TECHNICAL REPORT PRELIMINARY INVESTIGATIONS ON PREDICTING PROPERTIES OF BOTTOM PRESSURE FLUCTUATIONS RICHARD C. TIMME and FANNIE A. STINSON Applied Oceanography Branch Division of Oceanography JULY 1955 U. S. NAVY HYDROGRAPHIC OFFICE WASHINGTON, D. C. Pras. Ry M4 ee, TR-14 at Mine eng OE Se Oe iy. ee ABSTRACT The principal objective of this study is to determine which of the characteristics of pressure on the ocean bottom could be predicted if the sea surface wave spectrum is known. In particular, the distribution of negative pressure amplitudes, the distribution of pressure periods, and the distribution of half-periods for discrete negative amplitudes are investigated. An analysis is made of surface wave records to obtain the inherent power at various frequencies. The resulting surface power spectrum is attenuated to the bottom using classical hydrodynamic methods. The computed power at the bottom compared closely with the total power determined from a simultaneous bottom pressure record. Given total power, the distribution of pressure amplitudes can be determined based on a Rayleigh distribution. The distribution of pressure periods cannot be obtained analytically, but empirical methods give agreeable results. A method of determining the dis- tribution of half-periods for discrete negative pressure amplitudes is obtained for the case when this amplitude is zero. Comparison between predicted and observed pressure charact ristics is presented. — MBL/WHO!I ONE FOREWORD Recent developments in the study of ocean waves have stimulated interest in the prediction of pressure fluctuations at the ocean bottom caused by ocean waves. Although there is considerable literature on the exponential decrease in wave motion downwards from the surface, newer concepts of wave analysis at depth based on the power spectrum concept have scarcely been touched. This paper is designed to fill in some of this gap of knowledge and to provide a method for estimating bottom pressure from a given surface ae power spectrum. This developmental paper is an interim step in the U. S. Navy Hydrographic Office's continuing program for developing suitable wave forecasting methods for use by commercial and lcloan_ “pe Je B. COCHRAN Captain, U. S. 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Introduction Pressure power spectrum e ° CONTENTS Prediction of power spectra «. « « « Analysis of pressure power spectrum Remarks Bibliography . Pressure mesponse factors eis 6 s © © « © 6 6:66 6 «wl 6 Comparison between predicted and observed pressure power * 9? 6 ° spectra at a depth of 150 feet Behavior of cosh function ° e ® e ° ° eo ° eo e FIGURES 6. (e” (6° (ale er 64: (Oa ie Fey le.) e's ve: e ee ee Pressure power spectrum and associated Ep (M) curve 150 feet OY Oe LOY ONS) (Ore) (0-8 Te" 6% fe? "6-4 +07 (8! « ©: Comparison between predicted and observed amplitude ohisheal bhava 5 4 On ooo C 9: 8 58), Or © oo @ e ° eo e e Predicted and observed ogive curves for wave periods Predicted and observed ogive curves for wave periods Comparison between predicted and observed distributions wave periods ey 6). 4'-09 (@. is © Page 10 14 16 20 al 23 Wc Ta FIGURES (Cont'd) Ratio of expected number of maxima to expected number of waves as a function of the maximum frequency Fy ao Dt 0, 0 TABLES Predicted and observed values of pressure power spectrum at HAO mich Qtech OMOrO ClO 0 OO oO 6 0 66 O8D. 00 0-0 0 6 Theoretical values to be used in predicting wave amplitude distributions .« » » 6 e © © e © © © © © © © © © © © oo oe Rep determined for minimm \ Cn Our ksout Ou fr GeO 6 vi 25 15 26 A. INTRODUCTION Since January 195, the Navy Hydrographic Office has been engaged in the qualitative and quantitative analysis of wave and pressure data collected in 30 to 150 feet of water off the east coast of the United States. The analysis of these data is fundamental in that it has afforded an opportunity to initiate prediction techniques for determining various characteristics of bottom pressures. In some instances, the full signifi-~ cance of these properties have not been determined. Many of the features of surface waves have been explained by Pierson (1952) and Neumann (1953) on the supposition that a wave record as a func- tion of time is Gaussian. Employing the techniques of Tukey (199) and Tukey and Hamming (1919), the theory has been further developed to