VISUAL WAVE OBSERVATIONS ESA WILLARD J. PIERSON, JR. Department of Meteorology and Oceanography New York University March 1956 Cae: Published under the authority of the Secretary of the Navy U. S. Navy Hydrographic Office i Washington, D. C. Price 35 cents au Hen. ABSTRACT Visual wave observations will always be needed to supplement instru-. mental records. Ordinary visual observations are not adequate for estimation of the spectral components of the short-crested surface waves, but by the use of more precise visual observations utilizing all the waves passing a point, it is possible to determine the wave characteristics. The most pentane eRe neh wave characteristic is wave height. The theory of a wa he relationship spectrum. It is de derived both ributions using only wa es of recorded visual ob minimum of 50 ave height can 2 more difficult ve distribution ve length”’ can be estimated FOREWORD The Hydrographic Office recently published a new wave forecasting manual, H. O. Pub. No. 603, prepared by Professor Pierson and his colleagues at New York University under contract with Project AROWA. To supplement this manual the Hydrographic Office herewith presents “Visual Wave Observations,” also by Professor Pierson, as an ex- planation of the methods for obtaining wave observations in a manner compatible with the spectral forecasting technique. Wave records are of two major types. Those obtained from wave staffs, pressure recorders, and other mechanical devices are accurate and reproducible, but they are also expensive and limited in number. Visual wave observations are subject to error, but they are readily obtained from shipboard as often as desired. In spite of the errors inherent in subjective estimates of wave characteristics, several important types of data can be secured from visual wave observations which can aid in wave forecasting. This report outlines the theory underlying visual wave observations and indicates the data that can be secured from them. The Hydrographic Office is actively engaged in the development and operational testing of methods of wave forecasting. In order to increase the usefulness of the operational wave forecasts being issued by this Office, it is necessary to obtain more frequent and accurate synoptic wave reports. It is hoped that this report, which indicates how improved visual observations can be obtained, will encourage observers aboard ships to make observations in the method outlined. We Merle. H, H. MARABLE Captain U.S. Navy Hydrographer Ui TT 0301 0040 CONTENTS Page REO THEO Tee tered tosyereteioterote okereletatetorelevorsteleleketeicrelatcrelerovevetate tole roteleveleielalatelsicleiela|\eiaisials/<1@1s\«]slo\e)*/<\«0)= iii PN UTS) S etre mparestey re evetsvalerarcteterevetetareretslelerelete rel elote elereierelefeietalelalelereleieleleleve aiciela]ie/aiele\e nleleleleie/s/e\sieie's vi TAIDIIGS coseccanohoaasboousno oooann oandéq0Dbb00d00anso0ubb00DbOLODooSdo Hons bOOOddO bo oObOE vi I ITO GUC EVOMMr rer retere sleretelrecletecisicieielsioreletesieteisteiierelsieisieristeieielelotere oneictelsieiclerelcterianc's 1 II. Theoretical and Practical Aspects of Wave Height O)SBSIAVANBIOING sodoucos soboq00c0ddoonn500G0G00G0000 00000 odo ccobocobubaeGUD6uE 1 A. Techniques for Wave Height Observations................++++ 1 Bade he hheomynofeawWiasvie see CONG wisccseielonlole elocioielelselscisisielolsiele evetetociere 4 Cvalsulala eet hte @ b's © ravi tl OMS Me erernsclareloi fal s)olo : — > 7 “ei. >, S : ~<- Pte So ee “it. oe Se wave by stereophotography. That is, a dominant wave could be selected in the stereo pair, and it could be followed along the crest until its highest part was found. The crest-to-trough height would then be found at this point. A large number of such observations could be made. The average of these values would then give some sort of average height. Similarly, when a wave observer looks out over the sea surface, he tends to look at the highest part of each of the short-crested waves within his field of view. His eye skips about over the sea surface, and thus the values recorded are similar to the values described above. The theory of the distribution of the values obtained in the observation of the highest part of each of the short-crested waves in the field of view of the observer has not yet been solved. The theoretical proba- bility distribution of such heights is unknown, and it appears that it will remain unknown until some fundamental problems in time series are solved. If the theoretical properties of anobserved set of values are unknown, the observations are for all intents and purposes useless. To report that the average height of the waves as observed by this technique is so many feet does not permit an estimate of the higher waves or of their frequency of occurrence. Most observations of wave heights at sea do not even possess the property of being the average of the highest part of each short-crested wave in the field of view of the observer. They are even cruder estimates of the “significant” height as made by looking out over the sea surface and guessing as to a characteristic height of the waves. Such estimates are unreliable because they depend subjectively on the observer and on the type of ship from which the observations are being made since the scale of the waves relative to the size of the ship influences the observer's choice of the characteristic height. From the above discussion, it would appear that either just looking at the waves and assigning a characteristic height or just writing down a few heights of waves scattered about over the sea surface and computing the average is not an adequate method of visual ob- servation. Knowledge of the wave height distribution, of the errors inherent in the sample size, and of source of observer error must be developed theoretically in order to make the interpretation of observed wave heights reliable. Consider the observation of the heights of all waves that pass a fixed point. Such an observation could be made instrumentally by a wave-pole recorder, or it could be made just as easily by an observer if he knew that this was the correct procedure. Some of the theoretical properties of such a series of observations are known, and therefore their accuracy can be determined. The theory which is to be given is therefore based on the observation of the heights of all waves which pass a fixed point of observation. Once the distribution of the heights of all waves is known, it then becomes possible to omit the observation of some waves in a precisely defined way and still obtain reliable results. The theory can be extended to cover the properties of the heights of all waves that pass (or are passed) by a moving point. Thus a point fixed in azimuth and distance relative to a moving ship can be used just as well as a stationary point. The heights of the waves that pass a fixed point are lower than the heights of the highest part of each short-crested wave since the side of a short-crested wave can pass the point and the highest part can pass at a distance from the point of observation. Since these heights are the same as would be encountered by a ship under way, they are also of practical importance. B. The Theory of A Wave Record An ocean wave record is a sample from a quasi-stationary Gaussian process which is completely described by its energy spectrum. Much is known about the theory of such Gaussian processes since they have been studied extensively in electronics and in communication theory by Rice (1944), Wiener (1949), and Tukey (1949). 