THE UN IVERS IT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Final Report WAVE HINDCASTS VS. RECORDED WAVES Stanley J. Jacobs Assistant Professor of Oceanography John C. Ayers Project Director ORA Project 06768 under contract with: U.S. ARMY ENGINEER DISTRICT, LAKE SURVEY CONTRACT DA-20-064-C IVENG-65 -6 DETROIT, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1965

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT ix 1. INTRODUCTION 1 2. WAVE HINDCASTING METHODS 2 3. WAVE HINDCASTS AT THE LAKE MICHIGAN RESEARCH TOWER 9 A. General Considerations 9 B. Wind Data 10 C. Wave Data 13 D. Results 14 4. SUMMARY AND CONCLUSIONS 20 REFERENCES 21 ACKNOWLEDGMENTS 22 APPENDIX A. AUGUST DATA AND RESULTS 23 APPENDIX B. SEPTEMBER DATA AND RESULTS 29 APPENDIX C. OBSERVED SPECTRA 37 iii

LIST OF TABLES Table Page 1 Neumann and Pierson-Moskowitz Spectra 5 2 Correlation Coefficient Comparison of Wind Data 12 3 Student's t Comparison of Wind Data 12 A.1 Wind Conditions in August Hindcast Period 24 A.2 Hindcasts for August Hindcast Period 25 A.3 Comparison of Hindcast and Observed Wave Values for August Hindcast Period 27 B.1 Wind Conditions in September Hindcast Period 30 B.2 Hindcasts for September Hindcast Period 31 B.3 Comparison of Hindcast and Observed Significant Heights for September Hindcast Period 33 B.4 Comparison of Hindcast and Observed Periods of Maximum Amplitude for September Hindcast Period 34 Bo5 Comparison of Hindcast and Observed Largest Wave in September Hindcast Period 35 v

LIST OF FIGURES Figure Page 1. Wave hindcasting flow chart. 8 2. Calibration curve for spectra. 15 3. Location of Lake Michigan Research Tower. 16 4o Hindcast and observed significant heights: August. 18 5. Hindcast and observed significant heights: September. 19 C.1. Observed spectra: August. 38 C.2. Observed spectra: September. 46 C.3. Correction for hydrodynamical attenuation. 53 vii

ABSTRACT Wave heights at the Lake Michigan Research Tower are hindcast for periods in August and September of 1964. The hindcast is compared with wave heights measured by the U. S. Lake Survey. Hindcasts made with the Neumann energy spectrum prove to be superior to those made with the Pierson-Moskowitz spectrum and are in satisfactory agreement with observations. Results are presented in both tabular and graphical form. ix

1. INTRODUCTION The research program "Wave Hindcasts vs Recorded Waves", Contract DA-20-064CIVENG-65-6, calls for wave hindcasts at the Lake Michigan Research Tower for the month of October, 1964, and for the comparison of these hindcasts with wave heights recorded by the U.S. Lake Survey. The collapse of the tower during the storm of September 23-24 required a change in the hindcast period to August 1 through August 10 and September 13 through September 23, during which times necessary data was available. This report presents the results of the work carried out under the above contract. 1

2. WAVE HINDCASTING METHODS The statistical theory of water waves was set forth in the early post-war years by a number of authors and has been the subject of review articles by Pierson (1955) and Longuet-Higgens (1962). As the theory is well known, only major results will be presented here. Under the assumption that ocean waves are essentially random in character, the sea surface S(t) at a fixed point in space is represented by the sum ~(t) = ancos((nt + en) (1) n where the phases En are chosen at random. The sum of the squares of the amplitudes in a small element of frequency dp is an = A2()dB, (2) A2 (p) being the energy spectrum as defined by Pierson (op.cit.). The number E, defined by o00 0 is related to 5 by E a 2 a = Lim 2(t)t. (4) n 1/2 Thus (E/2)/ is the root mean square wave height. The properties of a process such as that given by equation (1) were investigated by Rice (1944, 1945) in connection with the problem of noise in electric circuits. Rice showed that if the frequency band of: is relatively narrow, the probability p(r)dr that the wave amplitude a (one half the crest to trough height) lies between r and r + dr is E(r)dr = 2r e- dr (5) 2

and that the expected number c(n) of waves occuring per unit time is 00 1~ci p p n) 1/2 c(n) = j _ A()dk (6) 2it E 2 E o J Longuet-Higgens (1952) showed that the probability distribution (5) can be used to find the mean amplitude of the highest pN waves in a record with N waves, that this amplitude is!~n1\1e P d(7) a~'p) = / (np t f e d3, ( 7) and that for large N the expected value e(amax) of the largest wave amplitude is (amax) = VE (n N)2 + 289(An N) 2 (8) An approximate result for the crest to trough height H is obtained by doubling (7) and (8). The mean height H is H = 2 x a = 177? (9a) while H / = 2,83 E' (9b) Hl/lo = 3.60 1 (9c) (Hmax) = 2 / (2 n N)2 +.289 (In N)- 2i (9d) Equation (9d) is especially useful in conjunction with (6), which relates the frequency of waves to the wave spectrum. The results given in equation (9) were compared with observations by Longu.et-Higgens (1952) and proved to be quite accurate o In order to forecast waves, the energy spectr-um must be related to causal factors5 such as the windc A number of authors have proposed spectra for fully developed seas, ioeoG'wave systems arising from constant or slowly changing winds

with large fetch and duration. Two such spectra are given in Table 1. In each the spectrum is related to the wind speed at only one elevation, the anemometer height of the research vessel. The Neumann spectrum was one of the first to be developed and has been used by Pierson, Neumann, and James in their wave forecasting manual, H.O. Pub. No. 603, Practical Methods for Observing and Forecasting Ocean Waves. The dependence of wave height on the 2:.5 power of velocity is a somewhat controversial feature of this spectrum; many other scientists believe the height to be proportional to the square of the velocity. Pierson (1964) has shown that consideration of the variation of the wind speed with elevation brings the Neumann, PiersonMoskowitz, and other spectra into better agreement, though the results definitely diverge at high wind speeds. It is still not clear which of the proposed spectra is most nearly correct. An energy spectrum derived for a fully developed sea in which all frequencies are present can be adapted for use in seas which are not fully developed. The predominant opinion is that in such cases the spectrum is modified by application of a filter which only allows a certain range of frequencies to be present. One such filter is used in cases in which the sea is either fetch or duration limited. For these cases there is an upper period Tu such that A2 = 0 for T > Tu, or p1 < 2T/Tu. In H.O. Pub. 603 curves are plotted which show the effect of this type of filter on wave heights. For a given wind speed there may be limitations due to either the fetch or the duration, with an empirically obtained Tu for each case. It is recommended that when waves are both fetch and duration limited the lower value of Tu is used, so that 00 2 E = ( A (4)dp (10) Tu is the smaller of the two possible values. A second type of filter must be used when the wind has stopped or decreased greatly. If F is the distance from an observation point to the windward edge of the fetch and t is the time after the wind shift, waves propagating with the deep water group velocity Cg = (g/4) T from the windward edge of the fetch reach the observation point at t = F/Cg 4