explain the methods of wave analysis for estimating the power spectrum of the steady-state sea surface. Various theories based on the analysis of noise, advanced by Rice (19) and 195), have been applied to surface waves. None of these theories have been carried completely or satisfactorily to the point of explaining how the surface waves are attenuated with depth. Yet the need for con- venient and efficient methods of predicting properties of pressure waves has made itself increasingly felt in research and development in fields where background pressure ( the variation in pressure on the ocean bottom as a result of the wave action at the surface ) is important in determining an index of effectiveness for pressure gear. In this paper some techniques are presented which have not been considered before for use in the theoretical and experimental investiga- tions of fluctuations in pressure at the bottom of the ocean. The limita~ tions of such procedures are discussed and some sample results are showne B. PRESSURS POWER SPECT RUM It is assumed that the pressure variations at a given depth are essen- tially reflections of the features existing at the surface of the oceane Consider then, the energy spectrum, namely the pressure power spectrum, formed as a result of the wave action on the surface. If the formation ts governed purely by attermation of the surface profiles, the pressure varia- tions will be given by aX ab Netw) = K* A Cp) “Gy where A’) is the power spectrum of the sea surface, K is the attenua- tion factor, and Ne (fe) is the residual energy at depth a. Theoretically, a family of curves can be derived from equation (1) by varying the depth. When the depth considered is equivalent to the depth from the still-water level to the bottom, the attenuating factor is ete MONS Oc (2) Cosh? (ard /L) where L. is the wave length associated with the spectral period, ay; and d is the depth of water. In practice, awd /\ is obtained from a knowledge of d J Le N where ee 5\aTtis the deepwater wave length. These quantities and other related factors have been tabulated by Wiegel (1948). The functional relationship between Ap( je) and A () defines the 2 -. Se | T= PERIOD (seconds) = aie Er E= ie | | i i i i x + = = eee == iS i — i i= Se! t ty t ie t= eal = Eee. Les oT ae ii | 4 444} 1 T Tht | {fl aaa a 0a BS | me me} #910 2 3 4 5s 67898 703000 8910" 2 3 4 5 6 78910" 2 3 4 5 6 7 8 910 +—FET—= 2 2 3 4 5 6 78910 2 Ee 8 FF SE EERE ot ee : SSSsesoe En Sh d= STILL WATER DEPTH (feet) 1 a2 03 04 05 06 07 08.0910 2 inca ery 2 cs) 2 0 © & 7 190100 200 se oe P=PRESSURE AMPLITUDE (inches) FIGURE |. PRESSURE RESPONSE FACTORS ™ sede Bus curve of the so-called pressure response factor from which cut-off values may be determined. The cut-off value 1s defined as that value of KE at which there is a two-percent response of the primal wave, or the fre~ quency at which the power is essentially zero. A set of such theoretical Curves, K=A,(uy/ A (pL), constructed for various values of the parameters y= QT] and d is shown in fisure 1, It is to be noted that the defini~ tion of the response factor is independent whether wave height or wave amplitude is used since the linear relationship, wave height equals twice the wave amplitude, is assumed to hold. Although a completely analytic solution of equation (1) is not possible, it can be solved numerically for any given depth, With AY) determined A from a wave-staff record and Ka given by equation (2), Apel em) can be determined. A set of data consisting of surface wave height measurements was used to obtained a surface power spectrum NADY ; For this purpose , values pity) were taken off of a 20-minute wave=staff record at inter- vals of At=4 seconds. These values were used to solve Tukey's chain of equations (Pierson, 1952) as follows: d Nee Qo = 2 /(N-p) vey a(t) alt -p) p=0,',..-m™ (3) Ly Uy, me-}j i frm (Qo+a ZL Qp cos Crph/m) + Qm costh — CH) ee h=o1)--. 7™ £43 Ly, +. St Ly +. 