Short-Term Wave Analysis

Transcription

1 Chapter 3 Short-Term Wave Analysis 3.1 Introduction In analysis of wave data, it is important to distinguish between shortterm and long-term wave analysis. Short-term analysis refers to analysis of waves that occur within one wave train 1 or within one storm (Figs..3 and 3.1). Long-term analysis refers to the derivation of statistical distributions that cover many years. To emphasize the difference between the two, they have been arranged into two separate chapters. Short-term wave analysis is discussed here and long-term wave analysis in Ch. 4. It was stated in Ch. that the complex sea surface appears to defy scientific analysis. A number of simplifying assumptions must be made to describe short-term recordings of the water surface and research has shown that a number of excellent approximations can be made (Goda, 197, 1985, ). Because there are many sizes of waves in any wave record we will need to resort to statistical analysis. We define z as the instantaneous water level related to a datum, and η as the difference between the instantaneous water and the mean water level; the values of z and η are functions of location (x, y) and time (t). A water level record such as shown in Fig. 3.1 therefore represents the process z(t) at a specific location. Water level records are normally not continuous, because they are recorded digitally. Thus z is only sampled at sampling intervals of t. A record of length t R then consists of N samples z j taken at times j t, where 1 j N. 1 Series of waves. 57

2 58 Introduction to Coastal Engineering and Management Water Level (m) T im e ( s e c ) Fig. 3.1 Water level record The water level record in Fig. 3.1 is one realization of the process z(t). We will call this z 1 (t). To understand the relevant terminology, imagine a basin of water with a wave generator at one end. We start up the wave generator steered by a certain drive signal, and after 5 minutes we measure the water level in the middle of the basin for 3 seconds. That produces a short-term record z(t) as shown in Fig Now we shut off the generator and wait for the water to become quiet. Then we start the generator up again (with the same drive signal) and after 5 minutes we measure another 3 second record at the same location. The second record is a second realization, z (t) of the same process. We could repeat this many times to produce an infinite number of realizations (records) of the process z(t), which in this case represents water surface fluctuations in the middle of a basin, after 5 minutes of wave generation. Three realizations of this process are shown in Fig. 3.. The complete set of K realizations z k (t) is called an ensemble. We can take all the values of z at t = j t in the k realizations and calculate statistical parameters such as ensemble mean z j and ensemble standard deviation σ j, where z 1 z and 1 z z K K j = k, j σ j = ( k, j j ) (3.1) K k = 1 K k = 1 Ensemble skewness and kurtosis can also be determined. If none of these ensemble parameters vary in time, the process is called stationary; if only the ensemble mean and standard deviation are constant, the system

3 Chapter 3 Short-Term Wave Analysis 59 Water Level (m) Tim e (sec) Water Level (m) T im e (sec) Water Level (m) Tim e (sec) Fig. 3. Ensemble of three realizations of a stationary and ergodic process

4 6 Introduction to Coastal Engineering and Management is said to be weakly stationary. If the time average for each realization z k (t) is equal to the ensemble average z k, where N K N zk ( t ) = z and z = z, N K N (3.) j k k j j= 1 k= 1 j= 1 the process is called ergodic. For the ensemble in Fig. 3., the process is stationary and ergodic. It is stationary because, for example, z j and σ j, do not increase with time and ergodic because z k (t) for each realization is equal to z k, the ensemble average. What does this mean in practice? A water level record is always only a single realization of the process to be studied. We have no other realizations. We cannot turn off and later re-play the same situation as we could in the wave basin example. Therefore any record is only an approximation of the process. Weak stationarity can only be inferred from this single wave record if z and σ do not vary with time (there is no trend in the mean water level and the wave heights). Finally, with only one realization, we can never show that the process is ergodic; we simply must assume ergodicity as also discussed in Kinsman (1965). 3. Short-Term Wave Height Distribution To determine wave heights, it is necessary to use η, the difference between the water level and the mean water level. It is usual to think of η as a superposition of an infinite number of small waves, each generated by its own wind eddies at different locations and at different times. The resulting sea surface is, therefore, the sum of a large number of statistically independent processes, and common sense would tell us that it is impossible to predict the exact value of η at any time or location. In other words, η is a random variable. The probability that η has a certain value is called the Probability Density Function (PDF), p(η). The Central Limit Theorem states that the PDF for a sum of many independent variables is Gaussian, which means that p(η) can be described by the Normal Distribution. The overall behavior of p(η) may be summarized by its mean, η, standard deviation, σ, and some additional statistical