TABLE 1 NEUMANN AND PIERSON-MOSKOWITZ SPECTRA Neumann Spectrum (Pierson, 1955) C e-2( g/~v)2 K(C) ="e P6 where C = 51.7 ft2 sec-5 and V = wind speed at 7.5 m; 00 E = A A2()dp= C ( ) o 22f2 g H1 = 2.83TE = 1.39 () f 10 where V is measured in knots; T = Average period = 1/e(n) -= (V/g) = 2.85 (V/10) sec, where V is measured in knots; Tm = Period of maximum amplitude of A2 = f6 T (V/g) = 4.03 ( V) sec, 10 where V is measured in knots. Pierson-Moskowitz Spectrum (Pierson and Moskowitz, 1964) 2 4 A (p) = 5' ewhere o =.0162, P = 0.74, and U = wind speed at 19.5 m; 00 E = A(,)du - = U4/O 5

U 2 H = 2.83 5E = 1.82 (U) ft., 3 where U is measured in knots; T = Average period = 1/E(n) ( 7 1/4 U/g = 2.67 (U/10) sec, where U is measured in knots; Tm = Period of maximum amplitude of A2 = 2jr ()1/4 /g = 75 (U/10) sec, 4p where U is measured in knots. Hence in this case Tu - =(4/g)(F/t) since waves with higher periods have already passed the observation point. Therefore the decay of sea into swell is described by the law 00 E - A (p)d. (11) gt/2F A number of filters applying to other situations are discussed in H.O. Pub, 603. They each serve to provide a period band which can then be used in conjunction with the energy spectrum to calculate E and the desired wave heights and periods. The process of wave prediction can thus be summarized as follows: (1) A wind field is obtained either from isobar analysis or from reported ship windso These winds must be corrected by empirical rules to provide the wind at whatever elevation is called for by the spectrum to be usedo 6

(2) The fetch and duration are estimated. If the sea is fetch or duration limited, an empirical filter is applied to cut out the lower frequencies. E can then be calculated for either fully developed or fetch or duration limited seas. (3) The past history of the wind system is studied to see if swell is present. If so, the contribution to E from swell is added on to the contribution from the local sea. (4) The composite spectrum of sea plus swell can be used to calculate the average period, the period range in which most (say 9/10) of the energy is located, and the period of maximum energy. (5) The composite value of E is used to calculate the various average heights. The expected maximum amplitude in a record of any given duration can also be calculated. A flow chart illustrating the method is shown in Figure 1. 7

Gradient Wind Ship Wind Past Corrected Correction Factor Correction Factor Past Corrected Wind Spectrum Spectrum Swell Filter Fetch or Duration Filter Wave P Statistics Average Period Period Band Period of Maximum Energy Average Height Maximum Height Figure 1. Wave hindcasting flow chart. 8

3. WAVE HINDCASTS AT THE LAKE MICHIGAN RESEARCH TOWER A. GENERAL CONSIDERATIONS The wave climatology of Lake Michigan is affected by its relatively small dimensions, about 275 nautical miles in length and 100 nautical miles in breadth, with a maximum depth of 923 feet and a mean depth of 276 feet (University of Michigan, 1963). The oceans, of course, are much larger and deeper. Since wind generated waves are fetch limited, with the minimum fetch necessary for a fully developed sea an increasing function of wind velocity, waves generated by strong winds have lower amplitudes on the lake than they would have for the same winds blowing over the oceans. The periods and wavelengths are also lower than in the oceans. An example of this effect is the case of a 44 knot wind, which for a fetch of 1000 nautical miles would produce waves with a significant height (H1/3) of 56.5 feet, an average period of 12.5 seconds, and an average wave length of 533 feet. The same wind blowing along a Milwaukee to Muskegon line, with a fetch of 100 nautical miles, would produce waves at Muskegon with a significant height of only 18 feet, an average period of 6.6 seconds, and an average wavelength of 210 feet. Depth effects are less important than might be expected. One's first inclination is to suppose that waves in Lake Michigan might have to be treated as shallow rather than deep water waves. In order to treat this possibility, an idealized problem was worked. This consisted of supposing that long crested waves of period T and deep water amplitude ao advance into shallow water of depth h with depth contours parallel to the wave crests. The object is to calculate the shallow water wave amplitude a. This problem ignores the effects of wave ray convergence but is still useful in deciding if the deep water theory is adequate. The result of the calculation is that the amplitude ratio (a/ao) as a function of the non-dimensional number gT2/h is approximately constant and equal to unity for gT2/h < 10, decreases to 0.91 at gT2/h = 41 and then increases, coming back to unity at gT2/h = 100. As h + 0, 1 2 1/4 (a/ao) * - g 24J h and the wave breaks. Of interest to the present study is the fact that for h = 50 feet, the water depth at the Lake Michigan Research Tower, gT2/h = 100 corresponds to 9

