Report 1755 HYDROMECHANICS Oo. AERODYNAMICS O STRUCTURAL MECHANICS O APPLIED MATHEMATICS O TICS AND GC 2ATION baw SIMULATION OF A LONG-CRESTED GAUSSIAN SEAWAY by Michael C. Davis, LCDR, USN HYDROMECHANICS LABORATORY RESEARCH AND DEVELOPMENT REPORT April 1964 Report 1755 April 1964 SIMULATION OF A LONG-CRESTED GAUSSIAN SEAWAY by Michael C. Davis, LCDR, USN Report 1755 TABLE OF CONTENTS ANALOG: PROG RAMMINGERE CHING QUIS ieee sce tetn ent: eke er ve eee SynthesiSolayNeumannys pee trellis hie yo eyes seeee eee nara a nee eee Synthesis of Inverse Wavemaker CharacterStiCS © -..2........:-.ccccccccsssecccecsesesecsseeessesevsese DIFFICULTIES ENCOUNTERED IN WAVE GENERATION ......-.-scecsceessseeeeeseeteeeeeeeees SUMMARY OF INFORMATION GAINED FROM SPECTRAL PROGRAMMING............... FUTURE: DIG VE OBRMEINGIY.oc:lrccr see inne c taleeh ad, 1, (wees ena ah 5) ane Generation of a Family of Unidirectional Spectra ..........:sscescecccsssssenesseceuecsesesresenns Approach to the Generation of Directional Spectra ..........-.cs-ccececeeceesccceceeceeecceeeceues AC KINOWDEEDGMIGNIDS wesosccts cscs sce sente evar stcsatvebsevstietes cet cudet iain neh teen ae Gk ae IRE BREIN@ EH Simms ccrsts sunt asta eenan mentee Un aee agatha adhe Am hentai Willa ete ee LIST OF FIGURES Figure 1 — Hydraulic Actuator Used to Control Flow of Air to Wasemakerb om ee ciin su cessasre irssecscor saeeeiiece: 82 obec ese hoon eases testes eal ee Figure 2 — Frequency Response of Actuator Wavemaking System as Determnedsfrom) SinusoidalWhxcitationmn se ener eee Figure 3 — Irregular Long-Crested Waves Produced by Hydraulically ControlllediWavemakersh tae eee ok ave cteesess ie csccesteascerean eee Figure 4 — Typical Wave-Height Recording of Neumann Spectral Simulation with Peak Frequency at 0.3 CPS ..............ccccecsescecerecececeeeceeeress Figure 5 — Spectral Analysis of Waves Produced with Actuator Installation to Simulate Neumann Spectrum with LOSES LAN RETO TOKE RAGA, Co} b (0)42) (CO) 24S). capchenncScononnocc hace oxissondbacdoedoceanoudeaseedaodabusnosheoaoooncane li Figure 6 — Spectral Analysis of Waves Produced with Actuator Installation to Simulate Neumann Spectrum with eakghire quencysotsOs Gon © bi sieessseesce see case eee Figure 7 — Neumann Spectrum Normalized with Respect to IPPECMEMNEG\ GinGl AVON OT IPERS ocococccoadssoogsoasencnce conencasonasonrcuscoseosAncnndcenGceace Figure 8 — Frequency Response of Linear System Synthesized on Analog Computer in Order to Insert Frequency Weighting of Neumann Spectrum with Peak Brequency, Of OLS ORS ccn.ssctavcsca cee ses coysece wets coreee ccc tinea ren ceen sc Secreneeoeene ones Figure 9 — Frequency Response of Linear System Synthesized on Analog Computer in Order to Remove Measured Frequency Response Characteristics of WERPCHDE IMS SIVSLEETM co nocaccosccaaacocnonasnenccooseoupoccacodcecenboseEIa godgenoacooseIBANoHoASOSON Figure 10 — Analog Computer Configuration Used to Insert Proper Spectral Characteristics into Wavemaker Excitation SOUSA ese ns a seltaa susecey. ve ccsutess Ss > 8 1S < el ro) = ie Fy i Bc} i=} 3 1S) 8 o ® n =) Figure 1 — Hydraulic Actuator in each wave length, or alternately, in each frequency band. Some general trends have been universally observed, however; for instance, when the sea disturbance becomes higher, the waves tend to become longer. One analytical form which is often quoted, probably because of its age rather than its close approximation to reality, is the Neumann spectrum of the form (C,/o°) exp (-C,/@7) , Where w is the frequency of the incremental wave com- ponent measured at a point and C, and C,, are constants. In selecting a goal for simulation in the Model Basin facility, it was first specified that a Gaussian random process be generated. Next, for lack of a generally accepted theo- retical form, it was decided to use the Neumann spectrum to adjust the relative weighting of power content in each frequency band. This choice is perhaps not a bad one since this smooth form results in a rather broad but grossly representative power distribution versus frequency and avoids extreme resonance effects which could result from a more narrow distri- bution interacting with the sharply tuned ship ‘‘system.’’ Obviously, it would be desirable to provide for directionality in the waves generated, as this is theoretically possible because of the independent control available for each of the eight wavemaking systems along the short side of the basin. However, it was felt that the initial work in this area should be confined to unidirectional or long-crested sea simulation in order to define and resolve the new problems anticipated with wave control using the hydraulic actuator system. The resulting long-crested waves would themselves be a very useful research tool, simulating rather well certain highly directional wave conditions at sea, and they might be expected to produce model motions generally representative of those observed at sea. In fact, one can argue that in certain cases it might be highly desirable to have long-crested waves in order to investigate nonlinear effects which are quite sensitive to wave direction. In summary, the goal of this exploratory development of wave-generation procedures was to reproduce a long-crested Gaussian seaway with wave lengths distributed correspond- ing to the Neumann spectrums BASIC APPROACH A very basic way of looking at the wavemaking operation is to consider it a “‘system”’ in the manner of the electrical engineer; that is, to consider as a ‘‘cause”’ the voltage signal which controls the hydraulic servo and as an “‘‘effect’’ the wave height that would be mea- sured with a suitable transducer in the middle of the basin. To a first approximation, this system could be considered linear, based on earlier studies of regular waves which indicated a roughly straight-line relationship between wave amplitude and the rate of air flow to the domes. A basic description of a linear system is its frequency response which, in this case, is a ratio of the wave height observed in the water to the sinusoidal voltage applied to the actuator controls. The first such measurement with the actuators (see Figure 2) was the starting point of the programming effort. 40 T 30 = 20 1.0 0.90 0.80 070 3FT-16 IN. DOME CONFIGURATION BLOWER SPEED = 700 RPM 050 c WAVE BARRIERS UP Q60 ;— ACTUATOR DEFLECTION = 41N./V 040 MEASURED DECEMBER 196! 0.30 —. =; = ae =e se WAVE HEIGHT IN INCHES PER VOLT INPUT 10 ses 1 aio 0.20 030 a4o aso 0.60 Qa7o aso aso 1.00 FREQUENCY IN CPS NR Ee ieee ae ey ee | ST Sa eet 500 150 100 60 50 40 30 20 15 6 10 98 7 G WAVELENGTH IN FEET Figure 2 — Frequency Response of Actuator Wavemaking System as Determined from Sinusoidal Excitation It is well known that the amplitude spectrum of a random process is multiplied by the square of the magnitude of the frequency response of a linear system through which the process passes, It was reasoned that if the voltage applied to the actuators had a power density spectrum with the desired Neumann shape, multiplied by a term inversely proportional to the square of the frequency response of the wavemaking system, the resulting wave height observed in the water would have the desired Neumann weighting. In other words, the charac- teristics of the wavemakers would be effectively cancelled out with the excitation signal. The problem then becomes the generation and recording of a random voltage with the desired power density spectrum, ieee, the Neumann shape times the inverse wavemaker char- acteristics. Fortunately, this is a well-established technique in analog computer simulation developed for use in statistical studies of complex systems. The standard source of randomness in the analog computation facility is the so-called ‘“‘white noise’? generator which produces a random signal with a relatively flat power spectral density over a large frequency range and which has the desired Gaussian characteristics. The steps in programming are: 1. A random noise source is used to excite a properly designed linear system on the analog computer, and the output of this system is recorded on magnetic tape. 2, This signal on magnetic tape is used to control all wavemaking systems in unison, creating a unidirectional random wave pattern. If the relationship between waves measured in the water and voltage applied to the wavemaking servos is approximately linear, then the waves will represent a Gaussian random process since it is well known that a Gaussian signal passing through one or more linear systems remains Gaussian. OUTLINE OF RESULTS Initial programming for the wave spectra was completed in June 1962. After several programming corrections and retaping, successful spectra suitable for ship-model testing were created and measured in MASK in September 1962. Figure 3 shows a typical wave pattern in the basin. Figure 4 is a typical recording of the wave heights observed in the water. Figures 5 and 6 depict the result of a spectral analysis of two of the most successful wave programs plotted for comparison with the desired Neumann weighting of power. It is obvious from these results that the basic goals of this exploratory wave- programming effort have been satisfied—the waves appear to the eye to be similar to those observed at sea, they are as close to a Gaussian random process as can be generated, and they have a controlled distribution of average energy in each band of wave lengths. As of January 1963, they had been successfully used in over 60 hours of model testing. The remainder of this report discusses in some detail the programming techniques used and the difficulties encountered; it presents a plan for the development