5 Chapter 3 Short-Term Wave Analysis 61 parameters such as skewness and kurtosis. Most often a two-parameter normal distribution is used, defining p(η) by η and σ only. By definition, η = and therefore 1 -η p( η )= exp σ π σ (3.3) where σ is the standard deviation of the process, η(t). It is equal to the square root of the variance of η. σ 1 η 1 η (3.4) N = t= tr η = limt dt R t j t = = R N j= 1 If the wave frequencies all occur within a narrow frequency band, (if the wave periods do not vary greatly) it may be shown theoretically (Longuet-Higgins, 195; Cartwright and Longuet-Higgins, 1956; Benjamin and Cornell, 197) that the PDF of the maximum instantaneous water levels is: η max -η max max exp p( η ) = σ σ (3.5) If it is assumed that for waves of a narrow frequency band the wave height H is equal to η max, then the PDF for H becomes: 1 H -H p(h) = exp 4 σ 8 σ (3.6) Equations 3.5 and 3.6 are known as Rayleigh distributions. The distribution of Eq. 3.6, multiplied by H is shown in Fig To determine the Cumulative Distribution Function (CDF) of wave heights, the Rayleigh distribution is integrated to yield the probability that any individual wave of height H' is not higher than a specified wave height H P(H < H) = p(h) dh = 1-8σ H -H exp (3.7)

6 6 Introduction to Coastal Engineering and Management (H).p(H), P, Q Q H/σ Fig. 3.3 Rayleigh distribution P (H).p(H) The probability of exceedence, the probability that any individual wave of height H' is greater than a specified wave height H may be obtained as -H Q(H > H) = 1 - P(H < H) = exp 8σ (3.8) The functions P and Q are also shown in Fig Research has shown that for practically all locations the wave height distribution is reasonably close to a Rayleigh distribution. One exception is in shallow water when the waves are about to break. The wave height with a probability of exceedence Q, may be determined from Eq. 3.8 as H Q 1 1 = 8 σ (-ln Q) = 8 σ ln = σ ln Q Q (3.9) To determine H Q, the average height of all the waves that are larger than H Q in a record or a storm H Q = H Q H p(h) dh Q (3.1)

7 Chapter 3 Short-Term Wave Analysis 63 Equation 3.9 and the numerical evaluation of Eq. 3.1 yield Table 3.1, in which a number of common wave height definitions are related to σ. Table 3.1 Commonly used wave height parameters Symbol Description Value H.1 Average of highest 1% of the waves 6.67 σ H.1 Height, exceeded by 1% of the waves 6.7 σ H. Average of the highest % of the waves 6.3 σ H. Height, exceeded by % of the waves 5.59 σ H.1 Average of highest 1% of the waves 5.9 σ H.1 Height exceeded by 1% of the waves 4.9 σ H s = H Significant wave height (Average height of the 1/ 3 highest 1/3 of the waves) 4. σ H Average wave height π σ H.5 Median wave height.35 σ Hmode Most probable wave height. σ Hrms H + H + H N σ Of all these definitions, Significant Wave Height (H s ) is the most important. It is defined as the average of the highest 1/3 of the waves in a wave train, H 1/ 3. In terms of significant wave height, four commonly used relationships based on the Rayleigh distribution are H = 1.7 H ; H = 1.67H.1 s.1 H =.63 H ; H =.77H s rms s s (3.11) The meaning of average wave height is self-explanatory. The modal or most probable wave height is the wave height with the greatest probability of occurrence. The median wave height has 5% probability, i.e., half the waves in the wave train are higher and half the waves are This wave height definition was historically chosen as significant because it comes closest to the traditional estimates of average wave height made by experienced observers before we had instruments to measure wave heights.

8 64 Introduction to Coastal Engineering and Management lower than this wave height. The rms wave height is the constant wave height that represents the wave energy. The probability of exceedence for the average wave heights may now be calculated from their values in Table 3.1, using Eq The results of this calculation are shown in Table 3.. Table 3. Probabilities of exceedence of average wave heights H H. 1 H. 1 H s H H mode Q(H >H) The expected value of the maximum wave in a wave train of N w waves could be estimated by setting Q = 1/N w in Eq A more accurate estimate is µ γ -3/ = ln N w + + O[(ln N w ) ] σ ln N Hmax w (3.1) where µ(x) denotes expected value of x, γ is the Euler constant (=.577) and O(x) denotes terms of order greater than x, i.e. small terms. Example 3.1 Calculation of short-term wave heights To analyze a wave record it must be stationary. Hence, it is normal to record waves for relatively short time durations (1 to minutes). A longer record would not be stationary because wind and water level variations would change the waves. Thus it is usual to record, for example, 15 minutes every three hours. It is subsequently assumed that the 15 min. record is representative of the complete three hour recording interval. Suppose the analysis of such a record yields T = 1 sec and σ = 1. m (3.13)

