oS (2) = i a = CE CRUISE Il. USS REDFIN (SS-272) P. S. DeLEONIBUS Submarine Systems Section Oceanographic Branch FEBRUARY 1961 PRICE 65 CENTS TECHNICAL REPORT POWER SPECTRUM ANALYSIS OF WAVE MOTION, SUBMARINE ROLL ANGLE, AND RELATIVE CROSS-FLOW VELOCITIES MBL/WH i} i ) TT smi ANA FOREWORD > The advent of the nuclear submarine with its unlimited cruising range has increased the urgency of acquiring and an- alyzing data required to assess the ocean environment. This report presents one example of the type of analysis required to understand some of the characteristics of random-type data. Such data are exemplified by continuous recordings of sub~ marine motions and relative flow across the deck of the hovering or slowly moving submarine, and digital recordings of the height of the sea surface above the submarine. The fact that so many interrelated variables can be sensed and recorded simultaneously is an indication of the submarine's potential as an oceanographic research platform. In addition, such simultaneous recording procedures make the data amenable to even more sophisticated analysis; namely, the estimation of cross-spectral densities connecting two continuously distributed random variables. Bo Go Sill? Rear Admiral, U. S. Navy Hydrographer Ta Vill. CONTENTS Page INTRODUCTIOND Ste, isl ai ae aes rn) gr ponies Lapeer OS OR D0 1 INSTRUMENTATION AND RECORDING PROCEDURES Aue lmstrumetntcitions i lel cn «te mente BA EET eh areca, )9 PROD ES 2 Bb, (Receteling Hkcieceliika 6 6 ols 6 6 0 6 4.0 go Cae Oe Sie eg &C THEORY AND METHOD OF POWER SPECTRUM ANALYSIS VRS PECINUMMANICNIYSISHG Nelkelissc se ea tejuetce) © Voulemie oMlieuy ouia a) Sune 6 Bre DtgitallEstimationioh bower Spectral s-\siee selene tol iscie «lt 5 9 IZ GeAnaleg) Esiimationiofihower Spectral imi) sisi ialte \lemusnmens) 1etsn te 16 D. Some Considerations in Estimating Spectra 1. Resolution and Statistical Stability) . ..........2.2-. 21 Dia PN CISAING ies ard kat Sct ors) Kos Neue ray ols sa: taly, No fosnFoin Yay Yon Mere hes Ueuteay Never ee 21 3... Comparison of Spectral Estimates. = = =. 2 2 5 6 1 «5 «6 « 4 « Ze POWER SPECTRA OF SURFACE WAVE, SUBMARINE, AND FLUID MOTIONS PEPE GENCE seagate ee hy a2 ie esha sont ey ol pnw a Gill sich a \ale Co oye comets 22 B. Surface Wave Height Spectra - - - - . - + 2-2 + es soe 6 os) C. Roll Angle Spectra RO! CO Oem On TON ec: SOWMLCHRION SOME Oe SOM! ECR Boum oo 2B D. Transverse Relative Flow Velocity Spectra... .......0.e. "26 E. Longitudinal Relative Flow Velocity Spectra... ....... Dy, SUMIMARNATO ESRESWEUS) te, sh. geval...) ton Wer ema Sey Ree ey wane ales 38 CONCHWSIOINS Wwenidt wpe temas otis! weve tee tee, don McMeel, Bio os Oe? FUTURE MESS ANDVANALEVSIS "BEANS ® Boe saa 4 .lsue sels @ ecu 43 AGKINOWEEDGEMEINTS aicncs vei leigh aie ote cm hue lence eh at ce. de 43 REREREINGES cqmemendieircurRnei iam. toutes het, Jasmibuaelp se hahs) hy paler 45 APPENDIX A. Surface Wave Height Data Obtained by The Sonic SUC CUSCONMEN Marcle UNeMe cists i553) (us) in Pair). le) 47 APPENDIX B. Power Spectral Estimates of In Situ Pressure Fluctuations . 55 an a - WwW FIGURES Oceanographic Instrumentation Locations, USS REDFIN (SS-272) . . Typical Variables Recorded on the Offner (Dynagraph Type R) 8-Channel Recorder - USS REDFIN, Crufsell] 2... .... Paps Example of a Wave or Ship Motion Record ......4 52000. ‘Surface Wave Spectrum - Transducer No. 1 .... 2... 5 se . Surface Wave Spectrum ’- Transducer No. 8: 9... 0.0. 1 210s es ee Surface Wave Spectra - Comparison of Transducer No. 1 and WransducersINot 85. Ss Ose Lad Salo eosin s Sater Sie iy aie Block Diagram of The Analog Wave Analysis System Used by The Submarine Systems Section, U.S. Navy Hydrographic Office... . Comparison of Energy Density of Roll Angle Estimated by Wave Analyzer at The Hydrographic Office and Digital Spectrum Estimated by Datatron, With Integrated Spectrum Superimposed, Run 4-1 . . . Digital Spectra of Roll Angle for Lags 30, 60, 90, and 120, Run 4-1 . Spectra of Surface Wave Heights, Run4 ..... BB eee (al OUR OPM D Spectra of Surface Wave Heights, Run6 ..... +++ Ries Ke Roll Angle Spectra for Different Relative Headings Under Hovering (Ceoiyattittel cy tiles ew i teen now omcitece Oulomce sclva ae 8b"Gr © » eae Roll Angle Spectra for Different Relative Headings Under Hovering Gonditions,. Run 4 ... « avingstes inna eewinl + certomleg ie lal aenl ve =a Roll Angle Spectra for Different Relative Headings Under Hovering (Senrehinenee Kul! on dea cree o SOME RAE cata abe hae u AAT aioe Roll Angle Spectra for Different Relative Headings Under Hovering Glelgra yakalnere GAA wtsy 8 ke. Cer Ne, BEE teat b OO rO duo 0 6 ool ig Sl eeae Transverse Flow Velocity Spectra for Different Relative Headings Under HoveringiGonditions, RUMmA sees cis ce ens ra cuis ke = Transverse Flow Velocity Spectra for Different Relative Headings Wnder Hovering: Conditions, RUNG)... 6. o. ey et suet oe eh ot ecu ells Longitudinal Flow Velocity Spectra for Different Relative Headings Under Hovering Gonditrons, (umG 0 2s eee ceieel uel 6 cs Br fay veh ai te Longitudinal Flow Velocity Spectra for Different Relative Headings Under Hovering Conditions, Run4 6 1 2 we ee ee ee ee : vi Page 35 FIGURES (Cont'd) Page Longitudinal Flow Velocity Spectra for Different Relative Headings Under Hovering Conditions, Run5 .... . PUA me hen piree gt Pls tea vio) Longitudinal Flow Velocity Spectra for Different Relative Headings \Uineler Inlexettineg) (Cemeniticns, INU 6 6 oe oo bo Oop Geo ed a S/ Block Diagram of The Analog to Digital Recording aes Used by The Submarine Systems Section .. « egtee olive a) ve velngeitaas) oll Fraction of a Unit Simple Harmonic Wave Height Remaining at Dkejettns, Cir ZA) colnvoh OOS |) o 4G ade SO oMbuOnO | og. uc) geeo Bo 8 Compressed Re-recording of In Situ Pressure Fluctuations .... . 