T = 12o5 secondso Since higher wave periods are seldom if ever found in Lake Michigan, the amplitude ratio is affected only slightly by bottom effects. By contrast, the higher period waves found in the oceans would be strongly affected by bottom effects in water of depth 50 feet. There may also be frictional and percolative effects, but not enough is known about these to make a quantitative estimate of how these modify wave heights. In summary, the small horizontal dimensions of Lake Michigan reduce wave heights and periods, but in so reducing the periods they diminish the importance of the small vertical dimension. Hence the waves may be treated as deep water waves except near the shores. B. WIND DATA For predictions made during the August period there were four possible sources of wind data: (1) measurements at the research tower, a mile offshore at Muskegon, (2) reported winds from ships, (3) corrected gradient winds, and (4) winds measured at shore weather stations. Wind measurements were measured at shore weather stations. Wind measurements were not made at the tower during the September period due to instrument malfunctions. The tower data consists of winds measured at 16, 8, 4, and 2 meters above the water surface. The 16 meter wind was used for hindcasts made with the Pierson-Moscowitz spectrum and the 8 meter wind for hindcasts with the Neumann spectrum, since the speeds should be close to those at 19.5 and 7.5 meters respectively. The ship winds were measured at elevations of from 16 to 22 meters and were used without correction with the Pierson-Moscowitz spectrum. For use with the Neumann spectrum they were multiplied by the ratio V/U as determined from tower data taken in 1963 and 1964 (Elder, 1965), and the result was rounded off to the nearest knot. The scatter in this data was reduced considerably by separating the measurements into stability classes, accordihg to the value of AT = (Twater Tair). The ratios as computed from Elder's data are V Wind speed at 7.5 m. U Wind speed at 19 5 m..85 AT < -5~F O95 -5~F < AT < +5~F 1.00 AT > + 5~F (12) Since mnost of the ships report AT as well as the wind at anemometer height, equation (12) is applied without difficulty. 10

The value V/U for neutral conditions, 0.95, is slightly larger than that given by other authors (c.f. Pierson, 1964) for the range of speeds measured by Elder, 5 to 25 knots, but is accurate enough for the purposes of the present study. Even under the assumption that the other ratios in (12) are correct, it is at best a crude approximation to assume that the wind speed at only one elevation provides all the information necessary for relating the waves to the wind. However, this is the best assumption one can make at the present time. The method used to find the wind speed from isobar analysis consisted of re-analyzing the maps provided by the Weather Bureau by plotting isobars at 1 mb. intervals, calculating the gradient -wind using standard methods, and then correcting the gradient wind to its value in the boundary layer over the water. Mr. Al Strong of the University of Michigan Great Lakes Research Division has gathered data relating the gradient wind to the wind as measured at anemometer height on ships and has separated the data according to stability classes. Strong's best straight line fit of his data together with the ratios given in (12) yield V = wind speed at 7.5 m in knots - 7.9 + o28 V AT < -5~F g (15) = 95 +.27 Vg -5~F < AT +5~F 1.3.1 +4- 31 Vg AT > 5~F where Vg is the gradient wind in knots. According to Strong, these results are most reliable when the gradient wind is greater than 10 knots. The fourth source of data is shore weather stationso It is unclear a priori how representative winds measured at shore are of conditions at sea. In order to test the validity of the sources of wind data, a number of correlation coefficients were computed and are given in Table 20 The sample in each case represents those occasions in the period August 1 through August 10 when the indicated comparison could be made, and the quantity compared is the wind at 7 5 m The correlation coefficients- exhibited in Table 2 indicate that ship and corrected gradient winds leave something to be desired as sources of data, but are still acceptableo They also seem to be superior in accuracy to the Muskegon weather station winds. In order to test the latter conclusion, the null hypothesis was made that the ship winds and Muskegon weather station winds are equally valid for use at the research tower and the hypothesis was tested using Student's t distributiono Similar tests were made comparing the corrected gradient winds and the weather station winds, and also the squares of the wind speeds. This latter comparison was made because it is the square or some higher power of V which actually enters into the calculation of wave heights. The results are given in Table 35.11.

TABLE 2 CORRELATION COEFFICIENT COMPARISON OF WIND DATA Correlation coefficient of Sample size Correlation coefficient 1. Vgrad and VS 23 0.72 2. VM and VS 23 0.18 3. VT and VS 7 0.72 4. Vgrad and VT 13 0.69 5. VM and VT 13 0.23 Vgrad = corrected gradient wind VS = ship wind VM = Muskegon weather station wind VT = research tower wind TABLE 3 STUDENT'S t COMPARISON OF WIND DATA Quantities t Degrees of Probability of t Compared Freedom Occuring as a result of Chance 1. VS and VM 1.35 6.25 2. Vgrad and VM o79 12.45 3. (Vs)2 and (VM)2 1G32 6.25 4o (Vgrad)2 and (vM)2 1.25 12.25 12

The probabilities in the last column are not low enough to state definitely that the weather station winds are inferior, but there is an inference in that directiono This is true despite the fact that the Muskegon weather station is so close to the research tower that if winds at sea are ever the same as shore winds, they are in this case. It was therefore decided to reject winds measured at shore weather stations for use in making hindcasts. C. WAVE DATA Wave heights were recorded during the August period by a pressure sensor maintained by the U. S. Lake Survey. During the September period a staff gage was also in operation. The output from these systems was analyzed by personnel of the U. S. Army Coastal Research and Engineering Laboratories using the electronic analyzer developed by the Beach Erosion Board. The properties of the analyzer have been described by Caldwell and Williams (1961) and its use in computing the spectra of waves from Hurricane Donna by Bretschneider (1961). The spectral curves give the frequency distribution of linear average and square average wave heights taken over a twenty minute period with a filter band width of 0.027 cps. Thirty-eight such spectral curves have been put at the author's disposal by the Lake Survey. Of these, fourteen represent analyses of waves recorded by the pressure sensor in August, twelve of waves recorded at the same times in September by the staff gage. In addition, continuous wave records from both systems were available. This data was used in the following way. The variance of the staff gage wave record was computed using University of Michigan facilities, the result being a continuous record of the variance for the preceding twenty minutes of real timeo Since the various average wave heights are proportional to the square root of the variance, the computation in effect provides a continuous record of the significant height and the other average wave heights. This record was used as the source of observed wave heights for the September period, while the spectral curves provide information on the periods of the waves. It should be noted that analysis of the staff gage record indicated no appreciable set-up. The wave record measured by the pressure sensor cannot be analyzed so simply because these waves experience a frequency-dependent hydrodynamical attenuation. However, spectra computed from the pressure sensor's output can be corrected simply for this effect. Accordingly, it was decided to calibrate the spectra computed from pressure sensor data and to use these spectra as the source of observed wave heights for the August period. The calibration was effected by correcting the spectra for attenuation effects and then numerically computing for each curve the integral over frequency of linear average wave height less linear noise. The significant wave height should be proportional to this integral (Bretschneider, op.cit.). 13

