Cross-spectral analysis of ship model motions: some experimental and computational problems

CROSS-SPECTRAL ANALYSIS OF SHIP MODEL MOTIONS:

SOME EXPERIMENTAL AND COMPUTATIONAL PROBLEMS

by John F. Daizell

Report 853 November 1961 DL Project GE 2245 SPONSORED BY BUREAU OF SHIPS

FUNDAMENTAL HYDROMECHANICS RESEARCH PROGRAM TECHNICALtY ADMINISTERED BY

DAVID TAYLOR MODEL BASIN CONTRACT Nonr 263(09) PROJECT S-R009 01 01

DAVIDSON LABORATORY

Stevens Institute of Technology

Castle Point Station

Hoboken, New Jersey

Approved by

P. Breslin

TABLE OF CONTENTS iii Wave System 10 Computational Techniques 11 Summary 1k Acknowledgements ₁₅ References ₁₆ Abstract Page V I. Introduction 1 A. General 1

B. Recording and Data Reduction 2

C. Towing Arrangements ₃

II. Effects of Towing Techniques 5

A. Test Setup ₆

B. Data Analyses ₇

ABSTRACT

This report presents the results of investigations of the low coherencies found in the first cross-spectral

analysis of ship model-tests. The major cause of these low coherencies was the computational technique used. The first

analyses failed to provide enough resolution in the rapidly varying cross spectra inherent in the experiment. Although

an optimum solution is not available, practical methods of improving the computational technique were developed.

I. INTRODUCTION

A. GENERAL

This report presents the results of a series of

experiments that were conducted to provide further

informa-tion on the causes of the low coherency that resulted when cross-spectral analyses were first applied to ship-model

tests, and forms a part of the attempt to substantiate the.

validity of spectral and cross-spectral analyses applied to ship-model motions (references 1 through

5).

Cross-spectral analysis was first used (in

₁₉₅₈₎

to

analyze ship-model motions in random seas (reference k) by comparing model transfer-functions derived from experiments conducted in irregular head-seas with. those derived from experi-.

ments conducted in reguiax waves. The. attempt was partially

successful; derived amplitude and phase responses (transfer

functions) were in reasonable agreement. However, coherencies

(a measure of the linearity of the systeiri') _{between wave data}

and heave or pitch data were obtained that were significantly

less than 1.0, though, the coherencies between heave and pitch

were about

_0.95.

_{These anomo].ous results cast serious doubts} on the validity of .analyzjng:shjp...modeis as linear systems.

Before the crOss-speotral analysis technique was

used to reduce large amounts of data It was studied in greater

detail to determine whether or. not the low computed cohCrencies

were a consequence of the experimental and computational

methods. . . .. .

*A coherency of 1.0 implies perfect linear dependence of

out-put on inout-put. A coherency of 0.0 implies, two incoherent .pro cesses--that is', different and essentially independent.

B. RECORDING AND DATA REDUCTION

The first consideration was to determine whether

the electromechanical system, which converted the motions of

the model and the wave elevations to digitized time-series

suitable for cross-spectral analysis, contributed any non-linearities or had any odd frequency response characteristics.

A concurrent general program of equipment

improve-ment at Daidson Laboratory showed that the electromechanical

transducers used previously (reference k), coupled with

ampli-fiers anda digital data processing system (reference 6),had flat amplitude-responses over the range of frequencies

encountered in the model tests. Quantitatively, these

trans-ducer responses were flat well within the resolution of the

digital system, which was one part in 127 for full scale. (This resolution is about the same as can be expected from

manual readings of oscillograph tapes.) The phaseresponse of all transducers was as good; no measurable relative phase

error between different transducers was found. The standard test technique is to take no data unless static calibrations are linear to a tolerance not exceeding the resolution of the

digital system. Therefore, the only distortion of results that were attributable to the sensing and recording systems were reading errors associated with the resolUtion of the

digitizer.

Reading errors manifest themselves in spectral

analysis as white noise. The following equation is an apprOx-imate formula for the coherency of a linear system with both input and output contaminated by incoherent white noise

(reference

7).

Coherency = I

1.

s(o)11

11+

Ill

where s(a) and

are the spectral densities of white

noise, and S(a) and s() are the spectral densities of the

input and output signals0

For the narrow-band processes involved in model work, an estimate of white-noise energy may be obtained by

noting that the scalar spectrum fails to go to zero at fre-quencies far removed from the significant band of energy.