83 bass (5) i Nua = Uy atm [at (b) N is the total number of values, & (ty) = [ p(t,) -p] 5 con~ sidered in the analysis, ee = Nate ) ear pal Ea and Pe= Wh/At m = an/® - The actual work of solving these equations was done by an IEM elect~ ronic computer at the Hydrographic Office. The apparent amplitudes, AY ( fy were then transformed into true amplitudes by modifying each dis- crete spectral amplitude. This was done by using wave=staff correction factors determined by the University of California (1945). The surface wave record was obtained by the H. 0. electric wave staff (Upham, 1955). This is a floating wave staff which requires a correction for its natural oscillatory characteristics. The set of individually-corrected values is the surface power spectrum. To obtain the background pressure at the bottom, which is at d ={50 feet, it is necessary to attenuate this surface power spectrum for discrete values of the frequency [, » Taking d= 160 feet, it is possible to determine the significant range of values for Al, = AW /T from figure 1, Clearly, the frequencies range over the interval am/9 to iT ]0. With these limiting fe As and corresponding values of Ke aaa A*(,) ythe pressure power spectrum follows directly from equation (1). The pressure power spectrum so obtained is shown in figure 2. The experimental spectrum obtained from analysis of a pressure record at 150 feet, taken simultaneously with the surface record, is also shown, Quan- titatively, the agreement in the significant range of frequencies between the observed and predicted values, given in table I, is considered good. The curves defining the confidence limits for the predicted spectrum are given in figure 2. These determine the ninetyepercent confidence level, faa OS! 40 Hidsd vo iv Vedosds YSMOd SYNSSSYd GSAYSSSO GNV GSLOGSYd NASMLAG NOSINVdWOD 2 JuNdIS (spuocss Ul) 1 GOIN3d SAVM 02 61 81 ZI OT ST v1 el ZI it Ol 6 GaAuasao ¥ \ da1910 aud —— \ / ana937 \ / ALISNAG ADSYANA JAILV143Y 4 2 (Spuodas=}}) Table I Predicted and Observed Values for Pressure Power Spectrum at 150 Feet h Me T AS) (predicted) ACH) (observed) 6 1/10 20.0 20450 20422 7 1/8.6 iyjoal 3010 ©2110 8 1/725 15.0 8612 «8902 9 1/6.7 1s) 1.2964. 1.2720 10 §9T/6.0 12.0 1.1260 ~ 8902 11 TT /505 10.9 5622 05084 2 “t/5.0 10.0 © 3165 2 eaIalo) 13 1/4.6 9.2 01185 0844, 1h, TW /4.3 8.6 0400 0422 15 Ti /4.0 8.0 sO 20422 That is, ninety percent of the time the spectrum will be expected to lie between the two dashed curves. Another set of calculations was made using a relationship between the average wave length te and the average period ale of a wave system, introduced by Pierson (1952). If it is assumed that Beery, (s.\2a) 7 * ) holds over a narrow spectral band width, then it can be substituted in the expression Carl foun aie (3) Ap (3 “cosh* | (44d/q) iteath (74/4 )| in which fez an/T and ————————— cosh” [ foed/9) vheoth Guid /4)\ is the attemating factor at depth d . Itcoth ( ped i) 4) is the iterated hyperbolic cotangent of peal4 >» and amd = (u'd/4) itecth ( w'd/4) : L This factor Ke » when applied to the surface power spectrum mentioned above, determined a pressure power spectrum much smaller than the true spectrum obtained from the analysis of the pressure record. In effect, making this substitution changed the attenuating factor from Ce te | eae +o neh Menace , where do = aud cosh" Uy CosWU.5) 4a The behavior of these functions is shown in figure 3. The results reported 9 1000 -- SLINN SLNIOSEV iO L FIGURE 3. BEHAVIOR OF COSH FUNCTION 10 — here indicate that the use of the average ) L 9 is not valid in this instance. It was considered worthwhile to give these results as an indication of what had been found for one particular case. In the following sections, the methods of predicting and analyzing power spectra will be described. The three characteristics of bottom pressure fluctuations of general interest will be discussed in some detail. These are the distribution of negative wave amplitudes , Ps» the distribution of wave periods colt and the distribution of a discrete negative wave amplitude, Ap ,with half-periods, + . All distributions are best expressed in ogive curves. These ogive curves and various other properties of pressure fluctuations will be determined and compared with observed results. C. FREDICTION OF POWER SPECTRA For a given wind velocity, V , when fetch and duration are considered unlimited, the so-called co-cumulative power spectra are defined and a number of such E(f) curves are given by Neumann (1953). These curves de-= fine the power spectrum for each wave system with total inherent energy Ewald (Ovo)? feos , (4) When either fetch or duration of wind is a limiting factor, the energy in the system also is obtainable from Neumann's curves. Basically, E is proportional to the summation of the amplitudes squared over the entire range of frequencies, and is equivalent to Q.