The significant height as computed from the variance of the staff gage record was then plotted against integrated linear average and the points were fit using the method of least squares. One spectral curve was unusable for this purposeo The resultant calibration curve, shown in Figure 2 and given analytically by HI = (17.1) j (Linear Average) d(^) + 1.3 ft., (14) ~~3~~~~~ ~ T intercepts the H1/3 axis at a positive value. This may be due to lack of response by the pressure sensor to low amplitude waves or to nonlinear calibration for low amplitudeso D. RESULTS The sources of wind data described in Section 3-B were used in making hindcasts for the periods August 1 - August 10 and September 13 - September 23 of 1964. For each synoptic time during the hindcast periods gradient winds and reported ship winds were entered directly onto a map of Lake Michigan. These winds were then corrected to provide winds at the desired anemometer heights. In each e asme a fetch was estimated by eye and the winds were then averaged over the fetchO When winds from the research tower were available these were used in preference to the gradient and ship winds, and hindcasts were made at the time of measurement of tower winds rather than at synoptic times Forty hindcasts were made during the August period and forty-five during the September period. The location of the research tower is shown in Figure 3. The wind conditions in August are given in Table A.1 in the Appendix. The winds during this period were generally light, from ten to twenty knots, and often were offshore. One would thus expect small amplitude waves. The results of the hindcast are given in Table A.2o The quantities tabulated are the significant height, the period band, and the period at which the theoretical spectra reach their maximum amplitude. For cases in which the waves are fetch or duration limited, only predictions from the Neumann spectrum are giveno For the range of wind speeds observed during this period higher waves are hindcast when using the Pierson-Moskowitz spectrum. In Table A,3 the hindcast significant heights and periods of maximum amplitude are tabulated along with the observed values taken from the spectra shown in Figure Col. Both of the hindcast methods predict higher wave heights than those observed, with the'waves hindcast using the Neumann spectrum in better agreement. For the thirteen occasions on which the observed spectra could be integrated to obtain the significant height, the average observed 14

- 0 0.0 Qi, 0 0 / LI-, 5 I 0 - 0 I I I Ii 0 0.1 0.2 0.3 0.4 0.5 INTEGRATED LINEAR AVERAGE (FEET-SEC-') Figure 2. Calibration curve for spectra.

1 -' -I- - Figure 3. Location of Lake Michigan Research Tower. 16

significant height was 1o7 feet and the average significant height hindcast using the Neumann spectrum, 2o5 feet. The hindcast and observed significant heights are plotted as functions of time in Figure 4o Given the variability of the wave heights, it appears that the agreement is fairo The September period was somewhat more interesting because of the stronger winds and accompanying higher waves. Fortunately, it was possible to make a more detailed comparison of hindcast and observed wave heights during this period. The wind conditions in September are given in Table B.l in the Appendix. At the start of the hindcast period the sea was clam. The winds gradually freshened, reached a peak on the morning of the 14th, and then shifted and became offshore. From early afternoon of the 14th to the morning of the 20th the winds were either weak or offshore. During the next few days the winds increased and then decreased. In the early hours of the 23rd a front passed over the lake and the winds rapidly increased, reaching a peak of 39 knots in the evening. At midnight the research tower collapsed. The result of the hindcast are given in Table B.20 As in the August period, higher waves are forecast when using the Pierson-Moskowitz spectrum. Hindcast significant wave heights and observed significant heights are given in Table B.5, and wave heights hindcast with the Neumann spectrum and observed wave heights in Figure 5~ Except for the period of light and/or offshore winds, the agreement is excellent. The decay of sea into swell starting at noon on the 14th was followed closely by the hindcast, as was the development of the sea during the storm on the 23rd. On a number of occasions the hindcast waves developed more rapidly than observed; presumably this was due to overestimating the duration of the wind system. The periods of maximum amplitude as interpolated from the hindcasts and those taken from the spectra shown in Figure Co2 are tabulated in Table B.4. The agreement is good here also. Another point of interest is the maximum expected wave. This was hindcast for the first two days and last four days of the hindcast period and the results compared with observed maximum waves taken directly from the wave record. The values are given in Table B.5o The excellent agreement serves as confirmation of the statistical theory. In passing it should be noted that the waves which caused the collapse of the tower were fetch and duration limited and had period bands much narrower than would ordinarily be expected for winds of this magnitudeo Whether or not this concentration of energy in a narrow period band was responsible for the tower's collapse is a moot pointo 17

10 - -- * Hindcast, Neumann Spectrum o Observed 0- - ULo5 o 0 0 0 I I I i i i I I I I II,,,I,, [, II,II [ [ [[I[I I 2 3 4 5 6 7 8 9 10 TIME (C.S.T.) (Date in August) Figure 4. Hindcast and observed significant heights: August.

10 -" — "S - Hindcost, Neumann Spectrum o Observed 55 UOI 0 4 o 0 o 0 000 0 -0 0 0000 000 00000 I _ I I I I I I, I... 13 14 15 16 17 18 19 20 21 22 23 24 TIME (C.S.T) (Date in September) Figure 5. Hindcast and observed significant heights: September.