1. The Envelope There are a number of ways to define the envelope of a wave record. For one way that is used, it can be shown that the envelope will touch each crest of the wave record only if the wave spectrum is narrow, and that the envelope is always distributed according to equation (1) as discussed below. For another way that is used, the envelope is defined to touch each horizontal part of the record, but then the probability distribution of the envelope reflects ripples and other minor (for this application) irregularities and only reduces to equation (1) for narrow spectra.* *Personal communication, R. A. Wooding; see also “Wind Generated Gravity Waves” by W. J. Pierson, Jr. (1954). 2. The Amplitudes If the spectrum of the waves is narrow, the probability distribution of the amplitudes is known (Rice, 1944). As in figure 2, a sufficient number of amplitudes read from the record will have a known proba- bility distribution function. If the spectrum is wide, the distribution is unknown. However, it would appéar from the theoretical results of Neumann (1953) that even a fully developed sea wave record will be approximately distributed in amplitudes according to this known distribution. Given, then a wave record and a set of wave amplitude observations which are from a long enough record, the amplitudes will be distributed according to the law given by equation (1) 2 g(x) dx= Ste */E ae (1) for 0 the significant height would be 28.3 feet. One wave out of 10 would be less than 6.4 feet high; compared with the more dominant waves, there would be a very strong tendency to ignore some of these low waves. When this is done, the problem is what to do with the observed values which have now become a sample from an unknown population, since the probability that the observer will ignore a given low wave is an unknown factor. The first thing that must be done is to truncate the theoretical distribution sharply. That is, equation (1) must be set equal to zero for all x less than a certain value and then correctly normalized. If the low waves are to be ignored in the tabulation, then all low waves must be ignored and not just some unknown and unspecifiable fraction of the values within a certain class interval. There are two ways to truncate the distribution. The first way is to discard a certain fixed percentage of the lowest waves of allthe waves that pass a given point. The second way is to discard all waves less than a preassigned height value. 7. Truncated Distribution at a Fixed Percentage The first way, namely discarding a certain fixed percentage of all the lowest waves to pass a fixed point of observation, is inherent in the concept of the significant height. The significant height is the average of the heights of the 33 percent highest waves to pass the point of observation. To observe the significant height correctly, the following procedure could be used. The observer would watch the waves pass the fixed point. If a high wave passed, he would note down its height. If a low wave passed he would simply make a check to note the passage of the wave. A series of recorded heights and checks would be the result. The total number of heights and checks, say M, would be counted up, and the sum divided by 3. The observed heights would then be put in 19 descending order, and the highest M/3 values selected. The average of the M/3 highest waves would then be the significant height. As an example, consider the series of recorded values given below as they may have occurred in an observation: Si fe So Oe vo O55 8, A, fh a/ 06 mn Ong 85 al l2i658,> O,eadis fs Hos “CaS 8 LO: y2i ye Ain Gata? 8! nb) baer fw, Silat wale £68% 6 6.5 465 @ Bp 48h DO Sing G89 Ghiw suas Sf yes Seb Bix wbne8, ot /esa/ Ae ey Aaa tend 6: iG, (een Ow 10 104) l@iu, Se n6;t26 ornament: G Grn Oru Ose Sine8 6, vr SiO sae, 98 66 Ah ho 6,10, ny Gf inf There is a total of 100 height observations. There are 37 check values for low waves less than 6 feet high. There are 31 six-foot waves, 20 eight-foot waves, and so on. The lowest two-thirds of the waves must be eliminated from the computations, so the sixty-seven lowest waves must be left out. The thirty-seven lowest waves automatically drop out, and then thirty of the six-foot high waves are eliminated. Thus the one-third highest waves consist of the values of the heights greater than six feet and one six-foot wave to make up a total of 33. The average of the one-third highest waves is then computed according to the following procedure: Height Number Product 6 1 6 8 20 160 10 l 70 12 3 36 14 1 14 16 1 16 TOTAL 33 302 302 Significant height = 33= 9.2 feet Wee VE = Soe ee 20 There are a number of disadvantages to this procedure. The total number of waves which pass must still be counted. How can the one-third highest waves be counted if the two-thirds lowest waves are not counted also, so that it will be known that the one-third highest waves are actually some one-third of a total number of waves? Also if the significant height is computed, a large number of the waves which pass cannot be utilized in computing a statistic about the height. Many fewer usable values are obtained during a given time duration for the observation. A lot of time is wasted doing nothing. With the aid of the truncated distribution for K equal to 33 percent, the mean of the distribution and the standard deviation could be found. Then the steps used above to determine confidence limits for samples from the complete distribution could be used on the mean and second moment of the truncated distribution to determine the confidence limits of a significant height determined from N observed values. The results would be more reliable for a given N because some (but not all) of the correlation effect would be removed. However it would take three times as long to observe the N elements of the sample. Exactly similar procedures could be used with any other percentage of the highest waves. However, in each case the total number of waves which pass must be observed. 8. Truncated Distribution at a Fixed Height The second way to truncate the distributionis to eliminate all waves less than a certain fixed height and observe every wave in excess of this fixed height. For a given state of the sea the observer might record all heights greater than, say, 4 feet. For a higher sea all waves in excess of 10 feet could be recorded. It is then possible to compute the average of the observed values and from this the true average of all the heights, including those which were not recorded, and any other desirable height parameter. The theoretical derivation is given in the following paragraphs. Let the minimum height recorded be equal to H_. . Then E us equals H a /2, and the theory will be worked out using amplitudes: The results must be doubled atthe finishto obtain the height parameters needed. (e8) 2 2 Since i EX e -x /E dx= I-e7 € min. /E (20) S iain 21 the truncated distribution is given by 2 g(x)= —2Xe NS. Gly Tonio SOOO: (20) E(j-e -€ min. JE ) The percent of the low waves omitted simply equals 2 foo (I-e 6 min Je). (22) The average amplitude of all of the waves that are higher than ee is given by the first moment of equation (21), and the evaluation of “the integral yields the following result for € * which is defined to be the average amplitude of all waves in excess of € moe Se falas a gents ©: witha 7 sis bat Fea ee? E(I-e min/' min / am n I-e § min/E min 2 © 2 (23) Ege emin/E, f atets dx Soe 2 [i-e7 & min. /E] The last integral in equation (23) isthe integral of the normal distribution between known limits. It can easily be evaluated from tables. Equation (23) is a function of three variables, €*, € min., and E. Any two determine the third. Suppose then that the heights of all waves greater than 4 feet are recorded, and that the significant height of the waves is 8 feet. The significant height determines E, and then €¢ * can be computed. Under these conditions the average height of all waves greater than 4 feet is 6.64 feet. The percent of waves omitted can be found from (22) and in this case it equals 39.4 percent. 