= LU, given, respectively, by equations (3) and (5). For all practical purposes, the power spectrum is given by 11 Ary d ECE) (10) A (4) Tf where E(f) is nondecreasing and the largest value is given by equation (9). Hence, once E (f) is known, the power spectrum of the surface follows directly from the operations indicated in equation (10). The adopted AR(£ mike saith 3 values of ( y over the significant band of frequencies can be attenu- ated to a given depth as outlined previously. The result will be the pressure power spectrum. The procedure for analyzing this spectrum to obtain some characteristics of pressure on the bottom of the ocean will be described next. D. ANALYSIS OF FRESSURE POWER SFECTRUM Writing equation (10) as A’ (+) df=d Ep (+) and integrating both sides of the equation results in ae a (iN) fe Ep(F) can be described as the area under the power curve for a give range of frequencies. More explicitly, Eo (fh) is the area under Ap ce) fron fae to f= . If it is assumed that Ep(f) starts with the value zero at fob the cut-off frequency, then Eo max Cf) will not differ appreciably from the sum of the amplitudes squared, obtained by integrating over the entire range of frequencies from £20 to f=cc, Thus Ep max (F) is the total area under Ap (f) over the significant band of frequencies. It should be noted that the expression Ep (mM) may be obtained by een f by its equivalent value pb [an ' 12 The total area under the pow2r spectrum may be obtained in a number of ways. The simplest method, of course, is to use a planimeter. The method more frequently followed, however, consists of dividing the frequency i axis into a set of intervals and determining the value of the power curve at the midpoint of each interval. The product of this value and the length of the interval summed over the entire range of frequencies is approximately Eprnax (1x) Figure 4 shows the experimental data Ke (a) at a depth of 150 feet. The associated Ep() curve was determined by integrating over the spectrum with a planimeter. The total area was found to be 59 st" over the signifi- cant band of frequencies, which approximates closely the value E pmax (u)= 204 i" already obtained by machine analysis. The numerical value B= Eo ax is the quantity used in forecasting the wave amplitudes. Once E is known, the ogive curve representing the number of waves with amplitudes greater than a specified amplitude, P. » can be obtained from table II, if it is assumed that the values therein hold for all wave records. Using E= .Qo4 $3 , ines VE = .5IK SH. the cumlative distribu- tion of waves with amplitudes greater than Ps inches was determined for the power curve shown in figure 4. Figure 5 shows the comparison of the predicted curve and the curve obtained from a hand analysis of the original wave record. The next step is to consider the distribution of waves with periods greater than a specified period, 1. Theoretically, very little has been done along this line because of the variation of periods from wave to wave. However, Rice (1944 and 1945) has derived an expression for the expected number of zeros per second which may occur in a random time series. It is assumed that the number of times the record crosses the mean line is the number of zeros; thus the expected number of zeros per second is twice 13 41445 0S! LV 3AuMND (7) 43 G3LVIOOSSY GNV WnYLoads YaMOd 3YNSS3Y¥d bv 3YNOIS 09 09 09 09 09 09 09 09 o9 6 Ft og 09 09 09 o< == a age, eae Sas a [Ee = =" — = = == yo 4 wor LST LyT LET LZ1 hai LOT ue l ug LL 29 ug h GZ os 9°8 26 O1 6°01 ral eEl GI VZ1 0z 09 y 91 GI +I €1 ZI Il ot 6 t 7 m ™ = ins) (spuodas) 2 L334 14 Table I] Theoretical Values to Be Used in Predicting Weve Amplitude Distributions Percent of waves 1 ith P. (inches of water) s amplitudes > ne 100 | 0.00 Pz 90 3.84 Vi 80 5.6, Va 70 7.20 Vi 60 | 8.52 Vi 50 9.96 Vs 40 11.52 Vu 30 13.20 Vu 20 15.24 VE 10 18.24 BE 15 LEGEND PREDICTED 4 OBSERVED] 0 2 4 8 10 12 14 16 : : WAVE AMPLITUDE Ps (inches of water) FIGURE 5. COMPARISON BETWEEN PREDICTED AND OBSERVED AMPLITUDE DISTRIBUTIONS 16 the expected nunber of waves per second. This may be expressed as follows: Ia, $2 ai(ty df Ewpected number of zeros per sec.