of a series of wave-generation programs which will meet the simulation needs for ship models in sea conditions up to State 7 severity as well as extending to the directional case ANALOG PROGRAMMING TECHNIQUES As will be recalled, the development of the wave program required that a random voltage be recorded on magnetic tape which has a power density spectrum equal to the product of the Neumann spectral shape and the square of the inverse frequency-response characteristics of the wavemaker. From linear statistical theory, this spectrum will result from excitation of a linear system by a white noise generator if the linear system has a transfer function or frequency response equal to the square root of the Neumann spectrum multiplied by the reciprocal of the wavemaker frequency response. The programming or synthesis of the proper frequency characteristics of a linear system on an analog computer is logically divided into the following two parts which represent two linear systems in tandem. n wy 2 g © > 8 = ao} ® a = 2 _ a e) 12) > a S By iS) = =) 3 iol ae} > eo) > 2 ae) ® i) =) aS) io) i= Au 2) © = & = oO ® ~~ n ® is o a0 = ° 4 i S — =} a0 ® i Ba — | los) © Lol =) a) om Fy NCO CIRO BS VA WON Vara epealin'g t fe), AAS A ' ron sad Tis it iF] Beetles is ee AUS ee aa AAT IN eal A WA jew aia ay ay evant A ENE een des aug ea sees ee RS EGE ESE A seeifeeslhituiplan ile F a Figure 4 — Typical Wave-Height Recording of Neumann Spectral Simulation with Peak Frequency at 0.3 CPS EXPERIMENTAL a —_—---- THEORETICAL ——- Os RMS=2.81" al POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) 0.1 o2 0.3 04 0.5 06 0.7 08 oO9 10 FREQUENCY IN CPS Figure 5 — Spectral Analysis of Waves Produced with Actuator Installation to Simulate Neumann Spectrum with Peak Frequency of 0.3 CPS + ——— EXPERIMENTAL —— THEORETICAL a5 }+— r RMS= 2.79" (o}] : ee | 02 a3 04 05 0.6 0.7 os o9 10 1S 2.0 FREQUENCY IN CPS POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) Figure 6 — Spectral Analysis of Waves Produced with Actuator Installation to Simulate Neumann Spectrum with Peak Frequency of 0.65 CPS SYNTHESIS OF A NEUMANN SPECTRAL SHAPE The basic building block of an analog computer is the integrator, with transfer function in the Laplace transform notation of 1/s. Through proper interconnection of these elements, an overall transfer function can be synthesized which is the ratio of arbitrary polynomials in s, assuming that the degree of the denominator polynomial is not less than that of the numer- ators The Neumann spectrum has the form N(o) =(C,/w° ) exp (-C,/@ 2), which is certainly not suitable for direct simulation using the analog computer. The Neumann spectrum has several interesting properties that are not readily apparent. Equating the derivative of the spectrum to zero, we find that the peak power density occurs at a, = VC,/3 with a value of C, exp (3/a?). Normalizing with respect to the peak value, we obtain N(a)/N(o,) = { Op /wt® fexp[3]} { exp [-3/(o/e,)*) , which is only a function of the ratio of frequency to peak frequency. This indicates that if plotted on log-log graph paper, all Neumann spectra would have the same shape; and if normalized to the peak fre- quency and peak amplitude, they would plot on the same curve; see Figure 7. This suggests that a proper presentation of experimental sea spectra should be in logarithmic form for comparison with the Neumann hypothesis. This normalization property of the Neumann spectrum also has significance for analog computer simulation, Because spectra should be provided over a range of peak frequencies in the basin to simulate various sea states, it is of considerable convenience that the shape as al POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) 0.05 0.4 os 06 07 08 09g 10 20 3.0 40 RELATIVE FREQUENCY, W/Wp Figure 7 — Neumann Spectrum Normalized with Respect to Frequency and Amplitude of Peak is a function only of w/ @»y since knowledge of an approximate polynomial expression for a certain peak frequency spectrum automatically yields an approximation for all peak frequencies by making a suitable linear scale change in s. By trial and error techniques common to the electrical engineer, a polynomial approxi- mation was synthesized for approximating the square root of the normalized Neumann spectrum: 4.85 §° (S? + 0.0625 S + 0.199) (S + 1.385) (S + 0.533) 12 (S? + 0.5368 + 0.199) (S + 1.124) The frequency response (at s = jw) of this approximation is compared with the theoretical expression in Figure 8 and is seen to agree quite closely except at extremely low frequencies. SYNTHESIS OF INVERSE WAVEMAKER CHARACTERISTICS The inverse of the wavemaker frequency response (relating the wave height in the basin to a sinusoidal voltage applied to the wavemaking system) was approximated, again by trial and error methods, by a polynomial expression: (S + 1.325)? (S? + 0.98 + 3.24) (S2 + 0.2958 + 8.71) (S* + 0.288 + 19.8) (S + 0.3)? (S2 + 1.448 + 3.24) (S? + 0.5028 + 8.71) (S? + 0.4458 + 1.98) (S? + O.87S + 33.1) (S2 + 1.748 4 8.41) (S? + 4,148 + 33.1) (S + 20)? (S + 50) 10 [POWER DENSITY SPECTRUM MAGNITUDE] ® (RELATIVE SCALE) © EXPERIMENTAL (ANALOG COMPUTER) THEORETICAL 0. 