69 os Comparison of Power Spectral Estimates of Filtered and Unfiltered Stigmals Umi m4 ii vel ich arr slfern trial: teuewtoe aie | « colqanie lates dy, witthiacte den sell 09, Band Pass Response of The Model 330-A Krohn-Hite Ultra-Low Frequency Band=Pass Filter . 2 2 1. 0 « 0 « « Siege ad's G16 6 1 ew) Power Specira of In Situ Pressure Fluctuations Compared to Attenuated Surface Wave Height Spectra, Run 4-4 . . «2... .-.s O61 TABLES Test Conditions During Cruise 11, USS REDFIN - 28 October to INovemibercl959 te) cooker heline inc) Me meiiterbehicen eenoy lpiomnannctieiptomte someones 7 Ninty Percent Confidence Intervals for Digital Spectra of Surface Wavy enhlelghts:. e.cc. tee tg te) ea jenites 1a) ia: ee a caitve badta ys. leytal eles Wt ay LO Ninty Percent Confidence Intervals for Analog Spectra...» 2 « © Boe27 Summary and Comparison of Results Derived from Surface Wave Seachits (Diteitrel|)/ ciel riiteleesie) 6 > 6 9 5 8 6 2 6 5 680 5 G6 fe) 6 Summary and Comparison of Results Derived from Roll Angle Spectra and Manual Computations. . . 2» 2 «+ ss +o « AO SO Men Gert eli Summary and Comparison of Results Derived from Transverse Relative Flow Velocity Spectra and Manual Computations . . . . . » . o « « - Al Summary and Comparison of Results Derived from Longitudinal Relative Flow Velocity Spectra and Manual Computations . . . . » » « « « « « 42 vil - ) ; *¥ tt a Porn: ati as at stee J wwe Re 7 pee . ‘ y a ae . iy dave. eles? ~ vs nena te act to bars i a oa ae a - Fl ative »". &. we $ a J ow A ba bang mnaHoviagtt ne ” Je AE Meio! sev? a , sR megs Fikaiee? wohow it en a 4 : hy at 7 > ae laren >) vs Picewais %. : ad it, sf pele “a A A: : woviuk tewenvnt ae et ee ue os <4 uA . 5 7 ‘th 4 . ’ A | spitaing Ladi aah Mii yea ier S ies L) 7 1 . a “ey will esas yy « sean tached Bit mn 2 Le) reid tt Cee 4 ; bs ea te oe, i oe me 7 bet pian aay 1% Vat! by nap ne Sar viper), , | ALAN sana tid mil tain it sive te pendtyien> teu P may Oe ee ae, ee a4 im Pe ee cd Mist ingrch t Bypsary, | i de, tah Ail | » docliel antispam? ‘mot aber hla Yo « ane me tits = : Th rs CRAG Ronei, hagutsote Ke orton’ a q re ey ) etMedaa bs pet treba tng Ae iluaell ty mnie? Dies Vo - . een) Hy erage? ecainet he evil verter Vow wot <9 a7 te 1, INTRODUCTION The submarine, USS REDFIN (SS-272), was utilized as a working platform by the Submarine Systems Section, U.S. Navy Hydrographic Office, to obtain digital re- cordings of surface wave motion and analog recordings of ship and fluid motions. This report is concerned with the spectral analysis of these recordings made while the REDFIN was hovering. One of the objectives of the Submarine Systems Section is to collect and analyze data needed to study environmental effects on submarine motion, and the characteristics of the relative water motion in the proximity of the submarine. Such data, recorded while the submarine was hovering or underway at the lowest possible speed, have been analyzed by computing power spectra of the recordings. Data presented in this report include composite graphs of spectra of surface wave motion (computed by digital estimation formulae), spectra of the submarine's roll angle, and spectra of the transverse and logitudinal relative flow velocity of water across the submarine's deck (obtained by an analog computer). Examples of spectra of in situ pressure fluctuations, estimated by both digital and analog methods, are presented. Differences in approach embodied in the digital versus analog method of power spectra estimation are discussed, and specific examples are presented to illustrate points of technique required for comparable resolution and statistical stability. Finally, spectra data presented in this report are summarized by presenting mean values of motion in tabular form. The objective of this particular test was to study environmental effects on the motion of the submarine. During this test period, wave conditions were generally mild with the result that some data were recorded near the sensitivities of the instruments. Thus, energy spectra and related statistical parameters are representative of conditions encountered under lower sea states. This report includes most of the data recorded on REDFIN Cruisel! during the period 28 October to 5 November 1959. During this time, the REDFIN was hovering at a keel depth of 100 feet, and operating in an area centered about 180 miles east of Wilmington, North Carolina. Hydrographic Office personnel participating in this test were Messrs. Q. H. Carlson (Senior Oceanographer), A. G. Alexiou, and D. E. Tidrick. Il, INSTRUMENTATION AND RECORDING PROCEDURES A. Instrumentation Wave, ship, and fluid motion observations discussed in this report were made from aboard the REDFIN. Instrumentation utilized and variables measured are included in the listing that follows. Locations of these instruments on the submarine are shown in Figure 1}. Pitch and Roll Angle David Taylor Model Basin stable platform and potentiometer Relative Flow Velocity Litton Electromagnetic Flow Meter In Situ Pressure Fluctuations Wiancko Pressure Measuring System (U.S. Navy Mine Defense Laboratory) Ship's Speed Ship's Electromagnetic Log Ship's Course Ship's Gyrocompass Depth (pressure) Vibrotron and Bourns potentiometer in the pressure gauge line Surface Wave Profile Westinghouse Sonic Surface Scanner Figure 2 illustrates a portion of a typical recording