The density of the scalar spectrum at these remote. frequencies

is the magnitude of the white-noise spectrum, which by

defi-nition is constant for all frequencies.

The estimates of the noise-energy to signal-energy ratio for wave- or model-motion spectra, obtained at Davidson Laboratory, seldom have exceeded 0.02 at frequencies near the peaks of the spectra and increased as frequency increased or

decreased.

Thus, the coherency expected in the previous,

experi-ments was about 0.96 over the range of significant input and

output energy. At frequencies outside those associated with significant energy content, coherency is expected to go to

zero. This behavior was exhibited by the heave-pitch

coher-ency (reference )-i-); coherencies of about 0.95 were obtained

in the significant range of spectral energy. However, in the

first application of cross-spectral analysis to ship model

motions, both the wave-pitch and wave-heave coherencies showed

a kO% reduction near the spectral peaks instead of a k

reduc-tion (reference k). Therefore, the effects of white noise could not be considered to be one of the major causes of the

anomolous coherency results reported in reference k.

C. TOWING ARRANGE TS

In the case of pitch and heave, model motions were directly measured with reference to a single vertical heave

directly abeam of the point in the model where heave was measured; therefore, the wave probe was mounted ahead of the

model. Because of this, the experimental system was set up

so that the input was the wave ahead of the model,

and

the

output was the heave (pitch) at the model (Figure 1).

The usual ideal system has the undistorted wave profile at amidships on the modelas the input. Therefore, the problem was to determine whether the experimental setup caused any coherencies lower than those of the ideal system.

Figure 1 shows the experimental setupused for the experiments reported in reference k. The model was permitted to drift relative to a constant-speed carriage and to oscillate

in surge under the influence of a constant tOwing-thrust. To

do this, (i) the model was towed by a sub-carriage that was permitted to move relative to a main carriage, and (2) towing weights were adjusted so that the mean speed over the run

equalled the carriage speed (Vc). A wave probe was mounted

on the constant-spèèd carriage.

The model moved relative to the main carriage afli

the distance (D) between the wave wire and the model was a

function of time--that is,

D = Do D1(t)

Do = Initial distance (constant)

D1(t) = oscillatory surge plus low-frequency drift, which

resulted from small changes in model speed during

the run. . ..

These experiments were conducted under moderate-wae.

condi-tions. During the experiments, no oscillatory surge was

apparent, however, low-frequency drift (speed change) was observed, which obviously was not linear with wave elevation In the sense of the assumptions used in linear systems analysis.

II.

EFFECTS OF TOWING TECH1'IQUES

When waves

are

assumed to be linear, the wave

eleva-tion at the model Oan be derived from the wave elevaeleva-tion

ahead

of the model. The amplitudes of all components do not change

and the phase is D2D/g (reference 1+) Thus, the theoretical transfer function of the wave at the model with respect to the wave ahead of the model is

Where, a is the wave frequency and D is a constant.

The transfer function of the response of the model

with respect to the wave at the model can be expressed as:

ió (w)

Z(We)e

z e

where, _We = w

for head seas

Z(CUe) = (response amplitude operator)V2

óz(We) = phase lag of response after the wave.

The theoretical transfer function of the model response with respect to the wave ahead of the model is

iE2D/g +

'

Z(We)e z

When D = + D1(t) is substituted, the phase relations between wave elevations ahead of the model and the model response can not be uniquely defined with the present methods of analysis,

except when D1(t) = 0--that Is, with normal towing techniques,

the model response, with respect to the wave ahead of the model, is nonlinear.

An experiment was devised to check whether this was the cause of the low coherencies reported in reference .

A. TEST SETUP

A destroyer model was towed in long-crested waves of moderate height (average height 1/60 L) Three

dif-ferent towing techniques were useth

Model permitted to drift relative to a constant-speed carriage, and to oscillate

in surge while under constant towing-thrust; wave wire fixed on carriage--that is,

D1(t)

0, V/ V, D

= 6.0 feet.

Model towed at constant speed, surge and drift restrained; wave wire fixed on carriage--that is, D1(t) = 0

V = V,

= 6.0

feet.

Model permitted to drift relative to a constant-speed carriage, and to oscillate

in surge while under constant towing-thrust;

wave wire fixed on a small sub-carriage and moved longitudinally with the model--that

is D1(t) = 0 V jt V, D

= 6.0

feet.

The tests were set up so that the effective mass

i'n longitudinal-motion of the model and sub-carriage system

was the same in technique 3 as in technique 1.