02 0.4 as 06 O7 08 09 1.0 0.3 FREQUENCY IN CPS Figure 8 — Frequency Response of Linear System Synthesized on Analog Computer in Order to Insert Frequency Weighting of Neumann Spectrum with Peak Frequency of 0.3 CP'S O AVERAGE FREQUENCY RESPONSE MEASURED ON ANALOG COMPUTER INVERSE WAVEMAKER SYSTEM INVERSE OF MEASURED BASIN FREQUENCY = 10} RESPONSE 3 FT- 16 IN. DOME CONFIGURATION o = BLOWER SPEED = 700 RPM a ai | > Z | = — s | Oo O5 a =x 2 _| = a we . ESS S i < w i 2 ce) a FA w c > 2 OF Oo 3 Ol rr c u w 7 [4 w > +—$—$—___—______— = 0.05 ee ell [ 0.1 0.2 03 0.4 05 06 07. 08 09 10 FREQUENCY IN CPS Figure 9 — Frequency Response of Linear System Synthesized on Analog Computer in Order to Remove Measured Frequency Response Characteristics of Wavemaking System 11 The complexity of this expression arose essentially from a desire to match every observed peak in the experimental frequency-response characteristics, as shown in Figure 9. In retrospect, it would have been better to use a much simpler approximation and to reserve a finer frequency correction for later application. The overall analog computer configuration is pictured in Figure 10. It was excited by white noise generating a random voltage which was recorded on magnetic tape in order to control the wavemaker system. DIFFICULTIES ENCOUNTERED IN WAVE GENERATION Having fixed the desired relative distribution of wave height for each increment of frequency, there are two basic goals in optamizing the controlling program: 1. To raise the upper limit of average wave height which can be created in the basin. This is constrained by the allowable range of travel of the flapper valve in the wavemaker and by the maximum rpm (1500) at which the air-blower motors can be driven. 2. To obtain the specified distribution of wave height at each frequency. Several deficiencies were obvious in the first set of programs developed in June 1962. The measured wave heights were much smaller than desired, the spectral distribution was greatly distorted, and the hydraulic servo systems were rate-saturated. It was immediately obvious from an examination of the spectrum of the taped excitation signal that a very consid- erable excess of power had been concentrated in the frequency range above 1 cps in order to compensate for the inefficient generation of waves at these frequencies. This excess energy was undoubtedly the reason for the observed rate saturation, and because of the nonlinear behavior, it contributed to the spectral deformation. Also, since the total power of a command signal determines the root-mean-square (rms) motion of the flapper valve, which cannot exceed a certain amount because of physical limitations on travel, this additional inefficient energy caused a large reduction in observed wave-height levels in the basin. To lessen the effect of this undesirable high-frequency power, the programs were later operated on by a low-pass filter, which had a flat frequency response to 1 cps and a 160-db/decade attenuation thereafter, synthesized on the analog computers Still another correction was introduced to increase the maximum wave heights which could be generated in the basin. In theoriginal programs, the average amplitude of the flapper valve motion, which was continuously variable, was adjusted so that the largest peaks in 20 min of recording were a little less than the mechanical stops on the valve. To increase this average valve motion, an electronic clipping of the subsequent taped program was made which sliced off occasional peaks at the same voltage. This permitted adjusting the running level so that the clipped peaks caused motion just within the mechanical stops and resulted in a considerable increase in wave height with a slight cost in nonlinearity. 12 43040934 3adVL [BUdIC UOTYBIIOXy JoyewoAeM OJUI SONSWoeyoeIeyD jeajoods sedoid jlesuy 0} pes uorzyeinsdijuoD seindulog sojeuy OT oinsty 13 Upon spectral analysis, the white noise signal which had been used to excite the analog computer circuitry displayed a disconcerting but predictable lack of smoothness, as shown in Figure 11. This irregularity was present to some extent in every excitation and wave spectrum observed throughout this wave testing, and it greatly complicated program correction analysis. Two sets of frequency corrections were made, using the analog computer to synthe- size correction networks. The final spectra developed in this exploratory program are shown in Figures 5, 6, and 12. These corrections were not uniformly