taken aboard the REDFIN. Most data were recorded on both magnetic and paper tape. Surface wave and in situ pressure fluctuation data, however, were recorded on magnetic tape only. A com- plete description of the instrumentation and measurement and recording procedures is given in the Hydrographic Office Technical Report No. 91 (Reference 10). Nearly all pitch angle recordings, heave acceleration, and sway acceleration were practically flat and will not be discussed further in this report. Thus, the only submarine motion of Interest is roll angle. Some cross-flow data with very low amplitudes were deleted. Although attempts were made to provide analog recordings of in situ pressure fluctuations measured by the Wiancko pressure gauge, most of the collected data were deleted because the amplitude of the recording's envelope frequently exceeded the dynamic range of the pressure gauge. When this occurred, the relief valve in the gauge would open, resulting in intermittent jumps in the continuous recordings. Attempts to filter these pressure data before spectral analysis were not too successful, and most of the spectra of the Wiancko recordings were not considered valid. (— TEMPERATURE PRORE | conoucrivity cet JX SOUND vELociTY METER FLOW VANE ANGLE INDICATOR ACOUSTIC FLOW METER SAMPLER DIFFERENTIAL PRESSURE TRANSOUCER \WUMINOMETE® PHOTO CELLS )) 4994 OL :yidaq BulseAoy 6S6L YSEWIAON | OL YSSO1DO 82 “NIJGIY SSN “Il ASINYD ONIWNG SNOILIGNOD JSaL “L a1aV1 POldeag UDEWy ?GOIYdd “SeADM 4seybiy pslyj-ouo ouy yo yyByey eBoseAY =] HOIFHx - - - - ral Ger peg Bg -0°'Z OF GZZ G 01z “lS. ANNL +9 - - - - - a eer | p89 Bg -0'%)O9'F GZZ S 01z -/0yl AONL €-9 - - - - - a e0rl peg Bg -0'% N'Y GZZ S 01z =/Z0| AON] | =) Z=9 - - - - - a OzEL Deg Bg -0°%'F GZZ S 01z “SEL AON IL L-9 - = - = a ZZ80 Pag OLI-0°E = S"9 060 Sul SLO ¥ “620 POOR ¢rS - - - - a 0220 P85 O'LI-O"E = $9 060 Z SZO y PO 0 E-S te ios ; > is £90 Z . S$ O°0I-S°%. -S°s G60 9 SZ0 y PO 0 z-S aa % x & . . S10 r z SO°OI-S°Z S"S 090 S SZO y PO Ok I-S Pas Eg -0'% 0'S OLo y 001 PO 6 S-F Bae wae . : "i cetoe elt ZS\ 3 S8°2-S1- O'¥ ZZ0 € PO 6 Fr Bag B°/ -G"| PO 6 -b (Lo), Bulppsy BALD By (285). (Lo) pudg Polsed UOlfD9IIG SJIDY) DIYGOUAS WO14 JSDDPUI}) SBAD AA SOAD/(A SODFANG (1o) uo!49a41q SUOI{DAJBSGO) [ONS A (P,44OD) 6661 YFGWIAON | OL YIPOLDO 8Z “NIAGIY SSN “Il ISINYD ONIWNG SNOILIGNOD Lsdl “1 FTV G@HO0UX NOLLOW dIHS YO SAVM V dO SIdWVXa °€ GuNOIA Pe HOR Reaee ZING ANE EAREP Aas | Sia ASE BER ARRTS Bes connecting the zeros of the process) can be given statistically by a probability density function. If the dominant part of the random process occurs over a narrow Interval of frequencies, amplitudes shown in Figure 3 can be given by the Rayleigh probability distribution: 2 Bell! E F (x) dx = e€ dx (1) Thus, the probability that a particular amplitude a, will fall between x and x + dx is given by equation 1. The first moment about the origin of equation 1 gives the re- lationship between the mean wave amplitude a and the quantity E: <= 0-866 VE- (2) The quantity E is given by the second moment about the origin of the Rayleigh distribution: eo 2x3 ee Be eae crm maak (3) 6 E_.9 Bare (4) where o” is the variance of the process. Now E is related to the energy spectrum of the random process by: = 2 2 = LAY dx and — = LAG (5) ° where (ROY is called the energy density. From the energy (power) spectra and related statistical parameters, the process can be completely described for practical purposes. The term "power spectra" is freely used by aero- and hydro-dynamicists among others, although much of the work dealing with the energy spectra of random processes originated in the field of communications engineering (Reference 5). 10 Recordings of submarine motion (roll angle) and fluid motions (transverse and longi- tudinal relative flow velocity) illustrated in Figure 2, may be visualized by considering them frequency band-limited random signals continually oscillating above and below an equilibrium level of zero signal. Very often (especially in the case of wave and ship motions), the spectrum of such motions is characterized by a well-defined peak value which is skewed toward lower values of frequency, The area under the spectrum curve is related to mean amplitude values of the motion, and from the theoretical work of Longuett—Higgins (Reference 2) et al, the following relations connecting mean wave amplitude a to the means of the one-third highest (a,) and the one-tenth highest wave amplitudes (a, ) were found: = il 3 10 an i 10 = 1.800 me (6) a (o] These amplitude relationships were verified on actual wave height data by J. K. A. Watters (Reference 12), who found the following mean wave height relationships H. from a sample of wave records: 1 & = 1.94 a2 = | ,58 (7) A From these we conclude that wave height values are given approximately by doubling the amplitude values. Finally, since: a@ = 0.866VE then ia =v =) 177 ee (8) from which we get: H, = 2.33 /E | (9) 3 H, = 3,60 JE (10) 0 VW Thus, equations 8, 9 and 10 hold for mean values of roll angle and relative flow velocity, since recordings of roll angle and flow velocity are assumed to be samples from the same type stationary Gaussian random process. B. Digital Estimation of Power Spectra The practical problem of estimating energy spectra has received widespread attention and can be viewed from two distinct but practically equivalent points of view. One method of estimating the energy spectrum of a stationary Gaussian random process is to sample the continuous (analog) recording of the process at discrete inter- vals of time, obtain an