4, SUMMARY AND CONCLUSIONS Hindcasts for the August period were made using corrected gradient winds, ship winds, and winds measured at the Lake Michigan Research Tower as input to the Neumann and Pierson-Moskowitz spectra. Due to the relatively small number of measurements at the research tower, the availability of tower data was less useful than had been anticipated. An examination of the sources of wind data reveals that the corrected gradient winds are more representative of conditions at sea than winds measured at shore weather stations. The sample on which this conclusion is based is fairly small, and a comprehensive study is needed on this matter. It might be possible to average the winds reported by all weather stations around the lake to get representative values. Such a procedure would be useful, since a large amount of labor is involved in computing the gradient winds. If reported ship winds are available these are definitely preferred. It is advisable to correct the ship winds to the anemometer height called for by whatever energy spectrum is used. Since this correction depends on the stability, as does the correction for gradient winds, the air-sea temperature difference should be obtained whenever possible. For the thirteen occasions in August on which hindcast and observed waves could be compared, the average significant height hindcast using the Neumann spectrum was 2.5 feet and the average observed significant height 1.7 feet. The correlation coefficient between observed and hindcast significant heights was 0.75. The agreement for this hindcast period is fair. For the forty-five occasions in September on which comparisons were made, the average significant height hindcast using the Neumann spectrum was 2 feet and the average observed significant height, 1.4 feet. However, during this period there was much better agreement when the waves were large, particularly during the storm preceding the collapse of the tower. This is reflected by a high value of the correlation coefficient, 0.96. The agreement between observed waves and waves hindcast using the Pierson-Moscowitz spectra was not as good. In view of the above facts, it appears that corrected gradient and ship winds should be used in preference to winds measured at shore weather stations. It also appears that acceptable hindcasts can be made using the Neumann spectrum and the procedures given in H.O. Pub. 6035 20

REFERENCES Bretschneider, C. L., 1961, Wave spectra from Hurricane Donna, Spectra of Ocean Waves, 267-272, Prentice Hall, Inc. Caldwell, J. M. and L. C. Williams, 1961, The Beach Erosion Board's wave spectrum analyzer and its purpose, Spectra of Ocean Waves, 259-266, Prentice Hall, Inc. Elder, F. C., 1965, An investigation of atmospheric turbulent processes over water, Report Number Two: Data, 1963 and 1964, Contract Cwb-10714, University of Michigan Report 05982-1-F. Longuet-Higgins, M.S., 1952, On the statistical distribution of sea waves, Journal of Marine Research, XI, 245-266. Longuet-Higgins, M.S., 1962, The directional spectrum of ocean waves, and processes of wave generation, Proc. Roy. Soc. A., 265, 286-315. Pierson, W. J., 1964, The interpretation of wave spectrum in terms of the wind profile instead of the wind measured at a constant height, J. Geophys. Res., 69, 5191-5202. Pierson, W. J. and L. Moskowitz, 1964, A proposed spectral form for fully developed seas based on the similarity theory of S. A. Kitaigoradskii, J. Geophys. Res., 69, 5181-5190. Pierson, W. J., G. Neumann, and R. James, 1955, Practical Methods for Observing and Forecasting Ocean Waves, H.O. Pub. 603, U. S. Navy Hydrographic Office. Rice, S.O., 1944-1945, The mathematical analysis of random noise, Bell System Tech. J. 23, 282-332 and 24, 46-156. Strong, A. 1965, Personal Communication. University of Michigan, 1963, Research and training in limnology, oceanography, and related fields. 21

ACKNOWLEDGMENTS The author wishes to acknowledge the cooperation shown by the sponsoring agency, the UoS. Lake Survey. Program Direction was provided by Professor John C. Ayers of the Department of Meteorology and Oceanography. The numerical calculations were performed by Mr. Fred Brock. D. Lee Harris of the U.S. Weather Bureau served as an unpaid consultant. The report was typed by Mrs. Pat Marchello. 22

APPENDIX A AUGUST DATA AND RESULTS

TABLE A.1 WIND CONDITIONS IN AUGUST HINDCAST PERIOD V = Wind speed at 7.5 meters U = Wind speed at 19.5 meters Time(C.S.T.) V(Knots) U(knots) <(~) Time(C.S.T.) V(Knots) U(Knots) <(o) 1:0600 15 16 19-20 6:0600 18 19 15 1:1200 16 17 20 6:1200 18 19 16-20 1:1800 14 15 19-22 6:1800 15 16 18-23 2:0000 10 12 18-21 7:0000 14 15 20-25 2:0600 13 15 23 7:0726 10.9 12.9 21 2:1200 14 16 25 7:1200 15 16 28-33 2:1926 9.8 11.3 23-25 7:1736 14.8 16 30 3:0000 11 13 26 8:0000 10 10 00 3:0600 9 11 32 8:0740 17.6 17.7 30 3:1136 6.7 7.7 30 8:1130 21.5 22.4 35-03 3:1636 12.8 15 32 8:2030 Calm 4:0000 8 9 03 9:0000 14 15 32 4:0600 9 10 31-33 9:0600 Calm 4:1200 9 9 34 9:1130 9.4 10.7 35 4:1800 5 6 03-10 9:1630 7.1 8.7 35 5:0000 17 18 02-08 10:0000 14 15 15-18 5:0600 15 16 02 10:0600 14 15 20 5:1406 16.6 17.5 35-04 10:1330 6.2 7.3(esto) 20 5:1800 15 16 04-05 10:1530 6.8 8.0(est.) 16 6:0000 14 15 15 10:1950 8.8 10o4(est.) 12 24

TABLE A.2 HINDCASTS FOR AUGUST HINDCAST PERIOD H1/3 = Significant height P.B. = Period band containing 90% of wave energy Tm = Period of maximum amplitude of energy spectrum Abbreviations for sea conditions: FD, fully developed, FL, fetch limited, DL, duration limited, Sw, swell. Time (C.S.T.) Sea Hindcasts (Neumann Spectrum) Hindcasts (Pierson-Moskowitz Spectrum) H1/3(ft.) PoB.(sec.) Tm(sec.) H1/(ft.) P.B.(sec.) Tm(sec.) ro 1:0600 FD 3.8 1.8-8.3 6.1 4.7 2.4-7.5 6.0 1:1200 FD 4.5 2.0-8.8 6.5 5.3 2.5-8.0 6.4 1:1800 FD 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 2:0000 Sw 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 2:0600 FD 2.7 1.2-7.4 5.3 4.1 2.2-7.0 5.6 2:1200 FD 3.2 1.5-7.8 5-7 4.7 2.4-7.5 6.0 2:1926 Sw 3.2 1.5-7.8 5-7 4.7 2.4-7.5 6.0 3:0000 FD 1.8 1.0-6.5 4.4 3.1 1.6-5.2 4.1 3:0600 Sw 1.8 1.0-6.5 4.4 3.1 1.6-5.2 4.1 3:1136 Sw 1.8 1.0-6.5 4.4 3.1 1.6-5.2 4.1 3:1636 FD 2.6 1.2-7.4 5.2 4.1 2.2-7.0 5.6 4:0000 Sw 2.6 1.2-7.4 5.2 4.1 2.2-7.0 5.6 4:0600 Sw 2.4 1.2-5 5.5 3.4 2.2-5.5 5-5 4:1200 FD 1.1 1.0-4.0 3.6 1.5 1.3-4.2 3.6 4:1800 Sw 1.1 1.0-4.0 3.6 1.5 1.3-4.2 3.6 5:0000 FL 0.5 2.3-2.5 2.5 5:0600 FL 0.5 1.5-2.5 2.5 5:1406 FL 1.4 2.0-3.0 3.0 5:'1800 FL 0.5 1.4-2.5 2.5 6:0000 FL 1.5 1.2-4.5 4.5...............-.L.,.L...