9 Chapter 3 Short-Term Wave Analysis 65 We want to calculate significant wave height H s, average wave height, H, average of the highest 1% of the waves H.1, and the maximum wave height in the record. From Table 3.1: H s = 4σ = 4. m, H = πσ =.5 m and H.1 = 6.67σ = 6.7 m With T = 1 sec, the average number of waves in the 15 min record N w = 9 and Eq. 3.1 yields Hmax.577 µ = ln (9) + = = 3.19 m σ ln (9) (3.14) or the expected value of H max = (3.19)()(1.) = 6.4 m. This calculated value of H max can be verified against the actual record. If the record is representative of a 3 hour recording interval, then T, H s, H and H.1 for the 3 hours would be the same as above. However, H max would be larger than 6.4 m. For the 3 hour recording interval, N w = 18 and Eq yields H max,3hrs = 7.8 m. 3.3 Wave Period Distribution In the above discussion the frequency bandwidth for the waves was assumed to be small (the wave periods are more or less the same) and in practice, wave period variability is often ignored. One attempt to define wave period distribution has been made by Bretschneider (1959) who postulated that the squares of the wave periods form a Rayleigh distribution. His expression for the PDF for wave periods is p(t) =.7 T T T exp T (3.15) From this, by integration, the expression for probability of exceedence of a certain wave period becomes 4 Q(T >T) = p(t)dt = T exp T T (3.16)

10 66 Introduction to Coastal Engineering and Management Wave periods are related to wave heights particularly in a growing, locally-generated sea, where high wave heights are always accompanied by long wave periods. A joint distribution of wave heights and periods can be postulated and that is normally assumed to be a joint Rayleigh distribution. 3.4 Time Domain Analysis of a Wave Record Wave recordings are time series of water levels that typically look like Fig They are discrete time series z(t), sampled at N short intervals of t. The water level recording must first be converted into a discrete time series η(t), the fluctuation about mean water level by subtracting the mean water level from the record. Although the record is assumed to be stationary, there may be a small change in mean water level with time, as is the case in Fig. 3.4, where the water level drops slightly. This could be, for example a result of tides. Because the record is short ( min), the water level fluctuation is assumed to be a linear function of time, z = a + bt. It is determined from the record by regression analysis and then subtracted, so that η = z z = z ( a + bt). Figure 3.5 presents the η time series for Fig The bottom graph is for the first 1 seconds and shows more detail Water Level - z (m) Time (sec) Fig. 3.4 Water level record

11 Chapter 3 Short-Term Wave Analysis 67 As an initial analysis of a record, we could simply calculate σ, using Eq The wave analysis program WAVAN was used 3 and σ was found to be.8 m for the record in Fig If a Rayleigh distribution of wave heights is assumed (Tables 3.1 and 3.), we can determine the values for the important wave heights for Fig H = 1.1 m, H =.7 m, H =.79m and H = 1.43m s rms.1 (3.17) η (m) Time (sec) η (m) Time (sec) Fig. 3.5 Wave record for Fig An alternate program ZCA was also provided with the software.

12 68 Introduction to Coastal Engineering and Management This initial analysis does not tell us anything about wave period, the other important wave parameter. The actual distributions of H and T may be obtained from the record by analysis of individual waves. Figure 3.6 shows a very short segment of Fig The earliest definition for wave height H is the vertical distance between a wave crest and the preceding trough (Fig. 3.6) where crest and trough are defined as a local maximum and a local minimum in the record. That definition would result in all the small ripples being identified as waves. How many waves are there between t = and 6 sec in Fig. 3.6? To define wave height more realistically, zero down-crossing wave height, H d, is defined as the vertical distance between the maximum and minimum water levels that lie between two subsequent zero down-crossings (in which η crosses zero on the way down). Similarly zero up-crossing wave height, H u, is the difference between maximum and minimum water levels between two subsequent zero up-crossings. These definitions are also shown in Fig They disregard the small ripples that do not cross the mean water level. Example 3. presents the zero crossing analysis of the wave record of Fig. 3.5 and compares the results with the values in Eq x.8.6 η (m) Eta (m) x H u x H x H d x Time (sec) Fig. 3.6 Wave height definitions