22 Table 4 The significant height, average height, and percent of waves omitted in terms of the average height of all waves in excess of 4, 10, or 20 feet AVERAGE HEIGHT OF ALL WAVES GREATER 458 6y015) $3 0S HON MOnL OM On64)) 7e22)e68) 880 9782.11.04 12.3 THAN 4 FEET AVERAGE HEIGHT OF ZOOS Sel Se Sm. Siu Oll 5.03) wOe25) i250) 8.6, 10-0) 1he3 ALL WAVES SIGNIFICANT HEIGHT 4 5 6 7 8 9 10 12 14 16 18 % OF WAVES OMITTED 86.6 72.2 59.0 48.2 39.4 32.9 27.6 20.0 14.5 12.4 10.0 AVERAGE HEIGHT OF ALL WAVES GREATER P2506 1152288) 13550) V4 Sar V6.6) 18.52) 2108 THAN 10 FEET AVERAGE HEIGHT GFZ Sige > Oke SaiGh LOROOMIIN Sim 12s Sir tS Olmal 78 OF ALL WAVES SIGNIFICANT HEIGHT 10 12 14 16 18 20 24 28 % OF WAVES OMITTED 86.5 75.1 63.9 54.2 46.3 39.4 29.5 23.2 AVERAGE HEIGHT OF ALL WAVES GREATER 22.7 25.4 27.5 30.4 33.4 36.8 39 43 THAN 10 FEET AVERAGE HEIGHT AND 22.55) 25.0. ABs Bile Ssh e3seh Or OF ALL WAVES SIGNIFICANT HEIGHT 32 36 40 45 50 55 60 65 % OF WAVES OMITTED 18.0 14.5 12.4 10 Ue) 6.5 5.6 4.6 AVERAGE HEIGHT OF ALL WAVES GREATER = _ 333.16 35.3 38.3 42 42.8 47.2 48.4 54.8 63 66 THAN 20 FEET AVERAGE HEIGHT OF 25 ZOnLY Se 3 S437 5830) 140n M43e7, 5051) 5663) 62:5 ALL WAVES SIGNIFICANT HEIGHT 40 45 50 55 60 65 70 80 90 100 % OF WAVES OMITTED 39.4 32.9 27.6 23.7 20 Nifesy, MNeV ey ites ko) Ue 23 Table 4 gives the data needed for typical significant heights. In an actual observation the number obtained is the average height of all waves in excess of a certain height, and the table then gives the true significant height, the true average height of the total sample, and the percent of waves omitted. In each entry, the average height ofall waves is less than the average height of all waves in excess of some fixed height. Thus the tendency to ignore the low waves can make a visual observation quite unreliable unless corrections for the omitted waves are made theoretically. Note also that as the percent of waves omitted becomes smaller the difference between the average of the truncated distribution and the average of the full distribution becomes less and less and less. 9. An Example The data obtained by the USCGC UNIMAK from 1800 to 1900Z on February 14, 1953 give an example of the procedures which can be employed in the use of the theory of a truncated distribution. The original raw data were first of all averaged to determine the average height of the reported waves. Then the value of VE was computed. From this and table 1 the theoretical distribution can be compared with the observed values. The result is given in table 5. Table 5 Data obtained by USCGC UNIMAK 141800Z to 141900Z of February 1953 Average Height 15.5 feet (uncorrected) Significant Height 24.5 feet (uncorrected) Limits Theoretical Observed Error Frequency Frequency Oheoy Bue 5 0 = Bo@ 6 472 5 9 +4 See OLS 5 6 +1 ORS wml 2 5 5 0 1234 -. 14.5 5 0 -5 14.5 - 16.8 5 6 +1 I.) 5 W)5s} 5 1 +2 NOES enreeee 5 10 +5 MATA cs PADS 5 6 +1 26.6 5 1 -4 TOTAL 50 50 24 From the table it is seen that the waves between heights of zero and 5.6 were simply not observed although there should have been about five of them in a sample of fifty values under the assumption that the true mean was 15.5 feet. Also the mean of 15.5 feet implies that there should have been five waves greater than 26.6 feet and only one such wave was actually observed. Now, the heights of all waves inthe original sample which are greater than 10 feet can be averaged. The result is an average of 17.1 feet, and by the application of the results given above it follows that a better estimate of the significant height is 20.9 feet and that a better estimate of the average height of all waves is 13.1 feet. The results are summa- rized in table 6. Table 6 Corrected data on the basis of the theory of truncated distributions Average of heights greater than or equal to 10 ft. = 17.1 ft. Significant height = 20.9 ft. True average height = 13.1 ft. Limits Theoretical Observed Error Frequency Frequency 10 26 9 -17 (not effective) 10-12 1a 9 -2 13-15 9.8 3 =6:8 16-18 8.5 10 +1.5 19-21 By) 9 Sie 22-24 3u3 5 +1.7 25-27 2.6 3 +0.4 27 (less than 0.5) 0 With a lower true average height the predicted number of waves with large height values agrees much better with the observations. Or, stated another way, table 6 agrees with the theory much better at the high end of the distribution than table 5. Another important point to note is that of the twenty-six waves less than ten feet in height, which in all probability actually passed during the time of observation, only 9 were observed. The effect of this omission must be to increase the computed average of the uncorrected results toa value greater than the true average. 25 0 S2-8I C= te a 0 ce sSil cs Ceol Clay Lena 0 OS sed Cs I1-8 0 US Lt+ Lileeal et O2-FI lor CiasiL a FESS *sqo Gz 10119 pueg rojoeyz Ayozes %06 I- 9- “sqo 0G 101190 pueg b2-61 T€-v2 02-91 02-91 O€ “EZ Ms AS Or-8 LL! 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During a two-week period in February 1953, the Weather Bureau observers on the various coast guard cutters of the Atlantic Weather Patrol made visual wave observations which were tobe used as a check of the forecasting procedures given by Pierson, Neumann, and James (1955). A check for internal consistency of the data showed that the observations had been carefully made. The height observation for a particular report consisted of the tabulation of 50 individual crest-to-trough heights. A considerable range of height values was reported for each observation. For a given observation, the reported height values would range from 10 feet to 30 feet, which appeared to agree with the theory that the wave heights were distributed according to equation (1). The average of the 50 reported heights was computed, and the significant height was found from the average height by multiplying the average height by 1.416/0.886 (or 1.60), which is the ratio of the significant height to the average height. Twelve of the reported observations were selected because of the high waves that were present. Forecasts based on the theory of the manual by Pierson, Neumann, and James (1955) were prepared without knowledge of the observed values. The results of the comparison of the forecasted values with the observed significant heights computed as described above are presented intable 7 under the heading, Observed Significant Height (no correction) and Forecast Significant Height. The results of the comparison of the forecast and observed values were most disappointing. Errors as bigas17feet resulted. The average forecast error was 8.6 feet. There was also a definite bias in that all but two of the forecast values were less than the observed values. The column labeled Error (uncorrected observations) shows these results. There was evidently something wrong! At that time, none of the work on confidence limits or on truncated distributions as discussed above had been applied to the data although the theory was a standard part of statistical texts as it is given, for example, by Cramer (1946) and Wilks (1951). The theories discussed above were then developed and applied to the observational data. A closer look at the reported height values shows that the heights did not follow the distribution given by doubling the coefficients in table 1. The low waves predicted by the probability distribution function were either missing or reported in far too small a proportion. 