Carriage speed was a constant 2.53 ft/sec and analogue and digital records of wave, pitch, and heave were recorded for each towing technique. Because of the

repro-ducibility of the irregular-wave program the detailed wave-time histories obtained for the three towing techniques

showed good correspondence. Records of the model drift relative to the constant-speed carriage for techniques 1

and ₃ were similar in character and generally agreed within

10% in magnitude.

B. DATA PNALYSIS

Because of the limited length of the tank, several runs were necessary to cover the full irregular-wave program. Each run contained about 20 wave encounters; during five runs, about as many enáounters occur as are usually observed at a fixed point during a full irregular wave program. The recOrds were sampled at an interval of 0.2 sec (Thkay rule of thumb),.

and 30 lags were used for the analyses which consisted of spectra and cross-spectra for wave and pitch, and wave and

heave. Tukey degrees-of-freedom for the analysis of each

run were about

5.5; 27.5

for five runs.

The energy spectra of wave, pitch, and heave,

obtained when different towing techniques were used. for tests

conducted under otherwise similar conditions, were almost

identical. The energy spectra and cross-spectra obtained from the five runs obtained with each test technique were averaged; amplitude and phase responses and coherency were calculated from these averages using the methods cited in

reference 4 (Figures 2 through 4).

Figure 2 shows the amplitude response of heave and pitch to wave obtained by taking the square root of the ratio

of the respective spectra. The data obtained with each of the test techniques are shown, as is regular-wave data.

Figure 3 shows phase lags of maximum upheave or

bow-up pitch after the time that the component .wave crest is at the LCG of the model. Results derived from the irregular-wave tests for the three test techniques and approximate

regular-wave results are shown.

Figure 4 shows coherencies. between wave-pitch and

C. DISCUSSION

Figure 2 shows that no significant, differences in

amplitude response occurred with the different test techniques. Data obtained during the irregular-wave tests agree

reason-ably well with regular-wave data. No different conclusions could be drawn from comparisons of the amplitude responses computed from the cross-spectra for the three test techniques. This result is in agreement with that from a similar investi-gation on the effect of towing technique on pitch and heave

in regular waves.

Figure 3 indicates that techniques 2 and ₃ yield

nearly the same phase results. One technique restricts surge and drift of the model completely; the other permits both. The phase results of technique 1 do not agree with the others. The results for technique 1 were obtained by using the nominal

distance D

= 6.0

feet as the dIstance between the wave wire

and the model. Actually, the recorded mean distance through-out the runs probably averaged slightly more than

6.0

feet.

If D had been assumed somewhat greater when the results for technique 1 were computed, there would have been better

agree-ment with techniques 2 arid ₃

Figures 2 and k indicate that the low coherencies were not óaused by model drift or surge nor were they caused by the varying distances between the wave wire and the model;

though this might be important under sea conditions more severe than those simulated in these experiments.

Under moderate wave conditions (similar to those

cited in reference )+):

Derived amplitude responses are not

sensi-tive to minor variations of speed during

the run or to the test technique.

Derived phase angles are not sensitive to minor speed changes during the run or

restrictions in surge; however, they are quite sensitive to variations in the

dis-tance between the wave wire and the model.

3.

Coherencies between wave and pitch or wave and heave displayed the same tendencies and magnitudes as those reported in reference ₅ regardless of test technique, restriction in surge, or varying distance between wave

wire and model.

As a result of these experiments it was tentatively recommended that future experiments be conducted with the

wave wire at a fixed distance from the model at all times.

Because neither the experimental techniques nor the data-gathering techniques appeared to be the cause of the low coherencies, the physical process was more closely examined.

III. WAVE SYSTEMS

The methods of analysis used in reference k depend heavily on a knowledge of the behavior of the waves between

the point of measurement and the model. Therefore, the

hypothesis drawn stated that the waves causing the response may not have been predictable from the measured waves.

Direct experiments were made to determine whether the theoretical transfer function relating two wave measure-ments a fixed distane apart would be verified by

ôross-spectral analysis. Two experiments wer:e made: the first with the wave probes five. feet apart, the second with the wave

probes 145 feet apart. The resulting scalar spectra were

essentially the. same for all wave probes, apparently verifying

that the amplitude part of the transfer function.was valid and

Indicating that the wave process was statistically stationary..

