 30 Figure 6. 10 then yields 182 / Basic Coastal Engineering gHs ¼ 0:043 W2 gTs ¼ 0:40 2pW or Hs ¼ 0:043(30)2 ¼ 3:9 m 9:81 Ts ¼ 0:40(2)p(30) ¼ 7:7 s 9:81 The smaller values, Hs ¼ 3:1 m and Ts ¼ 6: 4 s control and... probability distribution by Tr 1 ¼ r 1 À P(H) (6: 44) where r is the time interval in years between successive data points In order to plot the data a value of P(H) or Tr has to be assigned to each of the tabulated wave heights The most common approach is to use the following relationship: P(H) ¼ 1 À m N þ1 (6: 45) or Tr N þ 1 ¼ r m (6: 46) To use Eq (6. 45) or (6. 46) , tabulate the N values of wave height in . Center, 1984.) 164 / Basic Coastal Engineering Solution: From Eqs. (6. 3a) and (6. 3b) we have H rms ¼ 1:72 0:8 86 ¼ 1:94 m and H s ¼ 1:4 16( 1:94) ¼ 2:75 m From line b in Figure 6. 5 H 5 =H rms ¼. surface elevation T H Time, t Figure 6. 3. Typical water surface elevation versus time record. 162 / Basic Coastal Engineering authors (e.g., Goodnight and Russell, 1 963 ; Collins, 1 967 ; Chakrabarti and Cooley,. height–period distribution. 166 / Basic Coastal Engineering 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