successful indicating that at least mild nonlinearities in wavemaking were present, that the analog spectral analysis equipment was deficient in performance, or that the spectra depended to some extent on where in the basin the wave heights were measured. To investigate some of these possibilities, a series of wave measurements was made in January 1963 with variation in program running level (attenuation of the tape recorder signal), blower speed, and basin location. These resulting spectra are displayed in Figures 13, 14, and 15 where a severe variation in spectral shape occurs as a function of location, but only a mild variation develops for changes in the wave-height level. The controlling program, which was repeated during each wave measurement, was designed to have essen- tially constant energy content over the frequency range of interest and thus to have the shape of each spectrum proportional to the square of the frequency response of the wave- making system. POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) OM] E : | | 0.05 IL | 0.! 0.2 03 04 ohs} O06 07 08 0.9 i FREQUENCY IN CPS Figure 11 — Spectral Analysis of Signal from White Noise Generator with 30-Minute Recording Time 14 9 a | @) Eee aera! POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) ° fe} a —— Ol 0.2 03 04 0.5 06 07 08 o39 10 FREQUENCY IN CPS Figure 12 — Spectral Analysis of Waves Produced with Actuator Installation to Simulate Neumann Spectrum with Peak Frequency of 0.5 CPS 5.0 } RMS = 2.34" RMS=1.28" Os L POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) \ 7 TT zal s \ Ba oe | * eg PROGRAM AMPLITUDE = 0.877 \ ee -——— PROGRAM AMPLITUDE = 0.4385 \_ 7 PROBE IG6OFT FROM WAVEMAKERS \ oa BLOWER SPEED = 1400 RPM \ / 4 7 \ his ol 02 04 0.5 06 07 08 09 10 0.3 FREQUENCY IN CPS Figure 13 — Change in Wave Spectral Shape Because of 50-Percent Reduction of Excitation Level of Actuators Using Input Program with Flat Frequency Characteristics POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) POWER DENSITY SPECTRUM MAGNITUDE (RELATIVE SCALE) 5.0 RMS= 2.34" BLOWER SPEED =!400 RPM -<----- BLOWER SPEED =!!100 RPM PROBE 160 FT FROM WAVEMAKERS PROGRAM AMPLITUDE = 0.877 02 ee Ou 0.2 0.3 0.4 05 0.6 O07 og og 10 FREQUENCY IN CPS Figure 14 — Change in Wave Spectral Shape Because of 20-Percent Reduction in Air Blower Speed Using Input Program with Flat Frequency Characteristics At | 2 a 1 == RMS=2.34- 0 = y =a | # 7 | RMS=2.47~ oA TY NS y \ Zi Lie, y eer Zan | “ eZ a f f a = \ RMS=2.47~ 7 \ x \ IZ \ \ 1.0 ba 4 \ t | PROBE 80 FT. FROM WAVEMAKERS MA A ———=PROBE 160 FT. FROM WAVEMAKERS DS 4H == Us —--—PROBE 272 FT. FROM WAVEMAKERS N INSETS) N. PROGRAM AMPLITUDE 2 0.877 05 BLOWER SPEED = |400 RPM ‘i \ XN N \ \ | 0.2 = 0.1 0.2 0.4 [oke} 0.6 O7 08 0.9 1.0 03 FREQUENCY IN CPS Figure 15 — Change in Wave Spectral Shape as a Function of Location in the Test Basin Using Input Program with Flat Frequency Characteristics SUMMARY OF INFORMATION GAINED FROM SPECTRAL PROGRAMMING The following summarizes the results obtained from this exploratory effort: 1. MASK can simulate unidirectional Gaussian sea conditions which have a specified relative distribution of wave heights and sufficient wave amplitudes to conduct model tests over a range of sea states. 2. The hydraulic actuator installation is a flexible, reliable method of producing either regular or irregular waves with carefully tailored characteristics. 3. An important initial requirement for programming using the analog computer technique is that the white noise source have a flat power density spectrum. 4. Careful attention should be paid to the reduction of unneeded high-frequency energy in programming as well as to electronic limiting. 5. Because of some nonlinear characteristics in wave generation, the initial attempt to cancel basin frequency-response characteristics should be smooth and approximate. When the initial program is adjusted to generate waves of the desired intensity, spectral analysis will yield a much better second correction, assuming linearity at least about the operating points 6. Analog spectral analysis equipment should be used with great care since program cor- rections will be concerned with the difference between two relatively steep curves; this dif- ference is very sensitive to small errors in frequency determination. 