13 Chapter 3 Short-Term Wave Analysis 69 Example 3. Zero crossing analysis of Fig. 3.5 The zero up-and down crossing wave heights in Fig. 3.5 were determined using WAVAN or ZCA. The two estimates were virtually identical. The following wave statistics were obtained by averaging the up- and down-crossing results H = 1.5 m, H =.68 m, H =.76m and H = 1.3m s rms.1 (3.18) When these calculated values are compared with Eq it is seen that Eq overpredicts for this record. The wave heights were also grouped into 1 bins and the histogram of the wave heights is shown in Fig. 3.7a. This distribution can be compared with the Rayleigh distribution. It would also be possible to plot the cumulative distribution function (P, as in Eq. 3.7) or the probability of exceedence (Q, as in Eq. 3.8). However, since the wave height distribution is expected to be Rayleigh, it is best to compare wave heights directly with this theoretically expected distribution. Equation 3.8 may be re-written as Thus, H ln( Q) = σ (3.19) 1/ H R = { ln( Q)} = or H = ( σ ) R (3.) σ where R is called the Rayleigh parameter. A true Rayleigh distribution would plot as a straight line with zero intercept and a slope of σ on a graph of H versus R. Values of R were calculated for each wave height bin and H versus R was plotted in Fig. 3.7b. The solid line in Fig. 3.7b represents the initial analysis, combining σ =.8 m with Tables 3.1 and 3.. This analysis clearly overpredicts the actually measured values of H.

14 7 Introduction to Coastal Engineering and Management Number of Waves W ave Height - H (m ) y = H=.69R.6854x Wave Height - H (m) Rayleigh Parameter - R a) b) Fig. 3.7 Histogram and Rayleigh distribution The best fit line through the measured values has a slope of.69 and from this; using Eq. 3., another estimate for σ may be computed σ z =.69 / ( )=.4m. This version is called σ z since it was determined by zero-crossing analysis. An estimate of average wave period may be obtained by dividing the record length (t R = 1 seconds) by the number of waves (N w = 197) to find T = 6.1 seconds. Finally H max was found to be 1.56 m from the record. This can be compared to the theoretical value of H max = 1.9m, obtained from Eq. 3.1, using σ =.8 m. Using σ z =.4 m yields a more reasonable value of H max = 1.65 m. 3.5 Frequency Domain Analysis of a Wave Record A completely different type of analysis, based on wave frequencies, is called wave spectrum analysis. We use the statistical assumptions that the wave record is both stationary and ergodic. Although these assumptions are necessary to perform a statistically correct analysis, in practice we have no choice but to assume that records are short enough to be both stationary and ergodic. To express the time signal η in terms of frequency, we can use a Fourier series summation for each value of η j

15 Chapter 3 Short-Term Wave Analysis 71 η = a + a cos( π f t ) + b sin ( π f t ) Using Euler s relationship Eq. 3.1 becomes (3.1) j o n n j n n j n= 1 n= 1 e iψ j = cosψ + isinψ (3.) n= where C n is a complex coefficient n i π ( fnt j ) η = C e (3.3) 1 1 Co = ao ; Cn = ( an ibn ); C n = ( an + ibn ) = Cn * (3.4) Equation 3.3 expresses the time signal η in terms of discrete frequency components and is known as a Discrete Fourier Transform (DFT). The series in Eqs. 3.1 and 3.3 are infinite. However, a wave record, such as in Fig. 3.5 is neither infinitely long, nor continuous. The water level is sampled only at N specific times, t apart. There are N values of (η j ) at times t j = j t, where 1 j N. As a result, the frequency domain is also not continuous. The smallest frequency that can be defined from a record of length t R is f min = 1/t R. To provide the most accurate representation in the frequency domain (best resolution) we will use this smallest possible frequency as the frequency increment. Therefore f = 1/t R and we define f n = n f. The highest frequency that can be defined from a time series with increments t is the Nyquist frequency f N = 1 N N f t = t = This results in the finite discrete Fourier transform (FDFT) j N N n= + 1 r i[ π ( fnt j )] n (3.5) η = F e (3.6)