TABLE A.2 (Concluded) Time (C.S.T.) Sea Hindcasts (Neumann Spectrum) Hindcasts(Pierson-Moskowitz Spectrum H1/3(ft.) P.B.(sec.) Tm(sec.) H1/3(ft.) P.B.(sec.) Tm(sec.) 6:0600 FL 1.5 2.5-3.0 3.0 6:1200 FL 4.0 2.5-5.8 5.8 6:1800 FD 3.8 1.8-8.3 6.1 4.7 2.2-7.0 5.6 7:0000 FD 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 7:0726 Sw 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 7:1200 FD 3.8 1.8-8.3 6.1 4.7 2.4-7.5 6.0'7:1736 FD 3.8 1.8-8.3 6.1 4.7 2.4-7.5 6.0 8:0000 Sw 2.0 1.8-4.2 4.2 2.7 2.4-4.2 4.2 8:0740 DL 5.4 2.4-9.0 7.1 r) 8:1130 Sw 5.9 2.4-9.8 7.1 ~C\ 8:1930 Sw 2.4 2.4-4.0 4.0 2.4 2.8-4.0 4.0 9:0000 DL 2.8 1.5-5.8 5.7 9:0600 Sw 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 9:1130 FL 1.1 1.0-3.8 3.8 9:1360 FD 0.6 1.0-4.0 2.9 1.4 1.3-4.1 3.3 10:0000 FL 1.6 1.5-7.8 5-7 10:0600 FD 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 10:1330 Sw 3.2 1.5-7.8 5.7 4.1 2.2-7.0 5.6 10:1530 Sw 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 10:1950 Sw 3.0 1.5-6.8 5-7 4.0 2.2-6.8 5.6

TABLE A.3 COMPARISON OF HINDCAST AND OBSERVED WAVE VALUES FOR AUGUST HINDCAST PERIOD H1/3 = Significant height Tm = Period of maximum amplitude of energy spectrum Time (C.S.T.) Hindcast (Neumann Spectrum) Hindcast (Pierson-Moskowitz Spectrum) Observed H1/3(ft.) Tm(sec.) H1/3(ft.) Tm(sec.) H1/3(ft.) Tm(sec.) 2:1926 5.2 5.7 4.7 6.0 1.6 4.2 3:1136 1.8 4.4 3.1 4.1 1.7 4.2 3:1636 2.6 5.2 4.1 5.6 1.7 3.6 5:1406 1.4 5.0 2.0 5.6 7:0726 3.2 5.7 4.1 5.6 2.2 4.6: 7 7:1736 3.8 6.1 4.7 6.0 1.7 4.6 8:0740 5.4 7.1 2.9 4.6 8:1930 2.4 4.0 2.4 4.0 1.6 5.0 9:1150 1.1 3.8 1.4 6.2 9:1650 0.6 2.9 1.4 3.5 1.4 5.6 10:1330 5.2 5.7 4.1 5.6 1.9 5.8 10:1530 3.2 5.7 4.1 5.6 2.0 4.2 10:1950 3.0 5-7 4.0 5.6 1.7 5.o

APPENDIX B SEPTEMBER DATA AND RESULTS

TABLE B.1 WIND CONDITIONS IN SEPTEMBER HINDCAST PERIOD V = Wind speed at 7.5 meters U = Wind speed at 19.5 meters Time(C.S.To) V(Knots) U(Knots) <(~) Time(C.S.T.) V(Knots) U(Knots) <(~) 13:0000 Calm 18: 1800 17 18 12-15 13:0600 10 10 23-27 19:'-000 15 16 13-14 13:1200 12 13 18-25 19:0600 10 11 12-15 13:1800 11 13 21-23 19:1200 8 9 13-18 14:0000 14 15 21-24 19:1800 10 11 12-14 14:0600 16 17 23-25 20:0000 8 9 13-14 14:1200 14 15 27 20:0600 13 14 17-20 14:1800 21 22 03-06 20: 1200 15 16 16 15:0000 21 21 05-09 20: 1800 14 15 17 15:0600 18 18 07-09 21:0000 16 18 16 15:1200 14 15 07-09 21:0600 15 16 18-20 15:1800 13 14 03-05 21:1200 10 11 17-18 16:0000 12 12 13-18 21:1800 9 10 12-19 16:0600 12 13 12-15 22:0000 9 9 19-20 16:1200 9 9 15-18 22:0600 13 14 30-34 16:1800 10 11 16-19 22:1200 11 12 27-32 17:0000 11 11 18-23 22:1800 12 13 11-13 17:0600 9 10 15-25 23:0000 12 13 14 17:1200 6 7 18 23:0600 23 24 25-30 17:1800 14 15 14 23:1200 19 21 24 18:0000 15 16 15 23 1800 37 39 26 18:0600 13 14 15 24:0000 24 26 27 18:1200 17 18 12-15 30