estimate of the autocorrelation function, and then obtain raw or unsmoothed estimates of the energy spectrum by Fourier transformation of the auto- correlation function. The power spectrum Is then obtained by suitably smoothing the raw or uncorrected spectral estimates. Since the study of such processes is the study of the sample variance of the process, ft is essential that confidence intervals be given for each spectrum. The theory and procedures of digital estimation of power spectra are given in References 1, 3, and 6. If the process can be "quantized" or digitized at the transducer or in the recording process, much of the manual data reduction can be eliminated. For example, the Sonic Surface Scanner measures instantaneous height of the sea surface above the transducers located on the hovering submarine's deck, and these data are recorded directly in digital form on magnetic tape. These values are played back, put on punched cards, and programmed through a high speed computer to obtain the estimated spectra of the surface wave heights. The method of recording and playback is outlined briefly in Appendix A, As an example, consider Figure 4, which shows the power spectrum obtained from a one-half hour recording from transducer No. 1, located on the submarine's bow (Fig. 1). The dashed lines indicate the 90-percent confidence intervals (See Section C). For comparison, the spectrum of a simultaneous one-half hour recording from transducer No. 8 (located near the stern) is shown in Figure 5 with its 90-percent confidence intervals. Figure 6 shows the good agreement between the two measured spectra. Amplitudes of submarine motion during this recording were low. The heave accel- eration recording, for example, was practically flat. However, at higher sea states and lower wave frequencies, a submarine will move appreciably in response to surface wave motion. For example, heave displacement would be in phase with a pure swell, whose relative heading was 90-degrees, with the result that the fraction of the true recorded wave amplitude decreases with decreasing frequency (See Appendix A for a discussion of expected measurement errors). In general, surface wave spectra estimated from record- ings of the Sonic Surface Scanner, will have to be corrected for the hovering submarine's motion. Furthermore, if the submarine is underway, an additional correction will be 12 T °ON WaONaSNVEL = ESSE SAVM DOvTUAS ° ZaNold p-008/a45 2[_(a7)v ] kotsueg KBrSeuy 13 @ °ON YAONdSNVuL = ” HNALOGAS GAVM FOVMNMS °S TMs ERPeESTPUTITESEERS Ui G (E, Wd 5 2 S51 1 oe APES ae | HE j / : 288 Bgeno tees conme (Sasa ceaaeeees Viv tae Tetee aaee ae necessary for its speed and relative heading. Results of power spectra of surface wave motion obtained on the global circum- navigational cruise of the USS TRITON (SSRN-586)* indicate that such spectra can be measured successfully at speeds of 6 knots. At 6 knots, the TRITON's speed was equal to the phase speed of a 2-second wave. Thus, spectra given in terms of the frequency of encounter could be transformed in terms of the frequency relative to fixed coordi- nates. C. Analog Estimation of Power Spectra The digital method of estimating power spectra may not be practical If a large number of recordings are to be analyzed. This being the case, the recommended pro- cedure would be to record analog data on a multichannel tape recorder and compute energy spectra by commercially available wave analyzers. This was the method used to analyze ship- and fluid-motion data presented in this report (References 4 and 7). Most of the analog spectra of submarine and fluid motions presented in this report were computed at the David Taylor Model Basin. The Hydrographic Office has recently acquired the electronic components required to estimate power spectra shown in the block diagram of Figure 7. An example of an analog spectrum of submarine roll angle estimated by the system at the Hydrographic Office is shown in Figure 8. The analysis procedure is as follows: The random process to be analyzed is recorded aboard the submarine on magnetic tape at a tape speed of 1 7/8 inches per second. Each one-half hour run recorded at this speed is played on a 14-channel Ampex playback unit (Fig. 7) at 60 inches per second while being re-recorded on a 14-channel Precision recorder. The random signals are re-recorded also on an 8-channel Sanborn recorder so that the quality of the recordings can be inspected, and an indication of expected am- plitudes of the random signals can be obtained. This results in a compression of 1/32 the original recorded tape length; the frequency components in the random signal are all increased by a factor of 32. The compressed recording is formed into a continuous loop and threaded into a special loop attachment. The Precision recorder is utilized again to drive the loop attachment at a speed of 30 inches per second for an additional frequency increase of 16 times and a total frequency increase of 32 x 16 = 512 times the original frequencies. Thus, a frequency of 0.10 cycles per second appears on the analyzer as a frequency of 51.2 cycles per second. The reason for analyzing in this frequency range is that beat frequency analyzers commonly used can easily accommodate frequencies at higher ranges, but could not handle gravity wave frequencies (e.g. 0.03 to 0.30 cycles per second). “U.S. HYDROGRAPHIC OFFICE, Oceanographic data report for the global circum- navigational cruise of the USS TRITON (SSRN-586). 