16 7 Introduction to Coastal Engineering and Management Because F n = F -n *, Eq. 3.6 may also be written as: η The inverse of Eq. 3.7 is F n N 1 i[ ( fnt j )] j = Fn e π (3.7) n= N 1 i[ π ( fnt j )] η je N j= 1 = (3.8) Equations 3.7 and 3.8 form a FDFT pair that permits us to switch between the time and frequency domains. Equation 3.8 allows us to calculate the complex frequency function F n from a real time function η j and Eq. 3.7 permits calculation of the real time function η j from the complex frequency function F n. The complex variable F n may also be expressed as where F n i n = F e θ (3.9) b F = a + b and = a 1 1 n n n θn tan Substitution of Eq. 3.9 into Eq. 3.7 results in η n n n (3.3) N 1 i[ ( fnt j ) n ] j Fn e π = θ (3.31) n= In practice waves have only positive frequencies, only frequencies lower than f N can be defined, and η j is real. Therefore, using Eq. 3., we can rewrite Eq as where N 1 η = F cos( π ft θ ) j n n n= N / N / (3.3) = F cos( π ft θ ) = A cos( π ft θ ) n n n n n= n= A = F (3.33) n n

17 Chapter 3 Short-Term Wave Analysis 73 Here A n is called the amplitude spectrum and θ n the phase spectrum. In standard wave spectrum analysis only the amplitude spectrum A n is calculated. In effect, θ n is assumed to be a random variable -π < θ n < π resulting in the random phase model. This unfortunate assumption loses all phase relationships between the N terms, which means for example, that wave groups are not reproduced when calculating η j from A n using random θ n. Resonance and reflection patterns are also not reproduced. Parseval s theorem can be used to calculate σ from the amplitude spectrum A because n σ N 1 N / Cn Fn An n= n= n= (3.34) = η = = = Thus the variance at any frequency can be expressed as 1 1 S( fn ) df = An or S( fn ) = An (3.35) df where S(f ) is known as the wave variance spectral density function or wave spectrum. Variance is a statistical term. In physical terms, wave energy density is 1 E = ρ g σ (3.36) and hence wave energy distribution as a function of frequency is E( f ) = ρ g S( f ) (3.37) Wave spectra for Fig. 3.5 were computed using WAVAN and are shown in Fig Because we always have only one realization of the process and the record length (t R ) is finite, resulting in finite increments of frequency ( f ), the calculated value of S(f ) is always an estimate of the true S(f ). The wave spectrum for the record in Fig. 3.5, using Eqs. 3.3 to 3.35, is shown in Fig. 3.8a. This spectrum gives the maximum possible resolution and distinguishes between frequencies that are f = 1/t R = 1/1 =.83 Hz apart. It contains many closely spaced spikes of 4 An alternate program WSPEC was also provided with the software.

18 74 Introduction to Coastal Engineering and Management S(f) (m/hz) Frequency (Hz) Frequency (Hz) a) b) S(f) (m/hz) S(f) (m/hz) S(f) (m/hz) Frequency (Hz) Frequency (Hz) c) d) Fig. 3.8 Wave spectra of the wave record in Fig. 3.5 wave energy. Physically, such very local energy peaks are not possible. They are a result of the uncertainties in our estimates and therefore the wave spectrum is smoothed. Smoothing can be done by averaging S(f ) over frequency ranges longer than f so that m= M / 1 S( f ) = An + m where df = M f f (3.38) df m= M / where df is the resolution of the spectrum and M f denotes how many values of f are averaged. The results for M f = 6 and 1 (df =.5 and.1 Hz) are shown in Figs. 3.8b and c. Smoothing produces a more regular spectrum. The amount of smoothing to be used depends on the purpose of the analysis. If a general impression of a wave field is needed, Fig. 3.8c is most useful. If specific frequencies need to be identified, then

19 Chapter 3 Short-Term Wave Analysis 75 less smoothing such as in Fig. 3.8b may be more appropriate. Another method used for smoothing the spectrum is to divide the record into shorter sections, compute the spectrum for each section and then average the results. Figure 3.8d, presents the average of 4 spectra, each for ¼ t R. The resolution is now 4/t R =.33 and averaging the four spectra introduces some further smoothing. The value of σ, as computed from the frequency analysis by integrating the spectrum is denoted as σ f because it was derived by frequency analysis. Because Eqs. 3.4 and 3.34 both integrate η, σ f for all four spectra in Fig. 3.8, as well as σ found using Eq. 3.4, are the same and equal to.8. Only the basic principles of wave spectrum are presented here. There are other methods of computing the wave spectra and dealing with smoothing of the spectra. Further details may be found in the literature, such as Bendat and Piersol (1966) and ASCE (1974) and Janssen (1999). Frequencies that exceed the Nyquist frequency (Eq. 3.5) cannot be defined as separate frequencies. The energy in these high frequency waves is superimposed on the spectrum by a process known as aliasing. Figure 3.9a demonstrates aliasing in the time domain. A wave of frequency 1.1 Hz is sampled at t = 1., for which f N =.5 Hz. It is seen that the sampled signal (square points) does not have a frequency of 1.1 Hz, but of.1 Hz. The energy of such a wave component would therefore become added at.1 Hz in a wave spectrum. Aliasing in the frequency domain is depicted in Fig. 3.9b. Aliasing can be prevented by filtering frequencies greater than f N out of the signal. Alternately, if f c is the highest frequency that must be computed correctly (without aliasing), then it is reasonable to assume that f c = f N /. That defines the necessary sampling interval for the record as t 1/(f N ) = 1/(4f c ). For wind waves, if we wish to define the spectrum correctly for all frequencies f < Hz, t should be less than 1/8 sec.