However, the coheréncies computed from the experiment with the probes spacedfive feet apart werelow. The value of coherency

was about

_0.65

at the pa.k of the spectrum; the tendency was for coherencies to decrease with increasing frequency. This

result paralleled the coherency results between wave and model

reported In reference 14. The coherencles computed from the

experiment where the probes were spaced at 145 feet were so low

that the process could have been termed incoherent.

No immediate conclusions were drawn but it was thought that the mOdel could be temporarily eliminated from. the low-coherency problem and attention concentrated on the waves for two reasons;

l. The dual wave probe experiments showed the computed coherency to be strongly dependent

on the separation of probes.

2. Coherencies between model responses were. good (reference 14, pitch and heave).

IV. COMPUTATIONAL TECHNIQ,UES

The problem in the preceding section was brought to the attention of Professor W. J. Pierson of New York

University as a matter of interest arid he suggested a jOint

effort on the question of theadequacy of the cross-spectral computation technique prior to concluding that the waves generated in the tank did not approximate a linear system

This joint effort resulted in the major part of the solution to the low coherency problem (reference

5)

. The conclusion in reference 5 states: teThe loss of coherency

when computed by standard techniques from samples of vector Gaussian processes, where, theoretically,the coherency ought to be one, is largely explained by the lack of resolution in the cross spectra and the effect of the convolving filter made necessary by the nature of the:flnite sample. High

coherencies can be regained by modifying the experimental design so as to obtain less rapidly varying, cross spectra

and by increasing resolution."

Before proceeding 'with model tests, another check

on the validity of the methods in reference ₅ was undertaken.

In this case, instead of dealing with wave measurements at fixed points in the tank, a model experiment was simulated by replacing the model by a wave probe and moving both probes

at a constant speed of 2.92 ft/sec. Wave probe separation was 6.0 feet, and the translation velocity of both wires was

opposite to that of the Wave celerity. A moderate irregular wave program was used.

Three runs..at this speed 'were analyzed using the

data shifting method outlined an reference

5.

This method amounts tore-aligning the records so that the output which,

occurred T seconds later than the input is analyzed :aB if it

had been measured simultaneously with the input. The

theo-retical phase relationships involved are shown in Figure

_5.

The range of interest is that range where significant energy

exists in the wave spectrum. The line labeled w2D/g

repre-sents the theoretical phase lag of the wave component measured at the wave probe furthest from the source of the waves after

the component at the first wire. The theoretical phases,

introduced into the data by shifting, are also plotted. Where w = wave frequency and We= encounter frequency, w2D/g

_{- eT}

is the net theoretical phase after shifting.

Thus, if the wave process is linear and stationary, the modulus of the derived transfer function would be 1.0, and

the arguement would be w2D/g _{- WeTs}

Selected results of the analysis are shown in

Figures 6 through

_9.

Figures 6 and 7 illustrate the consider-able change in the form of the crossspectrum brought about by the shifting process and the resulting improvement of coherency for one of the three dual wave wire runs.

Figure 8 shows the scalar spectra resulting from the average of the three dual wave wire runs. The results

indicate that the spectrum encountered by the model will be virtually the same as that measured ahead of the model.

Figure 9 shows coherency between wave records

obtained at two moving points and the results of the derivation

of the, transfer function. The.coherency is 90$ or better

over the range where, the spectral energy is greater than 20$ of maximum. The deviations of the derived phase angles

from the theoretical are shown; an approximate 95% confidence

bound on the phase is indicated. (reference 8). In view of the width 'of the confidence bound, phase agreement with

theoretical is shown to be excellent where high coherencies

and significant energies prevail. Figure 9 also shows the derived amplitude response from scalar spectra and cross

spectra as well as 95 confidence bounds on the response derived from the cross spectra. Again, agreement with

theoretical assumptions is excellent. The average deviation from theory is about k, a very respectable accuracy for

this type of analysis.

It was concluded that by using the methods reported

in reference

_5,

the difficulties involved in deriving, the

coheröncy and transfer function of the Ideal wave to model system from the experimental system would be largely overcome.

It must be noted that neither the work cited in reference 5 nor that cited In this report has outlined an optimum procedure for crossspectral analysis. It is certain

that rapid oscillation of the cross spectrum is to be avoided, whatever the experiment, If the lengthof sample Is. limited

(reference ₅₎

An approach to optimizing the analysis would be to operate on the data so that the phase response of output to

input is a constant multiple of 7r/k. This would have the

effect that co and quadrature spectra.would have the same shape as the scalar spectra and that resolution optimum for scalarspectral analysis would be optimum for crossspectral

analysis. Because the exact phase between input and output is almost always unknown, this is Impossible in principle

and could only be. approximated by an iterative process.