16 February to 10 May 1960, by NR. Mabry (in press). Q *ON WHONGSNVUL ONY T °ON HEONGSNVEL D NOSTUYEHDD = = vaLoads SAVM Govauns % aun Ora qTPeY The pears sib Te josds wii wn ma : wn See eee eee Ayrvsueq Adaouy 15 z[(mvy] GOIMIO OTHdVYOOUGAH AAYN °S °n NOLLONS SHELGAS GNTEVWANS SHL Ad GaSN HELSAS SISZTYNV SAVM DOTVNV SHI JO KVaDVIC YoOOTd °L wunois *JtTe9 ‘euepeseg 6°09 Aspesoy Jepiosey I-X £4Tsusq AZ 13uy eseIOAY peyeisequy JO4eI9s WUT ianog re) IOJETTTOSO “FTLPO ‘soTesuy SOT "09 sqonpoig Teotuysey, (@ptm yout T sdooy 10,7) ae quemyoe} Fy dooy a °yJtTe9 S‘so[Tte9 ues °09 squsumiqjsuy UOTSTOSag W[eSd Jepsovsy edey, oryeus ey (pezt104s Tsuedy) Teuueug-F1 | *ssey “WeUqT eM Jopio0sey ULOQUES TeuuTuons qtuq yougseTd edey, aT yeUusTy Teuueyp=7L °FTTEO ‘£479 poompsy dao) xoedury 17 aici oe ‘6 T-] Nou *GasodWIdadNs WAULOAdS CHIVEOEINI HLM ‘NOULVAVG IA CULVHLLSA WNULOASS TYLIOIC GNY SOIAIO OIHdVUOOUCIH FHL Lv UAZATYNY GAVM V Za CaLYALLSa STONY TIO’ 40 ALISNYC ADWUNA JO NOSIuvdWod °g gunDId LL PER e PPE Peer ee eee | ee SRS SS REP aes ee 7 thy ee ai aa aT Do ae Nae = : eh 2a 7 an Arif a q u ate i EH : 7-098 /22ep ut co [(s) oh \ A£ytsueq ASTauTq 18 The random signal output is fed into the analyzer system consisting of an oscillator, analyzer, and power integrator. These three units transform the random time signal into a continuous plot of the average energy density versus frequency, which is traced out on a Moseley X-Y Recorder. This system also provides for estimation of the integrated spec- tra. The block diagram of Figure 7 illustrates an integrated spectrum curve superimposed on the energy density curve. The small inset curves in the lower right hand corner of the X-Y Recorder represent the spectral outputs of the filter used to resolve the oridinate scales of the graphs of average energy density and the integrated spectrum. Details and circuit diagrams are given in "Instruction Booklets for TP-625 Wave Analyzer System, " Technical Products Company, Los Angeles, California. A discussion of the problems associated with filter bandwidth, loop periods, time constants, and frequency scanning rates is given in Reference 4. Figure 8 is an example of the power spectrum of a submarine roll angle recording and the power spectrum of the filter used in the analysis. This illustrates a typical power spectrum of the filter used to calibrate the ordinate scale of the energy density graphs. The broken line curve is the digital spectrum of the same recording which is superimposed for comparison, The digital spectrum superimposed is one of four spectra shown on Figure 9 (See Section D on comparison of spectral estimates) . This analog spectrum was estimated by the wave analyzer in the Hydrographic Office, and all the digital spectra contained in this report were estimated by a Burroughs 205 Electronic Computer, also located in the Hydrographic Office. Such spectral estimates are distributed with a chi square distribution with 2N degrees of freedom, where N is the number of elemental bands covered by the filter. If the spectral density is not a fast changing function of frequency interval equal to the band- width of the filter, then N is approximately equal to the effective bandwidth of thé filter divided by the elemental bandwidth. The elemental bandwidth is 1/T, where T is the loop period, and the effective bandwidth used was 6 cycles per second (Reference 4). Thus, for the analog spectra in this report: effective bandwidth elemental bandwidth Number of degrees of freedom = 2N = 2 (11) Confidence intervals for the estimated spectra can be determined from a table of the chi square distribution. The 90-percent confidence intervals for the digital spectra are given by computing the number of degrees of freedom according to: 2 Number of degrees of freedom = TR (12) 19 Tet nnd *Gzt pue £06 f09 fo€ SovI wu STONY TIO“ H VALOIS TVLIDIA °6 Sana pee {Aaetpey ut ee a Gs 0°32 9°81 d 60t fieeael ae igs ine a WY Lome] Aqtsuogq A#areuq je) 1Oge 0 p—208 /- Hi 20 where Tn is the length of the record in seconds, m is the number of lags, and At is the sampling interval. For example, the spectral estimates in Figures 4 and 5 are distrib- cuted with a chi square distribution with 70 degrees of freedom. The 90-percent confi- dence intervals can be determined from a table of chi square distribution. For 70 degrees of freedom, the true spectral density Pe (f) is bounded by the estimated spectral density Pe (f) according to: PC) Pe (f) a 30 < PS (f) < wOu76 or 0.77P_ (FKP (F)<1.32P, (F) (13) Thus, if a single observed estimate with 70 degrees of freedom is observed to be 5 ft 2/sec™!, then we have 7 Uae confidence that ne true Igng-run value lies between 5/1. 30 = 3.85 ft 2/sec™! and 5/0.76 = 6.58 ft 2heec D. Some Considerations in Estimating Spectra 1. Resolution and Statistical Stability In estimating power spectra, whether by digital or by analog methods, a choice must be made between a high resolution of the spectral estimates over a chosen frequency band and the statistical stability of the spectral estimate in that band. The two considerations are mutually opposed in that fulfillment of one implies some sacrifice of the other. For example: a. Width of the "spectral window" determines the resolution of any peaks in the spectrum. Now, if the spectral window (i.e., the filter in the wave analyzer) is chosen too narrow, too much detail is obtained, and a truly significant peak may be overlooked in the resultant "blurring." However, if the spectral window is made too wide, the resulting spectral estimate may be too smooth, and here again some physically significant hump in the spectrum may be smoothed over. b. The statistical stability of the estimate depends on the width of the spectral window and the length of the sample, When high resolution is desired, fewer Fourier components are averaged over the narrow spectral window, and the spectral estimate will have wider fluctuations over the totality of ensemble estimates. Conversely, less reso- lution with a wider filter gives a value which is expected to deviate less from the true but unknown spectrum. 