20 76 Introduction to Coastal Engineering and Management η t a) S(f) (m/hz) Aliased Spectrum Correct Spectrum Frequency - f (Hz) b) Fig. 3.9 Aliasing of a wave spectrum by high frequency components 3.6 Parameters Derived from the Wave Spectrum The moments of the wave spectrum are defined as m h f = h = f S( f ) df (3.39) f = The zero moment (n = ) is therefore the area under the spectrum m o f = S f df σ f (3.4) f = = ( ) =

21 Chapter 3 Short-Term Wave Analysis 77 From the area under the wave spectrum, assuming the wave height distribution to be Rayleigh, the various wave heights of Table 3.1 may be estimated as in Eq To distinguish between significant wave height derived from time domain analysis and its counterpart derived from frequency analysis, the latter is called the Characteristic Wave Height or Zero Moment Wave Height. H = H = 4σ (3.41) ch mo and for Fig. 3.5, H mo = (4)(.8) = 1.1 m. The representation of the wave energy distribution with frequency is an improvement over the time-domain analysis methods discussed earlier. With this information we can study resonant systems such as the response of drilling rigs, ships moorings, etc. to wave action, since it is now known in which frequency bands the forcing energy is concentrated. It is also possible to separate sea and swell when both occur simultaneously (Fig. 3.1). f Sea S(f) (m/hz) Swell Frequency - f (Hz) Fig. 3.1 Wave spectrum with sea and swell The moments of the wave spectrum also define spectral bandwidth m ε = 1 (3.4) mom4

22 78 Introduction to Coastal Engineering and Management Cartwright and Longuet-Higgins (1956) show that for a narrow bandwidth (ε ) all wave periods in a wave train are almost the same and the distribution of η is purely Rayleigh 5. For ε 1, the distribution of η is random. This would obviously affect the wave height definitions used in Tables 3.1 and 3.. For the record in Fig. 3.5, ε was calculated to be.65. Since there are many wave frequencies (or wave periods) represented in the spectrum it is usual to characterize a wave spectrum by its peak frequency f p, the frequency at which the spectrum displays its largest variance (or energy). The peak period is then defined as T p 1 = (3.43) f Other spectrum-based definitions of wave period found in the literature are 1 m1 m p m m T = ; T = (3.44) Theoretically T is approximately equal to T, as obtained by zero crossing analysis. For the spectra in Fig. 3.8, f p = 1.3 Hz, if the narrow peak in Fig. 3.8a is discounted. Thus T p = 7.6 seconds and T 1 = 6.4 seconds and T = 6.1 seconds. T was also 6.1 seconds in Ex. 3.. In Fig. 3.1 we can distinguish peak f p and T p values for both the sea and the swell. Rye (1977) indicates that the moments for the spectrum are functions of the cutoff frequency (the highest frequency considered in the analysis) and thus ε, T 1 and T should be viewed with caution. Sometimes the angular frequency, ω, is used to define the frequencies in the wave spectrum. The total variance for the S(ω) spectrum is = 1 ω= σ ω ( ) S ω d ω π (3.45) ω= The results of the three methods of analysis for the waves of Fig. 3.5 are compared in Table The Rayleigh distribution is in fact based on the assumption that ε (Sec. 3.).