27 However, it was known that equation (1) was quite likely to be the true distribution of all wave heights as evidenced by the works cited above; and in addition the forecasting method which was being tested had worked well when compared with actual wave record observations. It was evident, therefore, that the observers had not been able to observe and record the low waves that had actually occurred. As shown in the UNIMAK example, with a significant wave height of 20.9 feet, 26 waves out of 67 should have had a height less than 10 feet. Only 50 waves were actually recorded, and 17 waves less than 10 feet high were omitted. Histograms of the data were plotted, and the observations were truncated at that height such that the distribution above that height resembled a truncated distribution as given by equation (21). From this truncated distribution the new corrected average height and corrected significant height were computed. The values which resulted are entered in table 7 under the heading Observed Significant Height (corrected). The result was to decrease each value by an amount which depended upon the nature of the original sample. Some heights were decreased by as much as six feet, and the average decrease was 3.75 feet. The decrease is tabulated under the entry labeled, Decrease. The forecast values and the corrected observed values then agreed far better than the forecast values and uncorrected observed values. Some of the largest errors were decreased a great deal. Seven of the twelve forecasts were within plus or minus five feet of the observed values. The forecast values still had a tendency to be lower than the observed values. When the truncated distribution is used to determine the significant wave height, the confidence limits determined fromthe full distribution, strictly speaking, should not be usedto obtainestimates of the reliability of the observations. The correct procedure would be to use the mean and second moment about the mean of the truncated distribution in a derivation similar to the one given above. However, such a derivation would have to be carried out for many different cases, and it is believed that the final results would not improve too much on the estimates obtained from the theory derived above as based on the full distribution. The confidence limits derived above can be applied to the results 28 obtained in table 7 with the reservation that the results are only approximate. In table 7, the 90 percent confidence limits are given on the assumption that the observations consisted of 50 independent height values. For example, for the observations made by the USCGC MENDOTA on 2 September 1953 at 1200Z, the Observed Significant Height (corrected) was 28 feet, and 90 percent of the time (under the assumptions which were made) the true significant height would be between 25 and 32 feet on the basis of many more observations. The forecast error as a departure of the forecast value from the closest value of the 90 percent confidence limits is then entered as the band error for 50 independent observations. The band error is a better measure of the discrepancy between the forecast and observed values because it does not penalize the forecast value for the unavoidable observational error which is due to the small sample size. Three of the twelve forecasts are within the 90 percent confidence of the observations. Four more are within three feet of the 90 percent confidence limits. When the band error for an assumed 50 independent height observations is studied, it is seen that the forecasts are quite accurate. The last two entries of the table show the 90 percent confidence limits on the assumption that the heights are really only 25 independent observations. This permits a spread of ten feet between the upper and lower bounds of some of the limits. For more precision it is evident that visual observations should consist of 100 observations at least, in order that it would be possible to be sure of somewhere near 50 independent values. Under these conditions, four forecasts fall within the 90 percent confidence limits. Five more fall within five feet of the 90 percent confidence limits. Under these conditions, though, the confidence limits are so broad that the observations are of little use in saying anything about the wave properties. One of the purposes of this paper is to show that reliable observations are needed and that they cannot be reliable if enough individual values are not observed. There is a consistent bias running through the data. The observed values consistently run higher than the forecast values. Much more data need to be collected before this bias can be established as real or false. There is, though, a possible explanation for this bias. It is that the observers did not keep an eye exactly at a fixed point on 29 the sea surface. If each observation had a little extra height added to it as the observer looked along the crest to the highest part of the crest, then the average of these heights would tend to be higher than the average of the heights of the waves passing a fixed point. ll. Other Errors There is finally the question of the reliability of the height estimates as made by visual observations. Can an observer estimate the wave height of a wave thirty feet high within plus or minus two or three feet? Any such error, if consistent, in the estimation of the individual wave heights would introduce errors in the reported values. Very little is known about the nature of such errors, but there does seem to be a tendency to overestimate wave heights when visual observations are made. A cheap easily used instrumental aid for the measurement of wave heights would be a very useful device to be supplied to ship's personnel taking wave observations if such an instrument could be devised. When the possibility of observer error, in addition to statistical error, is considered, it is seen that the results of table 7 are a good test of the theories given above and of the forecasting methods which were verified against the height observations. 