2. Aliasing a. If asignal g(t) contains no frequencies above f cycles per second, it is completely determined by giving its ordinate by a series of points spaced Sr seconds 21 apart. For example, if waves with frequencies greater than 0.5 cycles per second are absent, the wave record can be sampled every second. However, if appreciable wave energy is present at frequencies above f cycles per second, this energy will contribute (by aliasing) to the spectrum of the lower frequency waves. Thus, the sampling inter- val must be determined by the high frequency cut-off. b. The range of wave frequencies shown on the abscissas of Figures 4, 5, 10, and 11 represent the most important part of the spectral energy density associated with surface wave heights. Similarly, the range of frequencies in Figure 9 are associated with the dominant rolling motion of the submarine. Now, if a random process is char- acterized by oscillations about a slowly varying centerline, the spectral estimates at frequencies near zero may have appreciable values. However, since this spectral energy is not associated with frequencies in the gravity wave spectrum, which are the only frequencies of Interest in this report, the graph is arbitrarily cut off at the lowest important frequency. Similarly, spectral densities at the high frequencies were not plotted since spectral densities decreased rapidly as the Nyquist frequency was approached, 3. Comparison of Spectral Estimates When the estimated analog spectrum of a process is to be compared to a digital estimate of the same spectrum, the equivalent spectral windows must be the same. Figure 9 illustrates 4 digital estimates of the analog spectrum shown In Figure 8, The analog spectrum in Figure 8 had 58 degrees of freedom, and the four digital spectra had 31, 42, 63, and 125 degrees of freedom. Therefore, the digital spectra with 63 degrees of freedom were superimposed on Figure 8 for comparison. IV, POWER SPECTRA OF SURFACE WAVE, SUBMARINE, AND FLUID MOTIONS A. General Wave, ship, and fluid motion data analyzed in this report were taken while the REDFIN was executing a series of maneuvers which included hovering at different relative headings. During each hovering run, the REDFIN attempted to hold a keel depth of 100 feet at a fixed relative heading while making recordings of surface wave height (Sonic Scanner), roll angle, transverse relative flow velocity, longitudinal relative flow velocity, and in situ pressure fluctuations. Each recording was at least one-half hour in duration. 22 1] NOY “SIHOISH GAVM SOWMUNS JO VULOwAS °OT MMNDIA eft be ee Eb pm EF 23 es 098/35 Ur, [ (7) v | kytsueq &Brouy CEP ECE eee fas Medea AE (ah See ee 9 NOY *SIHDISH SAVM GOVAUNS JO VELOGdS °TT gunold V ic sz Bz be Oo £e bh € ib Ub 2, U9 bL OF 1 Oo Re Ee i ees FECES CET a 1 S SeSSSSS8\ 00 Hebi HH : oe dete 4 7 og To28S—== Asuenbatq ystabAy og ed ne G ¢ aro S/o43 UT 2 (7) v| kqtsueq kBreuq TS 24 Table 1 is a summary of test conditions for Runs 3, 4, 5, and 6 of Cruise Il. The first recording began with Run 3. Wave conditions were generally mild except for the first four recordings of Run 3. On four of the runs (Runs 4-1, 5-1, 5-4, and 6-3), large variations in depth occurred because of temporary loss of buoyancy control. However, such large depth variations were of very short duration, and the submarine generally was able to return to its original hovering position In about 7 minutes. All runs where this occurred were continued for 30 minutes after depth was regained, and only this portion of the record was analyzed. Submarine motions and relative flow velocity of water across the submarine's deck occur as a result of surface wave motion. Since motion characteristics of submarines will be required under various sea states, It is essential that some estimate of the surface wave spectrum be available. Such spectra can be obtained by hindcasting from the synoptic sea-level surface weather maps by methods described in H. O. Pub No. 603 (Reference 9). However, the exact functional form of the Pierson-Neumann spectrum and its dependence on such parameters as wind speed, wind duration, and fetch length are presently being Improved. Hence, It is essential to obtain a series of surface wave spectra over the open ocean so that differences between measured and predicted spectra can be reconciled. Although the Sonic Surface Scanner method js still in the experimental stage, it is a promising method of obtaining series of surface wave records in the open ocean. Sub- ject to corrections for the motion of the submarine, it should be possible to obtain reli- able spectral estimates of surface wave heights from such records. B. Surface Wave Height Spectra - Surfeice wave height spectra for Runs 4 and 6 are shown in Figures 10 and 11, . respectively. The sonic scanner was inoperative during Runs 3 and 5. Figures 10 and 11 indicate that wave conditions were stationary during each of these series of runs. No corrections have been applied to these surface wave spectra, since the amount of wave energy associated with high period waves is rather low. Table 2 presents the 90-percent confidence intervals for these spectra. Each of the submarine and fluid motion