23 Chapter 3 Short-Term Wave Analysis 79 Table 3.3 Comparison of analysis methods Analysis Initial Zero Crossing Frequency σ (m) σ =.8 σ z =.4 σ f =.8 N 197 T (sec) T = 6.1 T p = 7.6 T 1 = 6,4 T = 6.1 H s (m) H mo = 1.11 H (m) H rms (m) H.1 (m) H max (m) ε Uncertainties in Wave Measurements At this point, we reflect upon how well we know the basic wave parameters: wave height, wave period and wave angle. First we define uncertainty. It quantifies the combination of errors, randomness and general lack of physical understanding. For most physical quantities, errors increase with the magnitude of the quantity. For example, the absolute error in measuring a wave height of.5 m will be less than the absolute error in measuring a wave height of 5 m. For this reason we normally use a relative error to define the accuracy of our quantities. The errors in a quantity such as wave height H are assumed to have a normal distribution with H as its mean value and σ H as its standard deviation. The uncertainty in H is defined as its coefficient of variation σ H σ ' H = (3.46) H More detail about uncertainties may be found in Thoft-Christensen and Baker (198), Ang and Tang (1984), Madsen, Krenk and Lind (1986), Pilarczyk (199), Burcharth (199) and PIANC (199). From the definition of standard deviation, H is between H (1 ± σ ' ) 68% of the H

24 8 Introduction to Coastal Engineering and Management time, there is a 95% probability that H is between H (1 ± σ ' H), and virtually all values lie between H (1 ± 3 σ ' H ). Wave heights are based on measurements of instantaneous water levels, usually measured offshore, at frequent intervals (e.g. 1 Hz) over a recording period (e.g. 1 to Minutes). Zero Crossing or Wave Spectrum Analysis is then used to reduce the instantaneous water level measurements to one single wave height value (H s or H mo ) to represent the complete recording period. Along an exposed coast H s = 1 m would be typical. Even for very carefully measured instantaneous water levels, using the latest equipment H s = 1 m would contain an absolute error (standard deviation) σ H,Measured =.5 to.1 m (say.75 m). The uncertainty in a 1 m wave height would therefore be σ H,Measured =.75. The errors in measuring smaller waves would be less and for larger waves they would be greater. Therefore an uncertainty σ H,Measured =.75 would not be unreasonable for wave height measurements. This means that for H = 1 m, there is a 68% probability that.9 < H < 1.8 m, a 95% probability that.85 < H < 1.15 m and that almost all values of H lie between.78 < H < 1.. Such relatively accurate offshore wave height measurements are subjected to several conversions before they can be considered useful for subsequent computations. So far, the value of H s = 1 m represents 1 to minutes of record. For the 1 m wave height to represent a complete recording interval of 3 to 6 hours, it must be remembered that the environmental parameters such as wind speed and direction, water levels, etc. are not constant over the recording interval. This increases the uncertainty of the representative wave height values. The additional uncertainty depends on the variability of the conditions over the recording interval, but in most cases it would be reasonable to expect the uncertainty to double so that σ H,Interval =.15. Uncertainty of measured wave periods T is known to be greater than for H and reasonable estimates would be σ T,Measured =.1 and σ T,Interval =.. Wave direction (α) is notoriously poorly measured. Even the best directional instrumentation has difficulty to produce wave directions within ± 3 for large, well-formed waves and may be as much

25 Chapter 3 Short-Term Wave Analysis 81 as 1 wrong for smaller, more irregular waves. Estimates of wave direction by means other than directional measurement are much worse. Assuming the values of 3 and 1 to be maximum values of the errors in angle (assuming these values to be 3σ α removed from the mean), σ α can be estimated as 1 in the first case and 3.3 in the second. Thus an average value of standard deviation is σ α,measured =. This value of σ α is independent of the incident wave angle and hence we cannot define σ α. However, in order to complete our subsequent discussion about uncertainties, we will relate σ α to an incident wave angle of 1 with respect to the shoreline. In that case, σ α,measured becomes.. Many times the incident wave angle on a sandy shore is much smaller than 1, which would result in much higher uncertainty values. When the wave angle with the shoreline approaches (as is often the case), the uncertainty for wave angle approaches infinity and the whole discussion about uncertainty loses its meaning. For the longer interval and α = 1, a reasonable estimate is σ α,interval = 4. These uncertainty values are only general indications. They are heavily influenced by assumed average values for wave heights and periods, and particularly angles of breaking. The actual values are not as important, however, as the fact that they clearly indicate that the basic coastal data wave heights, wave periods and incident wave angles contain large uncertainties. Since these wave quantities are basic to all coastal design calculations, the effects of these uncertainties will pervade all subsequent calculations. The awareness of uncertainties is basic to our understanding of the fundamental issues of coastal engineering and management. For example, it explains why we can use small amplitude wave theory successfully for most design calculations. The discussion and evaluation of uncertainties will be extended in later chapters. 3.8 Common Parametric Expressions for Wave Spectra Since the measured spectra show considerable similarity (they basically consist of a peak and two curves decreasing toward f = and f = ), attempts have been made to formulate parametric expressions.