12. Summary In summary, based on the above results, the following rules can be given for the visual observations of wave heights: (1) The heights of the waves passing a fixed point should be observed, (The point could also be fixed relative to a moving ship.) (2) All heights should be recorded (or if this is too difficult, all heights in excess of a fixedlower bound should be observed and the theory of the truncated distribution then used). (3) At least fifty values, preferably one hundred values, should be recorded. (4) Table 3 then gives values for the confidence limits to be placed on the observations. The value is more exact theo- retically if all waves are observed, and it is approximately correct when a truncated distribution is used. 30 Ill. VISUAL “PERIOD”, “WAVE LENGTH”, AND “SPEED” OBSER- VATIONS A. Definition of Terms Part of the difficulty in making wave observations and wave record analyses lies in the loose interchange between theory and practice of two distinctly different meanings of the word, “period”. A period of a simple harmonic progressive wave is a number with a precise mathematical meaning. A true period will be underlined in this paper, and it will be designated by the symbol, T. The time interval between two successive characteristic points in a wave record, such as the wave crests or the zero up-crosses, is not a period in the exact mathematical sence since a wave record is not periodic. These time intervals will be called “periods”. A wave record has many different “periods”. A simple sine wave has only one period. “Periods” in this sense will have quotation marks around them. The individual “periods” will be designated T. as they are enumerated in an observation or from a wave record; arid the average “period”, that is the average of all of the observed “periods”, will be called T. Similarly, wave length, (L), and “wave length”, (L), will be discussed. For additional discussion of these terms, see Pierson(1954) and Pierson, Neumann, and James (1955). Figure 5 illustrates the analysis of a wave record for its various Figure5. The Definition of the "Periods" in a Wave Record. 31 “periods”. The “periods” are designated by T, T, and so on. In such a wave record it is theoretically wrong to’ equate these observed “periods” with the period of a simple harmonic progressive wave. B. Visual “Period” Observations 1. Theory of the “Period” Distribution The probability distribution function of the time intervals between successive wave crests is not known. Rice (1944) has given a formula that gives the mean of this unknown distribution in terms of an integral which involves the spectrum of the waves. Apparently none of the higher moments is known. Even if the distribution of these “periods” were known, it would still tell us very little about the true spectral periods in the exact mathematical sense of the word. In addition, the loose interchange of “periods” and periods in theoretical work leads generally to invalid results. 2. Method of Observation The observed statistical distribution of the “periods” and the average “period” for a given state of the sea are nevertheless useful values which can be obtained by the use of a stop watch in visual observations. Recommended procedures for observing the “periods” are given in Pierson, Neumann, and James (1955). A foam patch ora floating object can be used as a reference point. Two observers working as a team can make the observations more rapidly and efficiently. C. Calculation Of The Average “Period” In Terms Of Theoretical Spectra 1. Method of Calculation Neumann (1953) has shown good reason to believe that the spectrum of a sea grows from high frequency to low frequency with increasing duration or fetch. The spectrum of a given state of the sea is given by [aun] = ite e 2g /v"* for p > ph, (24) where »; is a function of the wind velocity and either the fetch or the duration, and v is the wind velocity. 32 The average “period” for a partially developed sea can be evaluated in terms of #; by the following procedure. Let 27f =p and 2mf. =p. Leal and let f = if ae. Then the use of a formula derived by Rice (1944) shows that the average “period” is given by 2 iN (ae df V5 ae u in | Oooo ts Wee) i mai df fi or by Tg 52@ 7 | = 4 /> Wii 2 +26) U iE Teac dT where qg=g2@/472,2 The integrals given in (26) can be integrated by parts until an integral involving the probability integral results. A change in variable under the final integral in order to put it into unit normal form yields, after several operations, the result that | /, J/3 20 Vv 2 Sem7yv Phe 202 (27) 29 - 2gin le /4m7°v fies G/2mv -¢72 4 ( e —__ 81,°9°/(27v)° da- 49°T7/(2ry)2 The ratio, gT./2rv, occurs everywhere in equation (27). This is the ratio of the phase speed of the highest spectral period present to the wind speed. It is usually designated by B.+ aT, /2ny (28) and then equation (27) yields | Vo F_ vSenv 2p: ion eae a 5 (29) 2 9 | Be , pene a - ape 8B; * i 33 As B, approaches infinity, this equation reduces to i ~ a/ 21rv Te > or (30) in c.g.s. units or to = 0.285v (31) where v is in knots. This is the average “period” of the fully developed sea as shown by Pierson (1954). The function of B,in equation (29) can be evaluated and used to determine T for a partially developed sea. Let the term in brackets in equation (29) be F (B,). Then equation (29) becomes T= 0.285y.F(B,)- The values of F (B;) are given below in table 8. Table 8 F (B;) as a function of B.. B. EF (B,) B. EF (B.) 0.1 0.09 0.8 0.67 0.2 0.18 0.9 0.72 0.3 OF2.0 1.0 0.78 0.4 0235 ibis 0.87 0.5 0.43 1.4 0.93 0.6 0.51 1.6 0.97 0.7 0.59 2.0 1.00 From the duration or fetch curves onthe co-cumulative spectra given by Pierson, Neumann, and James (1955), the values of f. and v fora particular weather situation can be determined. The use of table 8 above then permits a computation of the average “period” of the waves. If B. is low (less than 0.5), the value of T can be approximated from equation (26) in a more direct way. The exponential term under the integral can be expanded in series with the result that Garr ce) A Daven paar | ee I nl U dT /, ie) Siete aR a a (33) : p™ Oe (e1hatZalioreene: 2 > ULI i SEE SL n <6 n! ay dT 34 and that 7 T; (34) (2D eG ue =0 n! The first few terms of this expansion are | Diane) 2 4 y - = fo = (2) 5 oon B; 5 8: 2 5 cc Seb RMGEE OER Us, (35) | 2 2 2 --= Ete | 2 ean 3) 5 B: ~ B: and under the condition that B. be small, a reasonable approximation is that = oT Fie Ti e/B1S =O. Col ae as given by Pierson, Neumann, and James (1955). 2. Interpretation of the Average “Period” The average “period” as observed by stop watch, or as computed from a wave record, can be an extremely misleading statistic. It overemphasizes the short “periods” and neglects the long “period”. The maximum energy in the spectrum is always at a higher value than is indicated by the average “period”. The significant “period”, that is, the average period of the one-third highest waves, may equal the average “period” or it may be a trifle higher because of the neglect of shorter “periods” in the average. However, it is even more doubtful a statistic because its relation to the wave energy spectrum is not known. For either the average “period” or the significant “period”, the computation of the average wave crest “speed” or the average “wave length” cannot be carried out by the use of the classical formulas as will be shown later. The classical formulas apply only to the true period of a simple harmonic progressive wave. The average “period” can be used to determine the state of de- velopment of the sea for a given wind velocity. It can be used to check a given forecast of the wave spectrumifonly a sea is present. However, spectra of many different shapes can yield the same average “period”; and the average “period” and the significant height