V. SU1YINAPZ

The investigations into the effect of experimental techniques on wave-model coherencies disclosed no large

direct effects. However, the experimental setup previously used indirectly affected the coherency because rapidly

oscil-lating cross spectra were produced. This condition resulted in insufficient resolution of cross spectra when standard

computational techniques were used. Insufficient resolution in the analysis of the data, in turn, accounted for the low

coherencles of reference k. A practical method of overcoming

VI. ACKNOWLEDGEMENTS

The author wishes to acknowledge the contributions of Professor W. J. Pierson, Jr. of New York University who indicated the way to successful practical solutions of the

coherency problem. The author also wishes to acknowledge the assistance of those members of Davidson Laboratory who assisted

in these investigations; particularly, Mr. Wilbur Marks arid Mr. P. G. Spens.

VII. REFERENCES

Lewis, E. V. and Numata, E.: "Ship Model Tests in Regular

and Irregular Seas," September

_1956,

E.T.T. Report

_567.

Lewis, E. V. and Daizell, J.: "Motions, Bending Moment

and Shear Measurements on a Destroyer Model in Waves,"

E.T.T. Report

_656,

April

_1958.

Daizell, J: "Ship Model Tests in Irregular Waves with a Broad Spectrum," April 1958, E.T.T. Note

_k71.

11.. Daizell, J. F., and Yamanouchi, Y.: "Analysis of Model

Tests Results In Irregular Head Seas to Determine Motion Amplitudes and Phase Relationships to Waves," Ship Behavior at Sea, Second Summer Seminar, June

_1958.

PIerson, W. J., Jr. and Daizell, J. F.: "The Apparent Loss of Coherency in Vector Gaussian Processes due to

Computa-tional Procedures with Application to Ship Motions and Random Seas," Joint Report, Davidson Laboratory,NewYork University, Department of Meteorology and Oceanography,

September

_1960.

Spens, P.: "A Digital Recording System for Model Tests in Irregular Waves," DL Note

_550,

Presented at the 12th Meeting of ATTC, Berkeley, CalifornIa, August

_1959.

Coleman, T. L., Press, H. and Meadows, M. T.: "An

Evalua-tion of Effects of Flexibility on Wing Strains in Rough Air for a large Swept Wing Airplane by means of Experiment-ally Determined Frequency Response inctions with an

Assessment of Random Process Techniques Involved," NACA

TN k291, July 1958.

Goodman, N. B.: "On the Joint Estimation of the Spectra, Cospectruin, and Quadrature Spectrum of a Two-Dimensional

Stationary Process," March 1957, Scientific Paper No. 10, Engineering Statistics Laboratory, New York University.

MODEL

I

SUBCARRIAGE.

CONSTANT-SPEED CARRIAGE

FIGURE

I

TEST SETUP

II-vc

RAIL

TOWING

WEIGHT

I-I

Ui

=

ZO

LL cr.% LLJU)

.Q-w

WW

ctx

I-Ui

c

-j--0

I

C-)

I-0 5

40

30

20

(POINTS

FROM REGULAR

WAVE

TESTS SHOWN As: 0)

FIGURE 2

AMPLITUDE RESPONSES DERIVED FROM POWER

SPECTRUM ANALYSES (SURGE INVESTIGATION).

PITCH

0/

TECHNIQUE TECHNIQUE L.q.

\.__TECHNIQUE

'\ \ I 2

o

I

v, 3

r(

_./,.

\

4 I

-e'

RADIANS/SECOND I I'O I 12

.

14

-

D-.

--.' t

HEAVE

TECHNIQUE TECHNIQUE TECHNIQUE I. 2

RADIANS/SECOND

T

TECHNIQUE 2 TECHNIQUE 3

-340

-320

3.0

-280

-260

2

-220

4

--- 360

-300

-280

4

PITCH

REGULAR WAVE DATA

RADIANS/SECOND

I I I'O I 12

- 340

REGULAR WAVE

DATA--320

HEAVE

TECHNIQUE TECHNIQUE

_A

TECHNIQ

We, RADIANS/SECOND

I _:1 10 1 l'2

FIGURE 3

TECHNIQUE I

PHASE ANGLES

DERIVED

FROM

CROSS

.2

FIGURE 4

COHERENCES (SURGE INVESTIGATION)

.