26 8 Introduction to Coastal Engineering and Management Only the most common expressions will be presented here. Phillips (1958) postulated that for the equilibrium range (for f > f p ) the spectral shape S(f ) is proportional to f -5. He quantified his results as α P -5 Φ P = g f (3.47) 4 ( π ) where Φ p denotes the Phillips Function and the Phillips constant is α P =.74 (3.48) Pierson and Moskowitz (1964), added a low frequency filter to extend the Phillips expression over the complete frequency range where: Φ S PM ( f ) = Φ P Φ PM (3.49) PM 5 f = exp 4 f p The Pierson-Moskowitz spectrum is therefore S PM α ( f ) = ( π ) PM g 4 5 Commonly used expressions for α and β are: f e 4 4 ( ) (3.5) β / f (3.51) 5 4 αpm =.81; β = f p (3.5) 4 The quantity β was also related to wind speed U so that this spectrum can be used to hindcast waves from wind data (Ch. 5) g β =.74 πu 4 (3.53) where U is the wind speed. The Pierson-Moskowitz spectrum is valid for a fully developed sea condition. For developing seas the Jonswap Spectrum was proposed by Hasselmann et al (1973). It is essentially an enhanced Pierson-Moskowitz spectrum as shown in Fig

27 Chapter 3 Short-Term Wave Analysis 83 S(f) (m/hz) γ Jonswap Frequency - f (Hz) Pierson- Moskowitz Fig Jonswap and Pierson-Moskowitz spectra A developing seas filter, Φ J, can be assumed so that the Jonswap spectrum is where and Typical values of δ are S J( f ) = Φ P Φ PM Φ J (3.54) a e Φ J = γ (3.55) ( f f p) a = δ f p δ =.7 for f f δ =.9 for f > f The Jonswap expression is therefore p p (3.56) (3.57) e a e α J g γ S J( f ) = γ S PM ( f ) = 4 5 e ( π ) f a f f p (3.58)

28 84 Introduction to Coastal Engineering and Management The coefficient α can be related to the wave generating conditions gf α J =.76 U. where F is fetch length. Alternately, Mitsuyasu (198) states and The peak enhancement factor gf αm =.817 U 1 f p gf =.84 U /7.33 (3.59) (3.6) (3.61) S J( f p) γ = (3.6) S ( f ) PM has an average value of 3.3 and typically lies between 1 and 7. Mitsuyasu (198) postulates p = 7. gf γ U 1 1/7 (3.63) The above development traces the derivation of a parametric expression for a wave spectrum from the equilibrium spectrum (Phillips) through the fully developed sea spectrum (PM) to the developing sea spectrum (Jonswap). But waves have a limiting steepness. Thus, any wave in a wave train that reaches a limiting steepness will break. This is known as Spectral Saturation. Bouws et al (1985) modify the Jonswap spectrum to take spectral saturation into account and produce the TMA spectrum. where S TMA ( f,d) = Φ Φ Φ Φ (3.64) P PM J d 1 d tanh π d Φ = (3.65) n L

29 Chapter 3 Short-Term Wave Analysis 85 In deep water, the value of Φ d is one. In other words, the Jonswap spectrum through its own derivation takes into account the deep water wave steepness limitation and Eq modifies the Jonswap spectrum for wave breaking induced in shallow water. 3.9 Directional Wave Spectra Until now η has been considered to be a function of time at a single location and we learned to calculate S(f ) from such a time series. We also discussed some parametric expressions for such wave spectra. However, η is also a function of direction (of x and y). Measurement of wave direction involves correlating spectra for several synoptic, adjacent records of water levels, pressures and/or velocities. Some discussion may be found in Sec..3.1, but detailed discussion is beyond the scope of this book. An example of a Directional Wave Spectrum (a function of both wave frequency and direction) is shown in Fig. 3.1 and a good description of directional wave spectra may be found in Goda (1985). To describe such a spectrum, the simplest approach is S( f, θ ) = S( f ) G( θ ) (3.66) G is called the Directional Spreading Function and θ is measured counter-clockwise from the wave direction. A necessary condition is obviously θ = π G ( θ ) d θ = 1 (3.67) θ = π Two common directional spreading functions used are the Cos-Squared function π G ( θ ) = cos θ for θ < π G( θ ) = for all other values of θ and the Cos-Power function (Mitsuyasu, 198; Goda, 1985). (3.68)

30 86 Introduction to Coastal Engineering and Management S η (f,θ) θ f Fig. 3.1 Directional wave spectrum

31 Chapter 3 Short-Term Wave Analysis 87