do not completely characterize a given state of the sea. 35 In fact, there appears to be no way to obtain parameters which completely describe the seaway by visual observations or by the statistical analysis of a wave record or a pressure record. The partial characterization in terms of significant height and average “period” is, however, useful in many aspects if it is interpreted with care in terms of possible wave spectra and the meteorological synoptic situation. D. “Wave Lengths” 1. The Observation of the “Wave Length” Photographs of the sea surface, such as figure 1, show that it is composed of short-crested waves. There are medium waves super- imposed on the big waves and short waves superimposed on the medium waves. There are ripples on top of everything else. The waves in a photograph are much more irregular than a corresponding wave record. There appear to be more short waves in a photograph than there are in a wave record. Most of the time a dominant direction of travel can be determined for the waves. Then the length of the waves along this direction can be measured. The actual distance between successive crests must be measured. Procedures for measuring the “wave length” are given in Pierson, Neumann, and James (1955). The procedures involve towing a line with floats behind a vessel for use as a scale, and the use of the ship or other ships as a scale factor. The average “wave length” cannot be computed from the average “period” in terms of the classical formula. Stated another way, it is not true that the average “wave length” in feet equals 5.12 times the Square of the average “period” in seconds. For fully developed seas, the average “wave length”, if the theoretical spectrum which is assumed is correct, is given by a (37) ives he Sear For the theory of the derivation, see Pierson (1954). For nonfully developed seas the formula does not hold, and the derivation of the average “wave length” is more difficult. It appears that the p.d.f. of the “wave lengths” cannot be computed from the p.d.f. of the “periods” even if the p.d.f. of the “periods” were known. It would have to be computed by mapping the wave spectrum as a function of frequency and direction, into a frequency spectrum of the spectral wave lengths. Then, if the theory of the p.d.f. of the 36 —_—— —— “periods” is ever solved, it will be possible to determine the p.d.f. of the “wave lengths” by using the same theory on the new spectrum. Aerial photographs, if the scale is known, would be useful in the determination of the probability distribution of the “wave lengths” empirically. Observations which show that the formula given above is more nearly correct for the average “wave length” than is the classical formula are cited by Dearduff (1953). He states that “the observed wave lengths were as a whole much smaller than the calculated lengths based on the usual formula.” The value which was obtained from the analysis of observations made from Nantucket Lightship was half of the value which would be obtained using the classical formula. The theory on which equation (37) is based assumes that ripples on top of the more dominant cycles are not counted in the measurement of the “wave lengths”. The crest must be above sea level and the trough must be below sea level before the wave can be counted. A ripple or perturbation riding on top of a larger wave should not be counted. When such values are counted their effect is to decrease the average wave length to a value even less than the one given by equation (37). 2. Explanation of Theory of Equation (37) There is an idea prevalent in current wave theory that a wave record can be broken up into pieces of one wave per cycle and that each oscillation can be treated as if it were a sine wave with the use of the classical formulas for the piece. The theory can be sketched briefly as follows: Given a wave record as on the bottom of figure 6, the record is broken up into pieces at each zero up-cross and each fragment is treated as if it were a piece of a sine wave with a true periodequal to the length in time of the piece and with an amplitude equal to one-half the crest-to-trough height of the piece. If the above assumptions were correct, then the wave record could be represented mathematically as the sum of a number of functions of the form sketched on the top of figure 6. Such a representation is obviously absurd. If such a fragment were generated in a wave tank, it would alter in form completely before it could travel even a few feet. A Fourier analysis of one of the pieces shown in figure 6 would show it tobe composed of a very broad Fourier spectrum of frequencies so that it would not be correct to apply the “period” T. to one of the pieces. Such a small piece of a sine wave is not the same thing as a sine wave. 37 oie ee ee Figure 6. The Representation of a Wave Record as a Sum of Individual Sinusoidal "Cycles" with Different "Periods" in an Artificial Wave by Wave Analysis. 38 Figure 7. The Representation of a Wave Record as a Sum of Many Sine Waves with Individual True Periods. 39 The correct way tothink of a wave record is to think of it as composed of a very large number of very low sine waves with phases all mixed up and with different periods in such a way that figure 7 represents the wave record. If any one of the above pieces is generated for a long enough time in a wave tank, the waves propagate without change of shape and have classical wave lengths and classical phase speeds after initial transients have died out. Since, as a first approximation, the system combines linearly when all waves are produced simultaneously, the behavior of the sum equals the sum of the behaviors of the individual sinusoidal components. Figure 7 explains whyitis sodifficultto observe the visual properties of waves or to analyze a wave record statistically. The variation in wave amplitudes described at the start of this paper is caused by the complicated effects of phase reinforcement and cancellation of this large (infinite) sum of small (infinitesimal) amplitude true sine waves combined in random phase. It also explains the difficulties involved in determining the “periods”, since a “period” is the time interval between two successive zero up-crosses. When a sum of, say, fifty or sixty true sine waves is written out and when they are assigned amplitudes according to some spectral law and phases at random, it then becomes difficult, if not impossible, to solve for those times in the record produced where the record adds up to zero and to compute the time intervals between the zeros. These “periods” thus are produced by an interference effect. This is why the probability distribution function of the “periods” is not known theoretically. Mathematicians simply have not yet been able to solve this problem. Intuitively, at least, the reason why the average “wave length” is given by equation (37) in a fully developed sea can now be explained. If the wave crests were infinitely long, then corresponding to each sine wave in the sum as observed as a function of time at a fixed point, there would be a sine wave on the sea surface as a function of distance along a line. Each wave length in feet would be given by 5.12 times the square of the true period of the sine waves in the sum which goes to make up the sea surface along the line. The wave lengths are related to the square of the periods. The