WAVEPITCH

I A

/

/

- ...

_I

.

\

I ..'. : I . 1.0

9'.

.8 IStFIb

,.

4&

.7-

C.)

..:

___

_/

A:

_.. ... 0 .5 . TECHNIQUE TECHNIQUE TECHNIQUE I 2 3

3-L:J

.2-

.1-4

W,

6.1

8 I

RADIANS/SECOND

10

112

14

w2V

D WIRE SPACING

6.0 FT.

VSPEED

2.92 FT./SEC.

TRECORD SHIFT

1.1 SEC.

9

(ABSOLUTE VALUE)

ESTI MATED

RANGE OF INTEREST

,ppr

_{We, RAD.ISEC.}

8

12 16

20

I 1

FIGURE 5

(I)

w

I-U)

z

w -J

4

I-C) w 0 U)

0

C-) U) w

I-U)

z

Lii

4

a

-+15

-+10

CO SPECTRA ORIGINAL

DATA\\

2

4

I I I I I I I I I I I I I I 1 I I I I

\i

DATA PHASE SHIFTE I I I 10 12 I I

1'

I

I I I

/

' I ' '

AHEAD SEA" SPEED 2.92

-Ft/SEC. WIRE AHEAD

CONSIDERED AS "INPUT -WIRE FOLLOWING AS

"OUTPUT' WIRE SPACING

6FT., LtO.IO SEC., 40

-LAGS.

I

/

We , RADIANS/SECOND 16 18

20 22 24 26

I I I I I QUAD SPECTRA

DATA PHASE SHIFTED

,'-ORIGINAL DATA

I I I i

I!'

/.

\ /

We, RADIANS/SECOND 6

8

tO

(2 '/14

. 16 18

20. 22

24 26

H i

i.

.1 i i

.. I,,I

FIGURE6

..I

SHIFTED DATA

WAVE ENERGY>20% OF MAXIMUM

I

k

WAVE ENERGY>I0% OF MAXIMUM

-I-.

0e RADIANS/SECOND

UNSHIFTED \ I DATA I F'j

FIGURE 7

COHERENC.IES FROM

_{ANALYSES OF SHIFTED}

AND

UNSHIFTED

DATA.

RUN 0106

HEAD SEA SPEED 2.92 FT./SEC., WAVE WIRE SPACING

6 FEET. WAVE WIRE AHEAD CONSIDERED

THE INPUT,.

THE WIRE FOLLOWING, THE OUTPUT. M = 0.10 SEC.

4o LAGS.

-1.0

-.9

-.8

-.7

C.)

z

LU

-.6

LU

=

0

C)

-.5

-.4

-.2

8 12 16

20

24

I I I I

30

-25

-20

-'5

6

I

I A

It

FOLLOWING WIRE LEADING WIRE

AVERAGE OF RUNS 0106-0108

WAVE WIRES 6 FT. APART

"HEAD SEA" SPEED:

2.92 FT/SEC.

t= 0.1 SEC., 40 LAGS

DATA SHIFTED 1.1 SEC.

TUKEY "DEGREES OF

FREEDOM." ABOUT 24

We, RADIANS/SECOND

8 10 12 14 16 18

20 22

I I I I I I

24 26

FIGURE 8

WAVE SPECTRA: DUAL WAVE

WIRE

EXPERIMENT

COHERENCY

0.9---

0.8-0.7

+40 DEVIATIONS OF COMPUTED ARGUMENT FROM

THEORY-+30

95% CONFIDENCE BOUNDS

+20

-+10

1.0

0.9--RANGE OF SIGNIFICANT

WAVE ENERGY We, RADIANS/SECOND I 12 I'6

20

1 S.'.--,

A

MODULUS

/

/

FROM' RATIO OF SPECTRA-CROSS SPECTRA THEORY

'I'

\__j

\..,. 95% CONFIDENCE BOUNDS (GOODMAN)

.0.8--I I I I I

4

8 12 16

20

FIGURE 9

COHERENCY AND DERIVED TRANSFER FUNCTIONS

DUAL WAVE WIRE EXPERIMENT:

AVERAGE OF 3 RUNS, "HEAD SEA" SPEED 2.92 FT/SEC., M:

0.1 SEC., 40 LAGS, DATA SHIFT, 1.1 SEC., RUNS 0106-08, WAVE

WIRES 6 FT. APART.

We, RADIANS/SECOND

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