7 A significant limitation in the precision of simulation available is the spatial varia- tion in wave-height spectra in the basin. For future programming, an average spectrum over the normal running path of the model must be defined by making very slow passes with a moving wave-height probe over this track while the random waves are being measured. FUTURE DEVELOPMENT With the success of the actuator installation in generating random waves with desirable properties, two research programs are underway to make use of this capability. GENERATION OF A FAMILY OF UNIDIRECTIONAL SPECTRA The Neumann spectrum for fully developed seas causes every average wave height to be related to some frequency of peak wave height in the power density spectrum such that when the seas become rougher, the distribution shifts lower in frequency. An engineer desiring to simulate certain sea conditions at some scale ratio will select a basin rms wave height which is equal to the full-scale rms wave height divided by A and a basin peak fre- quency which is equal to the full-scale peak frequency multiplied by VX . These relations are summarized in Figure 16, which also shows the rms wave heights achieved to date with programs described in this report. From these wave-height results, it is planned to span 17 20 I I I I 7 10 PEAK FREQUENCY OF NEW FAMILY OF SPECTRA NEUMANN SPECTRUM SIMULATION IN MASK FACILITY RMS AMPLITUDE OF WAVES MAXIMUM VERSUS FREQUENCY OF PEAK OBTAINED ENERGY AS FUNCTION OF SCALE RATIO X OR SEA STATE MAXIMUM OBTAINED RMS WAVE HEIGHT IN INCHES (SINGLE AMPLITUDE) n 05 all LL 1 0.1 0.2 0.3 04 05 06 0.7 08 09 10 PEAK FREQUENCY IN CPS Figure 16 — Relation among Peak Frequency of Neumann Spectrum, RMS Wave Height, Model Scale Ratio A, and Sea State for Simulated Long-Crested Seaways in the Test Basin the useful frequency range of the basin with a set of six spectra, which will be measured so as to perform a spatial averaging. The peak frequencies of these spectra, as shown in Figure 16, are spaced so that a model of arbitrary scale ratio (within a large range) can be tested in successively higher sea states with each program. A typical model with d = 30, for example, would be able to test in a valid Neumann spectrum in States 4 through 7 seas. APPROACH TO THE GENERATION OF DIRECTIONAL SPECTRA With the present installation, there are eight wavemakers which can be independently controlled from eight channels of a magnetic tape recorder. The proper programming of these units to create an adequate simulation of a directional seaway is a challenging problem. One approach is outlined here. If a voltage sine wave of constant amplitude is used to excite each actuator but the phase lag between adjacent units is nonzero but constant, then in general a sinusoidal wave traveling at an oblique angle from the bank of wavemakers will result, apart from some second-order waves. The phase lag in degrees needed to produce a wave traveling at a fixed angle is a function of frequency. If, instead of a sinusoidal wave, it is desired to generate a program for an oblique but unidirectional random wave train, this could in theory be obtained 18 by passing some random voltage through a succession of seven identical linear networks, each having a constant amplitude but prescribed phase change over the frequency range (known as ‘‘ all pass’’ networks), and then by taping each of the eight random signals. Considering a directional spectrum to be made up of a large number of independent random wave trains, each traveling in a certain direction, an approximation to this spectrum could be made with perhaps five or seven such separate oblique wave programs produced as above and summed together on a composite tape. Assuming linearity of the wavemaking system, the desired directional properties could be well approximated both in frequency and direction through use of the techniques developed with the long-crested waves on each of the oblique components. Needless to say, such an ambitious program must rest on a founda- tion of experience with random unidirectional waves generated with the actuator installation. ACKNOWLEDGMENTS The author is grateful to Mr. Joseph E. Russ for his able assistance in all phases of the wavemaking tests and to the staff of the Motions Analysis Section for technical advice in the use of analog computing equipment. REFERENCES 1. Brownell, W.F., ‘“T'wo New Hydromechanics Research Facilities at the David Taylor Model Basin,’’ Paper presented before Chesapeake Section, Society of Naval Architects and Marine Engineers (Dec 1962). Also David Taylor Model Basin Report 1690 (Dec 1962). 2. Vossers, G., ‘‘Fundamentals of the Behavior of Ships in Waves,’’ International Ship- building Progress, Vol. 6, No. 63 (Nov 1959). 