=Q a » (a) Wort: from which is obtained the : $" Ns) dt Expected number of waves per sec= sla ce ) (13) AE) df E (o) Substituting f= 1/71 y equation (13) becones 9 "a \ + Ne 20 Expected number of Waves per sec. = [a (14) Sek el | TOO RCT) at (9) The reciprocal of this expression gives the average period be Bie. Vo pA a cy = | Yo (i 5) ms + Ad TUN AT ; re Neumann (1953) obtained an empirical spectrum for a fully-developed wind-generated sea, Ey on ee aC APY }») =Ch € ; where vy is the wind velocity. If Neumann's spectrum is applicable, equation (15) can be written y oo) ental a gs j Sacks T Jar | Co x _at +2 \ alan ce aT Q 17 where ae =4/ va ove This expression is valid for suface waves, where theoretically all periods between O and oO are possible. To extend this concept to depth it is necessary to transform the surface power spectrum to a pressure power spec~ trum and, considering the Pilbara action of depth, evaluate the integrals from the cut-off period T= 1, to Y=vo o First, to reiterate what is meant by a wave period, consider a time- series record of pressure variations on the bottom of a body of water due to the passage of surface wind waves. To ay | Ts ——— > Mean The time it takes. for the record to complete a cycle (dowmecross to downcross ) is called a period “Ij (ACY Akoce))6 The average wave period at a given depth (bottom) will be represented by Ya, |. T* A(T) seck Camtd/LY AT to ar = ° Cia) i" aot MT) seck®(atd/L) AT Th In general, it is not possible to solve this equation analytically, but a numerical solution will be velid. The wave measurements represented by figure ) were made at a depth of 18 150 feet, where periods less than 7 seconds are not represented. Therefore, the limits of integration for equation (17) are 7 and oo « Thus, the average wave period given by equation (17) is Ws tas seconds, This closely approximates the value T=19.10 seconds already obtained from a direct analysis of the wave data. Assuming a gamma=type distribution, prediction curves for wave period distributions can be determined from the mean period, the standard devia~ tion, and a measure of the skewness. This type of analysis has been done previously by Putz (1952). In fact, Putz gives relationships in terms of the average period for both the standard deviation Sy. and the skewness A, « These expressions are: —— S, = 0.318 T= 0-159. , A, = ~ 0.344 7 ads 24 A comparison of. these empirical prediction curves with the period distri« butions from several pressure records is illustrated in figure 6 and figure 7. The period for Putz's prediction curves has been defined in a different manner than the definition used in this report. Putz has defined a wave period as twice the time it takes for the record to complete a half cycle (trough to peak). This difference in measuring periods will have great effect in some cases. It is believed that the good agreement for the records shown is due to the relatively deep depths at which Putz's measure= ments were made for the empirical curves. As will be shown later, at these depths the ratio of the number of wave maxima to the number of waves ap- proaches one. 19 WAVES WITH PERIOD = T( percent) 100 80 60 40 20 | LEGEND | ——A—— OBSERVED | | —*— PREDICTED 4 6 8 10 12 14 16 18 WAVE PERIOD T (seconds) FIGURE 6. PREDICTED AND OBSERVED OGIVE CURVES FOR WAVE PERIODS 20 20 WAVES WITH PERIOD =>T (percent) LEGEND ——-A=—=— OBSERVED ——@— PREDICTED 4 ial x j AL 1 nw A | SS ) PRON AD OC eee = ——————#) 14 16 18 20 WAVE PERIOD T (seconds) FIGURE 7. PREDICTED AND OBSERVED OGIVE CURVES FOR WAVE PERIODS 21 For the example cited, in which the measurements were taken at a depth of 150 feet, it was found that Putsz's relationships were not valid. In fact, S4 given by the empirical relationship was almost exactly the square of the standard deviation computed from the wave record, and Xo was found to be three times lirger than the value obtained from the original wave record. Since As was small in absolute value, it was considered worthwhile to fit the normal distribution (the symmetric member of the gammamtype distributions) to the wave*period distribution. A sur- vey of the results indicates that the approach was valid, and the agree- ment is in order as shown in figure 8. This observation is a qualitative one and it should be emphasized that quantitative conclusions are not given much weight at this time. 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