more rapid oscillations in the record as a function of distance for periods less than the average “period” outweigh the effect of the much less rapid oscillations for periods greater than the —————— average “period”, and the result is that the average “wave length” is less than what would be computed from the average “period” by the use of the classical formula. When this effect is corrected for the short-crestedness of the waves, the result is equation (37) fora fully developed sea. As part of the erroneous method of wave analysis depicted in figure 6, it is frequently assumed that the “wave length” of the wave which passed during the time interval, T is given by ~ ~w 2 iy E174 (38) J J in deep water or by anappropriately modified equation in shallow water. This assumption is obviously dependent upon the assumption that the zero which passes at the start of the “cycle” does not disappear before the zero which passes at the close of the “cycle” finally arrives, and upon the assumption that a new zero does not form between the first zero and the point of observation and the old zero before the second recorded zero passes. (Similar remarks could be made about crests.) Since the wave forms of actual ocean waves do not propagate without change of shape, and since the crests of actual ocean waves are not conservative, these assumptions are not valid and the formula cannot be used. The average of the “wave lengths” as computed from the individual “periods” is always greater than the average “wave length” computed from the average “period”, and even this latter value is too big. Although it is unknown, suppose that the p.d.f. has the typical properties of all p.d.f.'s in that it gives the probability that a “period” within a band of “periods” will be observed. The p.d.f. of the “periods” is then g (T)dT with the properties that ans Olfor i; <0! see) g(T) 2 0 for FT >o, (40) a ~~ and di g(T) dT = ) (41) The average “period” then equals — @ ww ~ —- ur Nelare a) 41 The average “period” is estimated from a finite series of individual “period” observations such that Fs ne ose ts (43) Now consider the following form which is always greater than or equal to zero because it is the integral of an always positive (or zero) function. 2 2 if (F-F) g(F) dF 20 (44) re) It yields oO (0) Sue ~ = ~ a ws we. me @ i T a(F) at - at f Fg(hidteT[ g(T) dT 20 vse ) Oo 0 or 2 Oe ace Sec fe) n pees ee (47) Now let L* be the average “wave length” computed by computing the “wave length” associated with each of the observed “periods” and averaging the results. From equation (47), this “wave length” is given by (48) The wave length, L computed from the average “period” is found by averaging the observed “periods” and computing the average “wave length” from the average “period” according to equation (43). “ rhe Be, Le eXln = hh) ey But from equation (46), L* is greater than Ly, and from equation 42 ng eee, ae. Geers. = (37) L, is too big when compared with actual observations. Therefore the avérage “wave length” cannot be computed from the observed distribution of the “periods”. The individual “wave lengths” computed from the individual “periods” have therefore a very doubtful meaning. With reference to “wave lengths”, the only reliable formula is for the average “wave length” for a fully developed sea as given by equation (37). For swell, the average “wave length” is approximately given by the classical formula using the average “period” of the swell. For seas not fully developed or for cross seas, no convenient formulas, in general, exist. However, for newly generated partially developed seas in which B, is less than 0.5, it is _possible to obtain an approximate value for i Under these conditions, Ty is given by iS 2.56 T.” (50) The method for deriving equation (50) involves short-crested seas and employs approximations and procedures similar to those used in equations (33) through (36). E. Wave “Speeds” 1. Theory - A Contradiction The usual wave observation procedure has been that of observing the “periods” of the waves and computing the average “period”. The “wave lengths” and “speeds” of the individual waves are rarely independently observed. The theories given above suggest that the average “wave length” of a fully developed sea is two-thirds ofthe value given by the classical formula. Also some independent observations suggest that these theories are more nearly correct. In c.g.s. units, the two classical formulas for the speed of a wave crest are given by Gear (51) and Ga= oD /i2imin: (52) In terms of average “periods” and average “wave lengths” in 43 units, equation (37) becomes 2) ot 2m (53) Now suppose that the average speed is computed by assuming that the classical formulas involving the period and the wave length of a simple harmonic progressive wave hold for the average “period” and the average “wave length” of an irregular state of the sea. The results are that aa =i gT/27 (54) from equations (51) and (53) and that CG Sdetyear, (55) from equation (52). The result is two different values for the same theoretical quantity, and there is a contradiction involved. The contradiction lies in the assumption that the classical formulas can be applied to average wave properties. For an irregular sea, current theory tells us nothing about the average wave “speed”. Neither equation (54) nor (55) can be assumed to be the correct one. 2. The Observation of Wave “Speeds” Wave crest “speeds” must therefore be observed independently of the “periods” and the “wave lengths”. The “speed” of a given crest may not even be a constant. The wave crest “speeds” can be measured at the same time that the “wave lengths” are being measured by the methods givenby Pierson, Neumann, and James (1955). Such observations in a sea are very scarce, if any exist at all, and thus the present state of theory and observation can give no information on this problem. Data on this problem, when they become available, will prove to be very interesting. IV. CONCLUSIONS The visual observation of the properties of ocean waves will always be an important supplementary source of wave data. The data thus obtained can never be as adequate as wave records which are analyzed for their spectra, but they can be used if they are interpreted with care. | . : | . A series of wave height observations can be used to verify the theoretical probability distribution of wave heights. Even for irregular seas, the distribution may be fairly well approximated. The reliability of the average height can be estimated from the size of the sample and confidence limits can be assigned to the values observed. The theory of truncated distributions can be used to refine the values if the low waves are neglected. The average “period” is a misleading statistic unlessit is interpreted in terms of the wave spectrum. It gives a value which is shorter than the period where the maximum energy exists in the spectrum, It can be forecast and thus related to the spectrum of the waves. The average “wave length” cannot be computed with the use of the average “period” by means of the classical formulas. For a fully developed sea in deep water the theoretical value is two-thirds of the value that results from the classical theory. The “wave length” of an individual wave cannot be computed from the “period” of that wave as it passes a fixed point. The wave crest “speeds” are rarely observed, and the classical formulas cannot be used to predict the “speeds” from the “periods” and “lengths” in a sea. 45 BIBLIOGRAPHY ARAKAWA, H. AND SUDA, K. Analysis of winds, wind waves, over the sea to the east of Japan during the typhoon of September 26, 1953, Monthly Weather Review, vol. 81, no. 2, p. 31-37, February 1953. CRAMER, HAROLD. Mathematical methods of statistics. Princeton: Princeton University Press. 