spectra shown in the composite graphs (Figs. 12 to 21) was obtained in the same form as depicted in Figure 8. For ease of comparison and tdentification, however, a smooth curve was reconstructed from each analog spectrum. C. Roll Angle Spectra Figures 12 through 15 present analog energy spectra of roll angles for Runs 3 to 6. Each figure contains 4 spectra corresponding to a different relative heading. Each roll 25 angle spectrum has a typical peak, occurring between 16 and 20 seconds, which is associated with the natural rolling period of the REDFIN. The natural rolling period of the REDFIN is about 18 seconds. Secondary peaks in Run 3, occurring at periods lower than 10 seconds, are associated with the lower period waves observed during this run (Table 1). Table 3 presents the 90-percent confidence intervals for all analog spectra discussed in this report. TABLE 2. NINETY PERCENT CONFIDENCE INTERVALS FOR DIGITAL SPECTRA OF SURFACE WAVE HEIGHTS Degrees of Run No. Freedom Lower Limit Upper Limit Sa ee eS see i Sy ln he lal eres 4-] 80 .78 Pe (f) 1.27 PE (f) 4-2 80 .78 (NA7-7A 4-3 80 .78 27. 4.4 80 .78 Te27 6-1 60 74 1.39 6-2 60 74 1.39 6-3 66 .76 Wee 7/ 6-4 68 76 lmao D. Transverse Relative Flow Velocity Spectra Figures 16 and 17 present energy spectra of transverse relative flow velocity for Runs 4 and 6, respectively. Figure 16 presents 5 spectra while Figure 17 presents 2 spectra, each corresponding to different relative headings. Spectra for Runs 3 and 5 were deleted because the amount of energy associated with frequencies below the lower cut-off value (.03 cycles per second) of the wave analyzer made the determina- tion of parameters extremely doubtful. In fact, the excessive energy produced some spectra which appeared as "white" noise (flat spectrum) from 0.03 to 0.20 cycles per second (5- to 33-second periods). This problem will require further study, The two higher peaks in Figure 16, occurring in the first two recordings of Run 4, ore associated with periods of 8.3 to 9.0 seconds, corresponding to dominant periods of surface wave motion. Peaks at higher period values are associated with the cross- flow energy induced by the rolling motion of the submarine, which tends to roll at its resonant period. Similar peaks occur in Figure 17, with the peaks centered at lower periods associated with the prevailing lower period wave motion. 26 TABLE 3. NINETY PERCENT CONFIDENCE INTERVALS FOR ANALOG SPECTRA Degrees of Run No. Freedom Lower Limit Upper Limit 3-1 46 we Pe(f) 1.47 Pe(f) 3-2 35 68. 1.61 3-3 31 .68 1.67 3-4 32 .67 U ole? 4-] 58 5H) 1.39 4-2 49 72 1,45 4-3 4] 71 1.49 4-4 48 52 1.45 A-5 38 ./0 1.54 5-1 29 .68 1.64 5-2 Sy) 74 1.37 5-3 37 ./0 ieoZ 5-4 25 65 eae 6-1 28 .68 1.67 6-2 AO 71 1.49 6-3 37 ./0 1.52 6-4 24 66 2 E. Longitudinal Relative Flow Velocity Spectra Figures 18 through 21 present energy spectra of longitudinal relative flow velocity for Runs 3 to 6, respectively. Each figure, except Figure 20, contains four spectra corresponding to a different relative heading. Relative headings for the first two recordings in Figure 18 are designated with respect to sea because these relative headings associated with sea waves had values that could result in larger longitudinal flow components, even though swell was greater in height and period bandwidth (Table 1). In the last two recordings, relative headings with respect to swell are considered most important, although the choice is arbitrary. The extreme peak in Run 3-1, however, probably was caused by the joint contribution of both sea and swell. In Run 3-4, both sea and swell components were from abeam of the submarine, and this results in the lowest spectral values throughout the run because the longitudinal component is smallest. Di (320d) UT ra o S oa oO q ee * abt ROLL ANGLE SPECTRA FOR DIFFERENT RELATIVE HEADINGS UNDER HOVERING CONDITIONS, RUN 3 respect to 2L(e] ky tsued ADraug 28 > q Ss a " "q fe >q qd S q cd 6 NG O 1 _998/5(39q) ut > [(#)u ] Aqtsueq ASrouyq 29 1k _°2S/5(8ea) Ut 3 | (z)x] Ajqsueq Adroug 30 53\f Fe Tal OR in les Sec71 all O ih peel I i 1) O.D 0 rom = a + —— prt onees ay _ i : ‘ : “1 tb | : % ] _ i . a } i i ~\ . he ~ - = ee ee eee oe - acilin — eT . ‘ q a ,) : 1 4 ; i. : ; : I e | 7, ‘ malt : ¥ ‘ , 7 "| ° envio # - ee ee cont ' mh ealia b | 4 | b { Pe ! ; \ Pee sareny ft - b ohn onan oye Ps) “ if yi ’ a “LA gy ] , oe “ at om a *! i Fy " ’ / } i + ' oie es ‘eee MCE Pieaw pee Bac eee Be APPENDIX B POWER SPECTRAL ESTIMATES OF IN SITU PRESSURE FLUCTUATIONS 55 MMQVIRGSA. Ov ARR SIO : + 1 Hi Fi “70 IRDA, BTA i ‘¢ APPENDIX B ~- POWER SPECTRAL ESTIMATES OF IN SITU PRESSURE FLUCTUATIONS A. General One of the varlables the Submarine Systems Section attempted to record was pressure fluctuation, as measured by a Wiancko differential pressure gauge. This gauge was mounted on the sail with the diaphragm parallel to the deck of the sub- marine (Fig. 1). Thus, when the submarine hovered at a keel depth of 100 feet, the gauge's rubber diaphragm was actually 60 feet below the surface. Total pressure change occurring at the diaphragm of the differential pressure gauge of the hovering submarine will be composed of, (1) pressure change due to surface wave motion, (2) linear motions of the submarine (heave, sway, and surge), and (3) angular motions (roll, pitch, and yaw). Pressure changes at the gauge due to surface wave motion alone are independent of the submarine's relative heading. Total motion of the submarine, hence total pressure change assoclated with the submarine's motion, is definitely a function of relative heading. Even under stationary wave