TABLE B.2 HINDCASTS FOR SEPTEMBER HINDCAST PERIOD H/3 = Significant Height P.Bo = Period band containing 90o of wave energy Tm = Period of maximum amplitude of energy spectrum Abbreviations for sea conditions: FD, fully developed, FL, fetch limited, DL, duration limited, Sw, swell Time(C.S.T.) Sea Hindcasts (Neumann Spectrum) Hindcasts (Pierson-Moskowitz Spectrum) H1/3(ft.) P.B.(sec.) Tm(sec.) H/3(ft.) P.B.(sec.) Tm(sec.) 13:0000 0 0 13:0600 FD 1.2 1.0-6.0 3.8 1.6 1.4-4.4 3.5 13:1200 FD 2.2 1.0-7.0 4.8 3.1 1.9-6.1 4.9 13:1800 FD 1.8 1.0-6.5 4.4 3.1 1.9-6.1 4.9 14:0000 FD 3.2 1.5-7.8 5-3 4.1 2.2-7.0 5.6 14:0600 FD 4.5 2.0-8.8 6.5 5. 2.5-8.0 6.4 14:1200 FD 3.2 1.5-7.8 5.7 4.1 2.2-7.0 5.6 14:1800 Sw 3.2 1.5-7.8 5-7 4.1 2.2-7.0 5.6 15:0000 Sw 1.7 1.5-3.9 3.9 2.0 2.2-3.9 3.9 15:0600 Sw.9 1.5-2.6 2.6.9 2.2-2.6 2.6 15:1200 FL.5 1.5-2.0 2.0 15:1800 FL.5 1.2-2.0 2.0 16:0000 FL.7 1.0-2.4 2.4 16:0600 FL.7 1.0-2.4 2.4 16:1200 FL.9 1.0-3.8 3.8 16:1800 FD 1.4 1.0-6.0 4.0 2.0 1.5-5.0 4.4 17:0000 FD 1.8 1.0-6.5 4.4 2.2 1.6-5.2 4.1 17:0600 FD 1.1 1.0-5.5 3.6 1.6 1.4-4.4 3.0 17:1200 FD.4.8-3.0 2.4.9.9-2.8 2.3 17:1800 FL.9 1.5-2.5 2.5 18:0000 FL 1.3 1.8-3.0 3.0 18:0600 FL 1.2 1.2-3.0 3.0

TABLE B.2 (Concluded) Time(C.S.T.) Sea Hindcasts (Neumann Spectrum) Hindcasts (Pierson-Moskowitz Spectrum) H1/3(ft.) P.B.(sec.) Tm(sec.) H1/3(ft.) P.B.(sec.) Tm(sec.) 18:1200 FL.9 2.3-2.5 2.5 18:1800 FL.9 2.3-2.5 2.5 19:0000 FL.9 1.8-2.5 2.5 19:0600 FL.8 1.0-2.5 2.5 19:1200 FD.8 9-3.5 3.2 1.5 1.-4.2 3.4 19:1800 FL.8 1.0-2.5 2.5 20:0000 FL.6.9-2.4 2.4 20:0600 FD 2.7 1.0-7.0 4.8 3.6 1.9-6.1 4.9 20:1200 FL 2.8 1.8-5.5 5.5 Nu 220:1800 FL 2.8 1.7-5.9 5.9 21:0000 FL 3.0 2.0-5.0 5.0 21:0600 FD 3.8 1.8-8.3 6.1 4.7 2.4-7.5 6.0 21:1200 FD 1.4 1.0-6.0 4.0 2.2 1.6-5.2 4.1 21:1800 FD 1.1 1.0-6.0 3.6 1.6 1.4-4.4 3.6 22:0000 FD 1.0 1.0-6.0 3.5 15 1.3-4.2 3.4 22:0600 DL 2.0 1.2-4.8 4.8 22:1200 FL 1.8 1.0-6.5 4.4 22:1800 FL.9 1.0-2.5 2.5 23:0000 FL.9 1.0-2.5 2.5 23:0600 DL 2.8 3.5-4.0 4.0 23:1200 DL 5-7 2.8-7.0 7-0 23:1800 FL 9.0 5.8-6.0 6.0 24:0000 FL 9.0 3.7-8.5 8.5

TABLE B.5 COMPARISON OF HINDCAST AND OBSERVED SIGNIFICANT HEIGHTS FOR SEPTEMBER HINDCAST PERIOD H1/3 = Significant height Ne Neumann spectrum P-M: Pierson-Moskowitz spectrum 0: Observed Time (C=S.T.) Hi/3(ft,) Time (C.S.T.) HL/5(ft.) N P-M 0 N P-M 0 15:0000 0 0 0 19:0600.8 0 1530600 1.2 1.6 0 19:1200.8 1.5 0 15:0900 1.5 19:1800.8 0 1531200 2.2 3.1 1.8 20:0000.6 0 13:1800 1.8 3.1 1.6 20:0600 2.7 3.6 0 13:2100 1.6 20:1200 2.8 6 14:0000 3.2 4.1 2.7 20:1800 2.8.8 14:0600 4.5 5, 3 3.2 21:0000 3.0 2.0 14:1200 3.2 4.1 3.8 21:0300 3.8 14:1500 2.7 21:0600 3.8 4.7 2.3 14:1800 3.2 4.1 2.0 21:0900 1.4 15 0000 1o7 2.0 1.1 21:1200 1.4 2.2 1 1 15o0600.9.9.9 21:1800 1.1 1.6.9 15:1200.5 0 22:0000 1.0 1.5.6 15~1800 5.7 22:0600 2.0 1.3 16:0000 o7 0 22:1200 1.8 1.3 16:0600 o7 0 22:1800 9.6 16:1200.9 0 23:0000 9.6 16o1800 1.4 2o0 0 23:0300.9 17:0000 1,8 2o2.7 2350600 2.8 2.8 17:0600 o 1 1.6.9 23:0900 6.6 17~1200 o4 o9 -6 23:1200 5~7 6.3 17:1800 9 0 23'1800 9.0 7 5 18 o0000 1.3 0 25:2100 8.7 18:0600 1.2 0 23:2330 8.9 18:1200 o9 1.6 24:0000 9,0 18:180 o9 o 6 9 o 0000.9 1.3 33

TABLE B.4 COMPARISON OF HINDCAST AND OBSERVED VALUES OF PERIOD OF MAXIMUM AMPLITUDE FOR SEPTEMBER HINDCAST PERIOD Tm = Period of maximum amplitude of energy spectrum N: Neumann spectrum P-M: Pierson-Moskowitz spectrum 0: Observed Time(C.S.T.) Tm(sec.) _N P-M 0 21 0100 5.2 4.8 21 0500 5.9.6 21:0900 5.4 4.8 21 1400 3.9 4.9 23 0700 4.5 5.7 23 1100 6.5 7.0 2531500 6.5 7.8 23 1900 6.4 8.2 23 2000 6.8 8.3 23 2100 7-3 8.6 23 2200 7-7 8.7 23 2300 8.1 8.9 34