19 - 4 a ve a oT @ en By ¥ ; wi Gat 45 gn Saree ‘ 4 at TA re i i > | b ae wet Ihe , Lit ihe Laney aT duh ak hit ate die Pah eck wade veer ahead Pei ae ihren w meaegaey mull he vs opted clan thin Lelie gh A cual ued! i ae Ty EM ey Ty BA fet wd Wr cy ME bingtal: ae ia ht ” - A prodhhveg rntd tae | pation? fad; eget in il : vals A bt (it shy ) ne } pulldoe ie a hil A ey Ain) SALMO ts yas r Wei. F ze ort Fey ‘aa vate . y ish ’ bowie & Sc a eee Be ; at, Ley ul rik rN i Nal hatin OO LATE: tel ween x t ee . / t Md beth mae a a i foes) ae i n Rm aes ty a) iy tay a. ae iwi Pa te i ee hap th fe gibt weal cr ; : i y ite “hit rn rit ty pitts af of Bike e 54 ( i 5 = Lin rahi ve moe, = IG | ee TTP abril ry oF AKI: ree niitess Os Eee he PMO Ra eee BD } i ; viz os a 1) ae x ae ty) ana ace a CEL ate te eee ey AP bt (Po ye ied oh { nt X = « 0 £4, + peivhi feed Te aati af Ht CONSE St i i A \ ’ & : ‘ & "Ea Le i) I si) ae tha kta ante Tiptree - ' ‘ 4 7 ‘i eat Y See oy ae! SAL eo ce , ng ae sie ve j Pik easy + La ee R48 i SOE VRE Rise Falta | : , y i Zz ae ne ah — Se > AS Roa pa : Relea Rist et Ee Pe tere Ae a aie yr r 3 % ” ee pide é ‘ s i rs Reade pl Ae ; reve, ME ROS? i A E Ateae) rs ‘ 3 fir shes in| iy Syed hat 5 = ay iy - is j t j Y + & ‘ ah \ ‘ fs y: 1ehs ced Copies 9 INITIAL DISTRIBUTION CHBUSHIPS 3 Tech Lib (Code 210L) 1 Appl Res (Code 340) 1 Lab Mgt (Code 320) 1 Prelim Design Br (Code 420) 3 ASW Support Br (Code 373A) Attn: Mr. B.K. Couper CHBUWEPS 2 Aero and Hydro Br (Code RAAD-3) CHONR 1 Naval Applications (Code 406) 1 Math Br (Code 432) 1 Fluid Dynamics (Code 438) 1 Undersea Programs (Code 466) ONR, New York ONR Pasadena ONR, Chicago ONR, Boston ONR, London NAVSHIPYD NORVA NAVSHIPYD BSN NAVSHIPYD PTSMH NAVSHIPYD PUG COMSURASDEVDET CDR, USNOL, White Oak DIR, USNRL CDR, USNOTS, China Lake CDR, USNOTS, Pasadena CO & DIR, USNMDL CO, USNUOS SUPSHIP, Quincy SUPSHIP, Camden 1 Mr. J.W. Thompson, Nav. Arch (Design) DIR, Langley RESCENHYDRODIV CDR, USN Missile Ctr, Point Mugu Copies 1 1 1 20 CO, Frankford Arsenal Off Air Res DIR, NATBUSTAND BUWEPSREP, Eclipse Pioneer Div, Bendix Aviation Corp, Teterboro DIR, DEFR&E CDR, DDC , Attn: TIPDR ADMIN, MARAD 1 Mr. Vito L. Russo, NNSB & DD Co, Newport News 1 Asst Nav Arch 2 Dir, Hydrau Lab Dir, Hydrau Lab, Univ of Colorados Boulder Dit, Hydrau Res Lab, Univ of Connecticut, Storrs Dir, Scripps Inst of Oceanography, Univ of California, La Jolla Dit, Fluid Mech Lab, New York Univ, New York Dir, Robinson Hydrau Lab, Ohio St Univ, Columbus Dir, Hydrau Lab, Penn St Univ, University Park Dir, ORL, Penn St Univ, University Park Dir, Woods Hole Oceanographic Inst, 1 Dr. Columbus O’D. Iselin Dir, Hydrau Lab, Univ of Wisconsin, Madison Dir, Hydrau Lab, Univ of Washington, Seattle Admin, Webb Inst of Nav Arch, Glen Cove Dir lowa Inst of Hydrau Res, St Univer of lowa, lowa City 1 Dr. L. Landweber Dir St. Anthony Falls Hydrau Lab, Univ of Minnesota, Minneapolis Dir of Res, Tech Inst, Northwestern Univ, Evanston NSO “AGOT $°O [eByoIW ‘stad I used [2° pow 1OpAB], PIABd--°S*()--SuIseq duidooyees-duLIOANeuBW °F durmueidoig --[ONUOD--SIOYBWOABM °E suie}sAs. [01}U09 =-UOI}B[NWIG--SOABM UBODQ °F suru -Weid01 g--Ssule}sKks [oIU0D, --S]epoul sves xefduiog °T NSN “UGOT “0 [evyoIw ‘stavq *T uIseq [9 pOW 1ojke], plAwd--"S*--suiseq dutdooyees-dulleAnousp °F sulMuUsidoig --[0ONUOD--SIOYBWOABM °E suiaysks [oIyuOg --UOI}B[NWIS--SOABM UBADQ °Z surw -uwidoig--suieyshs [o1]U0D, --S[epoul sves xejdwog °T *pejueseid O18 UOIZB[NUIS BOS [BUOI}IEIIP 0} UOISUe}xe puB JUeUIdOTEAOp JOYNJ OJ SUB[q “POSSNOSIp O18 [01]UOD JOYBUIEABM 10j JeqnduIooD dopeue ue JO osn 9y} YIM peyeloosse SoN[Noyjip pue senbiuyoey, *uOIBIeUET OABM 10j WoySAS [OUD OI[NeIpAYyoIzO9Ie UB JO asN Aq 3ut}s9} [epoul 10j poonpoid oom odeys jerjoeds uueuINeN ayy jayeurtxoidds yorym SABMvosS UBISSNEH WopuByY cAqI]IO¥y dutdeoy -80G pue SULIOANOUBP| S1OpUNeS “| P[O1BH{ ey} UI S¥es posal -JuO] 1e[NFeut Jo uoTB[NUIs ONSI]BoI oy} SuIOOUOD yloded SIYJ, GalaIssv'IONN esjoa ‘sydead ‘esnqyt deg ‘ttt “P96T dy “stawq °D [eByorW Aq ‘XVMVUS. NVISSQVD GHLSHAO-DNOT V 40 NOILVTANIS. GGZ| Hoday ‘ulsog japow 40;AD) piang *peyuesoid O18 UOIJB[NUIS BAS [BUOI}OEIIp 0} UOISUe}xe puB JUeUIdO[eAep JOIN] 10J SUB[ “POSSNOSIp aIB [01}U0D JOYBWOEABM 10} JeynduUIOD do[eus ue JO sn oy} YIIM peywloosse Saty[NOjIp pue sonbiuysey, *uoTwseued ABM 10j We4SAS [OX}UOD OI[NBIPAYOI}0e[e UB Jo osN Kq 3urys9} [epoul 10J poonpoid osm odeys [ejoeds uueuNeN ay) ayeutxoidds yorym sAemuos uBissney wopusy *AjI]I08qy dutdeoy -8o0G pus DULIOANOUB), S1OpUNLS "| Pole] oy} UI S¥aS pojSeJoO -dUO] I8[NFeUL jo UOI}B[NUIS INSI]Bel oy) SUIed.UOD 410de1 SITY], GaIdIsSV TONN esjor ‘sydead “esnqyt “dgg ‘ttl “F96T dy *staBq °D [eByoIW Aq ‘XVMVGS. NVISSNVD GHLSTAO-DNOT V 40 NOILV TANS. “GGLL Hoday = “ulspg japow 40j4Dy p1ang NSN “AGOT §°O [owyoIW ‘staeq °] used [9 poW 1ojAe], PIAB=-"S*()--suIseq dutdeeyves-dutieanousW °F sulMUBId01g --[ONUOD--SiayVuleawm °¢e suie_sks [ou --UOI}B[NUIG--SOABM UBBDQ °% suru -weidoig--swieysXks [o1WU0Z. --S[epoul sees xefdwog °T NSn “AGOT “°O [owyorN ‘stavq *] uIseq [epoW JopAB], PIABG--"S*/--suiseq sutdooyves-suteAnous °F durwweidoig --[ONUOD--SIoYxBWOABM °E swiesks [oIyu0D +-U01}B[NWIG--SOABM UBEDQ °G suru -uiwido1g--sule}sAs [oU0Z: --S[apoul Ses xefdwog °T *pequesoid eiB UOIB[NUWIS Bes [BUOT}DEIIP O} UOISUe}xe puB yUeUdojeAep JoyjJINJ 10j SUB[ “PpoSSsNdSIp oe1B [OIWUOD JeYBUWIEABM 10} JeynduIoD do][eus ue JO oSN dy} YIIM