575 p. 1946. DEARDUFF, R. F. A comparison of observed and hindcast wave characteristics off southern New England, Bulletin, U. S. Beach Erosion Board, vol. 7, p. 4-14, 1953. LEWIS, E. V. Ship modelteststodetermine bending moments in waves, Transactions of the Society of Naval Architects and Marine Engineers, vol. 62, 1954. LONGUET-HIGGINS, M. S. On the statistical distribution of the heights of sea waves, Journal of Marine Research, vol. 11, p. 245-266, 1952. NEUMANN, GERHARD. Zur charakteristik des Seeganges (On the nature of sea motion), Archiv Fur Meteorologie, Geophysik und Bioklimotologie, Series A, Meteorologie und Geophysik, vol. 7, p. 352-377, 1954. PIERSON, W. J., JR. A unified mathematical theory for the analysis, propagation, and refraction of storm-generated ocean surface waves, parts I-II. New York: New York University, College of Engineering, Department of Meteorology. Prepared for U. S. Beach Erosion Board, @ontractino~ Nonr=285) (03); pt: 1) S36.p; pti; V25ip- 19527 --- An interpretation of the observable properties of sea waves in terms of the energy spectrum of the Gaussian record, Transactions of the American Geophysical Union, vol. 35, p. 747-757, 1954. PIERSON, W. J. JR., NEUMANN, GERHARD, and JAMES, R. Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics, Hydrographic Office Publication No. 603. Washington: 285 p. 1955. RICE, S. O. Mathematical analysis of random noise, The Bell System Technical Journal, vol. 23, p. 282-332, 1944; vol. 24, p. 46-156, 1945. 46 SEIWELL, H. R. Results of research on surface waves of the Western North Atlantic, Papers of the Physical Oceanography and Meteorology. vol. 10, no. 4, p. 30-56. 1948. TUKEY, J. W. The sampling theory of power spectrum estimates, p. 47-67. In: U.S. Office of Naval Research, Symposium on Applications of Autocorrelation Analysis to Physical Problems, 13-14 June 1949, Woods Hole, Massachusetts. 79 p. 1950. WEAGT OT ERG tem, Sin At Distribution of height in ocean waves, New Zealand Journal of Science and Technology, sec. B, vol. 34, p. 408-422, 1953. WEIGEL, R. L. An analysis of data from wave recorders on the Pacific coast of the U. S., Transactions of the American Geophyscal Union, vol. 30, p. 700-704, 1949. WIENER, N. Extrapolation, interpolation and smoothing of stationary time series with engineering application. New York: John Wiley. 163 p. 1949. WILKS, S. S. Elementary Statistical Analysis. Princeton; Princeton University Press. 284p. 1948. WOODING, R. A. An approximate joint probability distribution for wave amplitude and frequency in random noise, New Zealand Journal of Science and Technology, sec. B, vol. 36,no. 6, p. 537-544, May 1955. 47 ADDENDUM Since the preparation of this paper and of the wave forecasting manual (H.O. Pub. No. 603), two papers of interest in connection with this paper have come to the attention of the author. The first paper, by Arakawa and Suda (1953), gives some data on the measurement of the average wave length of a wind-driven sea. The second paper, by Wooding, gives information on the meaning of the significant period. Arakawa and Suda (1953) summarize and discuss some wave obser- vations made by the Japanese Navy during a typhoon which occurred on September 26, 1935. Table 6 of their paper is reproduced in full anda paragraph referring to the table is quoted as follows: “Table 6 shows that comparisons of measured and computed values for the MIKUMA gave rather unsatisfactory results. This may indicate that the state of the sea as observed by the main squadron was, to some extent, uncertain. Comparisons of measured and computed values for the wave length and the wave period from the cruiser NACHI on the other hand gave fairly satisfactory results.” _ It. should be noted that according to the notation used in this text, L, T, and C are probably what were really observed. Table 6. Observed and computed values of velocities, lengths, and periods of wind waves in the typhoon area, Sept. 26, 1935 Wave velocity C, Wave length L,m. Wave period T, sec. m. sec. computed computed from from Observed Observed Observed Mikuma 1250 JMT, Mikuma 1445 JMT Nachi 1500 JMT Nachi 1550 JMT |About 8 .7| 14.0 | About 120 126 |About 9 For a simple sine,wave, the formulas, L= CT, and L = gT Ghee imply alisomthatielan=i2mCm ie Cu=n/ ele) 2maGe= gai mk =./ena) py and — = J/2rC/g. Thus if L or C or T is observed, the other two quantities can be computed from it. In table 6, the comparison shows that C could not be predicted from either T or L. The value of L when computed from C is much too low. When L is computed from T the NACHI observations agree, but the computed wave length is considerably greater than the observed wave length inthe MIKUMA observations. Whenthe formula for the ayerage wave length L, in terms of the average period, a. namely ic = £ oF /27, is applied to the MIKUMA observations, the period of 13 seconds yields a value of 2/3 of 264 meters or 176 meters as compared to an observed wave length oi 180 meters. The second set of observations yields a value of 184 meters as compared to an observed value of 200 meters. The percentage error withrespect to the observed average wave length is about 2% with the new formula and 47% with the classical formula in the first case. In the second case, the errors are 8% and 38 %, respectively. It is most interesting that two of these four sets of observations obtained in 1935 should agree with the newly derived formula. Since the other two do not, it can be added that observations in a towing tank in which Gaussian waves were generated, confirm the theoretical basis of the derivation of the new formula.* Wooding (1955) has derived an approximate joint probability distri- bution for wave amplitude and frequency (period) inrandom noise, and he has applied the results to the interpretation of wave observations. The results show that the time interval between the successive upcrosses in a wave record has a higher probabiliy of being large if the wave is high than if the wave is low. Thus the average time interval between successive crests of the one-third highest waves shouldbe greater than the average time interval betweenallthe crests. Or, stated another way, the significant “period” is greater than the average “period.” It should be possible to derive a formula for the significant “period” in terms of a theoretical wave spectrum using the results of Wooding (1955). If an average wave length were obtained using the “significant” period and the classical formula, the error would be even greater than that obtained by using the classical formula and the “average” period. In view of the difficulty of observing the significant wave height discussed in this paper, it is believed that the observation of a true * See Lewis (1954). 49 significant “period” would be even more difficult, and that the con- clusions of this paper with respect to visual wave observations should still be adhered to substantially. 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