conditions, power spectra of the Wiancko recordings will, in general, be different for different relative headings. B. Spectral Estimates of Filtered Signals The problem of recording pressure fluctuations on the hovering REDFIN was complicated by the limited dynamic range of the Wiancko differential pressure gauge. To prevent damage to the gauge, a relief valve eliminates differential pressure in the interior chamber of the gauge when the dynamic range of the gauge Is exceeded. A typical in situ pressure recording is shown in Figure B1. The small oscillations are pressure changes due to surface wave motion and linear and angular components of the submarine motion. The very long "period" oscillations are associated with the slowly changing dive angle of the hovering submarine. The amplitude ratio of the large to small oscillations is about 20 to 1. This highly compressed recording represents approxi - mately 8 minutes of recording. Many of the Wiancko pressure recordings were charac~ terized by breaks or "jumps" which occurred when the amplitude of the long-period oscillations became too large. When this happened, the relief valve would relieve the differential pressure momentarily, and the recording would begin anew at the center of the chart. Power spectra of the recording shown in Figure B1 were estimated by the wave analyzer system at the Hydrographic Office and are shown in Figure B2. In Figure B2, three different estimates are superimposed to show the effect of filtering. Curve A is the spectrum of the unfiltered signal while Curves B and C are the resultant spectra of first passing the signal through a Krohn-Hite Band=Pass Filter, Model 320A. Cut-off limits were arbitrarily set at 0.08 to 0.5 cycles per second (periods of 2 to 12.5 seconds) for Curve B and 0.09 to 0.5 cycles per second (periods of 2 to 11.1 seconds) for Curve C, Filter response curves for the frequency band used to obtain Curve B are shown in Figure B3. It should be noted that the shape of the filter response curve allows a fraction of the energy associated with frequencies less than 0.08 cycles per second to pass, but blocks off the very low frequencies. Lb i out $ Minutes aad iS oR me 5 i cE FIGURE Bl, COMPRESSED RE-RECORDING OF IN SITU PRESSURE FLUCTUATIONS A digital spectral estimate of this same run was obtained by re-recording the filtered run with pass-band limits set at 0.05 to 0.5 cycles per second. In Figure B4, the dashed curve is the power spectrum of this filtered recording; the solid curve, surface power spectrum; and the dashed-dot curve, the residual power spectrum at 60 feet obtained by attenuating the surface power spectrum according to: 2 a Zo feet (- = LAY, where Z and L are given in Appendix A. 2 urface The difference between the in situ power spectrum and the residual spectrum at 60 feet due to surface wave motion alone is attributed to pressure fluctuations associ- ated with the submarine motion; i.e. roll angle and cross-flow. 58 Tet] tind *STVNOTS GHYEUITANN ONV CHHALIIE JO SAIWATISH TVELOBAS sod 40 NOSTaVEHOD ‘zd mUNOTs S| Be crear ae Same ooo" et oe Por ECP err aie meres Pee a | q-298/at5 ut a mere A4tsueq f2rt38euy SECC EEE CEE EEE OMS TEs or elie |) Eee ee fo ee ee Gain In Decibels FIGURE B3. eW) 206 e 34 ° 50 LO Frequency in Cycles| per Second BAND PASS RESPONSE OF THE MODEL 330A KROHN=HITE ULTRA=LOW FREQUENCY BAND=PASS FILTER 60 Fluctuatio a) n at 6 O-foot} Depth). face Wave Hefight tu Pressure nuating S ressulre Gau Surf fa ; ce | ) ace Sranne In S fferential P rum ae power Sipect by Atte Power Spec (Sonlic S Powelr Spec (Wianeko Dit esipual P (Sbesines eo [aS N ce, eons 2 “a an iv 2v ar 2 2 27 5} 3 i 10 58 3\6 34 in Rafians/Bec. oS Frequ eee POWER SPECTRA OF IN SITU PRESSURE FLUCTUATIONS COMPARED TO ATTENUATED SURFACE WAVE HEIGHT SPECTRA, RUN h-h FIGURE Bh. q-288/a35 UT al (Mv Ay Tsueq ABxrouy 61 OOL-Yl “O “H sngjuoayeq *S*q :doujnD *SB1J]D0]9A MO|4-SSO1D SAIJDjay pup ‘a|Buy |joy aulipwigns 7uo}Jow eAD/M JO sIsAjDUY winsjoedsg JamMog :8]414 (222-Ss) NI4a3¥ ssn UOLJOW SADA aulinwgns - Aydpi6ouns29, AydpiBouned9 - eulspwigns OOL-¥l “O °H snqjuoayeg *S*q :ouynD *$914]20]9A MOj4-SSOJD BAIJD|Oy pup ‘a|buy |jOy aulsipwiqns ZUO|JOW @ADAA JO sIsA|DUY winiyoads JaMoy +8]414 (222-Ss) Nl4aa¥ ssn UOIJOW SADA auyipwiqns - AydpiBoune29, AydoiBounes6 - aulsnwqns —-NO DT “Wl *uasAiB 24D DyDp jub4)nsas jo sysAjpuD jD4y9eds Jamod jo syjnsey *sBulppay eAj4jDje4 yUa1E}4Ip 4D BuliaAoy sDM auLIDWGNs D 38|!4M eapoW a1aM sBuipsooed esoy) *Ajaatyoadsas ‘suoljow pjnjj pub diys pup uoljow BADM adD}Ns JO sBulpsooe4 BojpuD pup |p4|5ip JO UoIssnosip D sjuasaid Uol4yDO1\Gnd sty) (O0l-YL “O °H) “Ado Zz ‘seyqns Z ** S615 ZZ “dig “1961 Atonige4 ‘uoje|ppiw *f “W pur sngiuosyeq *s “d Aq “{ZZZ-SS) NI4Ga¥ SSN “II ASINYD ‘SAILIDOTSA MO14d-SSOUD JAILV14y GNY ‘3IDNY 1104 SNIYVWENS “NOILOW JAVM dO SISATVNY WnuldadS YIMOd B10 a1 ydouBoupé 4 Arony *S *A *ueaAiB aip DyOp jubj|NsaJ jo sjskjDUD |D4y9eds Jamod jo syjnsoy *sBulppay 9Aj4Djed jUeJA4}1p 4D Bul4sarcy som AULIDUIGNS D ajJ4M epDW a4eM sBuIpsooed asoyy *Ajaatjoadses ‘suojjyow pjnjjy pub diys pup uojyou @ADM 2dD}.ins jo sBulpiooe4 Bojpup pup |b4)6ip JO Uolssnosip D syuaseid uolyoo1\qGnd sjyy (O0l-¥l °O °H) “doz ‘seyqn4 Z ** S815 ZZ *-d 19 “1961 Avonigqe4 ‘uose|pplw “f “W pur snqiuoayeq “Ss “d Aq “{Z/2-SS) NIddau SSN ‘Il ASINUD ‘SAILIDOTSA MO14d-SSOUD AAILV13Y GNV ‘JIDNY 1104 JNIBVWENS “NOILOW JAVM 4O SISATVNY WNULDadS YAMOd 22130 21 ydouBospAyy AADN “s'/n OOL-Yl “O *H SHG} PCE EG] OS “cl *SO14J0]aA MO|4-SSO1D BA}4{DjOy pup ‘ejBuy joy euliowgns ‘uo! 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