poeyeloosse son[Noyjip pue senbruyoey, *uolBieUed BABM 10} WeySAS [ONUOD oI[NeIpAYyoI}09[e UB JO asn Aq Sut}s0} [epow 10j peonpoid e1em odeys [esjoeds uuewneN au} eyeutxoidde yorum SXAeMBes UBISSNeH WopuBYy *AzI]IOBy Surdeey -B0G pue JULOANOUBW SiepuNs “| P[olef] ey} UL SBes paysaio -du0] Je[Ndeut Jo uoe[NUIS ONSI[BeI ey} SUIoOUOD yoded sty J, GaI4ISSV TONN *sjoa ‘sydead Sesnqyt dgg ‘ttt “p96T Idy *staBq *O jeByorW Aq ‘XVMVGS, NVISSQVD GHLSHYO-DNOT V JO NOILV TAWIS "GGZL HOday = ‘ulspg japow 10;ADy_ prang *pejueseid 918 UOlIB[NUWIS BES [BUOI}DIIP 0} UOISUS}xe puB JUeuUIdoJeAep JOY}ANJ 1OJ SUB] “PeSSNOSIp aI¥B [OIUOD JoYBUIEABM 10j Je;nduIoD doBuB uw JO OSN oY JIM pazBloOsse SoN[NOYJIp puw senbruysey, *uOI}BoUed GABA IO} WeySAS [ONUOD OI[NBIpAYoI}Oe]Je UB JO Asn Aq 3u1}80} Jepow 1o0y peonpoid 1am edeys jeOeds uuBUNEN ey} eqyeutxoidds yorym SAsMves UBISSNYH WopuBy *AqI[IoBy Fuideey -80G pUB DUIJOANOUBW SIOpuNBS “| P[oleA{ oy} UI SBes paqselo -dUO] Ie[NFeU! JO UOI}B[NUIS ONSI[BeI OY} SUIEOUOD WIOdoI SIYT, GaI4ISSV TONN esjoi ‘syduid S*snyjt °dgz ‘ttt p96T dy “staBq *O JaByoIW Aq ‘XVMVGS, NVISSQVD GALSAAO-DNOT V AO NOILLY TNWIS "SLL Hodey —-uispg japow 40j404 piang NSN “UdOT §°O [evyoIW ‘stAvq “I uIseq [oPOW 1ojAB], plABg--°S*()--SuIseq Sutdooyvos-suLleAnousp °F surmueidoig --[ONUOD--SIOYBWSABM °E suiess [oUu0D --UOI}B[NWIS--SOABM UBBDQ °% suru -ureido1g--swieyshs [o1yUu0D --S[epoul Sves xe[dwog °T NSN. “Ud “*O [evyoIW ‘stavq “I uIseq [OPOW 1oO[ABY, PIABG--"S*()--SuIseq dutdoeyses-duLeAnous °F durmueidoig --[ONUOD--SIoyxBUOABM °¢ swejsks [01]U09 +-UOI}B[NWIG--SOABM UBODQ °% dur -uwid01g--suieysAs [oU0D, --S]epoul sves xejduog °T *pejuesaid 08 UOI}B[NUIIS BES [BUOT}EIIp 07 UOTSUe}xe puB JUoUdoTeAep JOYJINJ 10} SUB[d “POSSNOSIp av [OUOD JeyeWIEABM 10} JeyjndWoo Soyeuse ue jo asn 9y} YIM peqwloosse Seljy[NojIp pue senbiuysey, *uOTB1eUeT GABA IO} Wo ISAS [OUOD OI[NeApAYOI}Oe]0 UB JO OsN kq gunsey Jepou 10J poonpoid o1am odeys [ejoeds uueuNneN ey} ayewtxoidds yorym Skemues usissney wopusy °AyI]I08y dutdeey -80G pUB DULIOANOUB SiOpUuNBS “| P[OIvH{ ey} UI S¥es pojsoaIO -3UO] IV[NFZOLUI JO UOIZV[NUIS ONSI[Bel OY} SUIODUOD j10deI STY I, Gala ISSVTONN esjoi ‘sydvad “esnqt -dZg ‘IHt P96T dy “slaw °D TeByoIW Aq ‘XVMVUS. NVISSAVD GHLSTUO-DNOT V 40 NOILY'TANIS, “GGLL Hoday = “ulspg japow 40j4Dy piang *pequesoid 038 UOIB[NUIS BAS [BUOI}DEIIp 0} UOISUa}xe puB jUeUdoTeAep JOYJINJ 1OJ SUB[d “*POSSNOSIp ai¥ [0I]UOD JayeWeABM 10} JeyndWoo SojBue ue jo osn oy} YYIM peyeloosse SeN[NoyjIp pue senbruysoey, *uolywioued CABM IO} WeySKs [O1}U0D OI[NBIpAYOI}O9]9 UB Jo esN Aq 3utyse} [epoul Joy poonpoid elem odeys [esjoeds uuBUINEN oy} ayeuixoidde yotym SABMeOS UBISSHEH Wopusy °AI[IO’yY durdeoy -80g pus JUTIOANOUBA SIOpUNBS *|] ploIs_] OY} UI SBes poqselo -dUO] IB[NZeU JO UONB[NWIS INSI[BeI EY} SUIODUOD yIOdeI SIYT, GaI4ISSV TONN esjoi ‘sydead ‘esnqpr “dg ‘ttt “P96T Ady *staBq °D [eByoIW Aq ‘XVMVGS. NVISSAVD GHLSAAO-ONOT V 40 NOILV TANIS. 'SG/| HOday ‘ulspg japow 1ojAny plang NSn “UGOT §°O [oByoIW ‘staeq J uIsegq [9poW IO[ABT, PIABG--°S°()--suIseq Suidooyvos-sulleAnousy °F durmwuBisoig --[0NU0D--SioyBUeaBy °¢ suieyshs_ [017U09 --UOI}B[NWIS--SOABM UBBDQ °° surw -weid01g--suleysKs [01jU09. --S[epoul Svas xefduog °T NSn “AGO ‘0 [eByoIW ‘stauq “I uIseq [°pow 1o[AB], PIABG--"G*()--SuIsEq sutdeoyees-dulloAnousy °F durmusisoig --[ONUOD--SIOYBUIEABM °¢ suieisAs [oud --U01}B[NWIG--SOABM UBEDQ °Z suru -weidoig--sulayshs [01009 --S[epow sves xejdwog °T *pequeseid O18 UOTB[NWIS BOS [BUOT}OEIIP 0} UOISUa}xe puB UeUIdO;oAep JoyjINj 10} SUB] “PpoSSNOSIp e1B [ONUOD JeYBUIEABM 10} JaynduloD dojeue ue jO osn 8yj YIIM peqeloosse son[noyjip pue senbruyoey, *uoI}BieUed BABM IO} WeySAS [ONUOD OI[N¥ApAyoI}OeTe UB JO asn Aq 3u13s0} Jepouw Joy peonpoid e1om odeys jesjoods uueuneN eu} eyewixoidds yorym sXemees uBISsNey WopuBy *AqI[IO8q Surdeey -80G puw DUIeANeUBW SlepuNes “| p[olsf{ oY} UI SBes peqseio -dUO] IB[NFeUt jo uOIB[NWIS ONsI[BoI OY) SUIOdUOD yOdeI SIT, GaldISSV TONN esjoa ‘sydvad Sesnqyt "dzg ‘ttt p96T Ady *staBq °D JeByoIW Aq ‘XVMVGS. NVISSNVD GALSANO-DNOT V AO NOILV'TANIS "GSZL HOday = ‘ulspg japow 10jAD] plang *pequeseid 018 UOIIB[NWIS BES [BUOT}OEIIP 0} UOISUa}xe puB JUeUIdOT;eAep JOYJAN] JO} SUB] “*PpeSSNOSIp eB [OIUOD JayBWEABA 10j Jeynduloo do]Bue uv JO ESN OY) YIM poyBloOSse SoN[NOYJIp pue senbruyoey, *uoKBieued OABM 10} Wo\SAS [ONUOD dI[NBIpAYoI}IeTe UB JO esn Aq du1}se} Jepow 10J peonpoid o1em oedeys [ejoeds uuBuNeN ey} ayeutxoidds yorym SAeMves uBIssNey WopuBy *AqI[IOBy Fuideoy -80G pUB DJUIIOANOUBW SJopuNnes *q pfolBy oy) UI SBes pajsalo -dUO] JB[NFEUL JO UOB[NWIS ONSI]BeI OY} SUJOOUOD WOdeI SITY, dal4Issv TONN esyoi ‘sydvad fesnyyr °dgg ‘ttt P96T Ady “stag °O JeByorp Aq ‘XVMVGS. NVISSQVD GHYLSAAO-DNOT V AO NOILY TANIS "SLL Hodey —“uispg japow 10)404 piang b. \ 4 u oy athe siy Apa a: j Y ‘ ' t AN tes ( :