TABLE B.5 COMPARISON OF HINDCAST AND OBSERVED LARGEST WAVE IN SEPTEMBER HINDCAST PERIOD Hm = Largest wave in twenty minute period preceding given time N~ Neumann spectrum 0O Observed Time (C.S.T.) Hm(feet) N 0 1531200 359 3.8 13o1800 3 2 3.5 14:0000 5o7 5.0 14s0600 7.8 6.4 14 1200 5.7 7.1 14 1800 5-7 3.8 15:0000 3.1 2.2 15:0600 1.4 2.2 20:1200 5-0 1.4 20:1800 5.0 1.5 21: 0000 5.4 3.4 21: 0600 6.7 4.9 21:1200 2.5 2.3 21' 1800 2.0 1 9 22:0000 1.8.8 22.0600 3.6 2.6 22:1200 3.2 2.6 22:1800 1.6 1.9 2350000 1.6 1.1 2350600 5 1 5.2 23 1200 10.1 11.2 235 80 14.0 14.2 24:0000 15.9 15.7 35

APPENDIX C OBSERVED SPECTRA

Figure C.1. Observed spectra: August. 38

1.5 August 2, 1926 (CST) 1.2 LIJ UJ U0.9 I 0.6 03 NOISE 0 3 3.5. 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5 August 3, 1136 (C.S.T) 1.2 HLUJ U 0.9 I w 0.6 I 0.3 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 39

1.5 August 3,1636 (C.S.T) 1.2 LLU | 0.9 0 F3 LLJ I 0.6 0.3 - NOISE ore - I_______I ~ I I I i I I i IiI i i i i 3 3.5 4 5 6 7 8 9 10 12 15172025 30 PERIOD (SECONDS) 1.5 August 5,1406 (C.S.T.) 1.2 iI u 0.6 0.3 - NOISE ~1 L I i I - I I I I I I III itl I I 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 40

1.5 August 7, 0736 (C.S.T) 1.2 UJ uS 0.9 IU 0.6 r 0.3 -NOISE 3 3.5 4 5 6 7 8 9 10 12 15 17 20 2530 PERIOD (SECONDS) i.5 August 7, 1736 (C.S.T.) 1.2 LU LU LL 0.9 -. I LJ' 0.6 NO ISE 3 3.5 4 5 6 7 8 9 10 12 15 17202530 PERIOD (SECONDS) 41

1.5 August 8,0740(C.S.T.) 1.2 IL LL 0.9 I(2, Lu 0.6 0.3- NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5August 8, 1130 (C.S.T) 1.2 w LLJ u_ 0.9IU 0.6 - 0.3 - NOISE 0 *__ I I I I-I 111t111111 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 4~2

1.5 August 8, 1930 (C.S.T.) 1.2 IUi Li u_ 0.9 I I 0.6 0.3 NOISE 01 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5 August 9, 1130 (C.S.T.) 1.2 LLu U. 0.9 I w 0.6 0.3 NOISE 0 t o I I I I I I I I I I I l I I I I 0 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 43

1.5 August 9, 1630 (C.S.T.) 1.2 LiJ 0.9 I 0.6 0.3 NOISE O I I I I I I I111111 I1 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5 August 10, 1336 (C.S.T.) 1.2 IL 0.9 I 0 I 0.6 03 -NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 44

1.5 August 10, 1530 (C.S.T. ) 1.2 LJ w 0.9 U0 i 0.6 I 0.3 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5 August 10, 1930 (C.S.T ) 1.2 LJ u_ 0.9 i0o W 0.6 0.3 NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 45

Figure C.2. Observed spectra: September. 46

1.5 September 21,0100 (C.S.T.) 1.2 LU LuL 0.9 I 0 0.6I 0.3 NOISE 0.3 3 3.5 4 5 6 7 8 9 10 12 15172025 30 PERIOD (SECONDS) 1.5 September 21, 0500 (C.S.T.) 1.2 u 0.9 i 0.6 I 0.3 0- NOISE 0.3 0O _____________I I I I I I I I I I III 1 1 1 1 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 47

1.5 September 21,0900 (C.S.T. ) 1.2 ILuJ LL 0.9 (D,i 0.6 I 0.3 - NOISE 0. 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 1.5 September 21, 1300 (C.S.T.) 1.2 LLJ uL 0.9 H I(. I LUJ 0.6 - 0.3 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 48

3.0 September 23,0700 (C.S.T.) 2.4 w w PERIOD (SECONDS) wi 1.2 6.0September 23, 1100 (C.S.T. ) 4.8L.J E 3.6 I (0 w 2.4 I / \ 1.2- NOISE 0 I I I I 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 49

6.0 September 23, 1500 (C.S.T) 4.8 Iw LJ L 3.6 w 2.4 1.2 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 6.0 September 23,1900 (C.S.T) 4.8 w w Lu 3.6 - Iw 2.4 1.2? NOISE 3 3.5 4 5 6 7 8 9 10 12 15172025 30 PERIOD (SECONDS) 50

6.0 September 23,2000 (C.S.T.) 4.8 u 3.6 I / \ iU 2.4 \.2 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 6.0 September 23,2100 (C.S.T) 4.8' 3.6 I / \ u 2.4 1.2 - NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 51

6.0 September 23,2200 (C.S.T.) 4.8 LL36 I u_ 3.6 I 2.4 1.2- NOISE 3 3.5 4 5 6 7 8 9 10 12 15172025 30 PERIOD (SECONDS) 6.0 September 23, 2300 (C.ST) 4.8 / \ LLU 3.6/ \ I 2.4 1.2 NOISE 3 3.5 4 5 6 7 8 9 10 12 1517202530 PERIOD (SECONDS) 52

30 25 0 20 UJ w (f) 0 0 15 (r'J w 10 5 0~ ~ ~ ~ ~~~~~~~~~~~~. 0 —- — 21 0 3t0 CORRECTION FACTOR Figure C3.. Correction for hydrodynamical attenuation.

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