On the directional sea measurement using a circular array

On the Directional Sea Measurement

Using a Circular Array

By

Shigesuke ISHIDA*

SUMMARY

For estimating the directional spectrum of short-crested seas Circular Array Method

(CAM) is proposed. CAM is an array-dependent method, distributing wave gauges ön

a circle and one on the center of the circle. Through this array arrangement the Fourier

coefficients of the directional distribution function can be evaluated by a simple calculation,

inverse Fourier transformation of cross spectrum. But for higher orders MEM extrapólation is used.

Fourier coefficients are calculated directly and distinctly with simple equation, so

it is easier than other methods (e.g. MLM) to estimate the reliability of the result and

to find the reason when we get an unexpected directional distribution. For a single-peak distribution the resolution of CAM is satisfactory with a skillful filter, but for complicated distribution there is a room for improvement of MEM extrapolation process.

..Standing wave effect caused by a reflection wall can be a serious problem for exper-iménts in model basins and for field measurements. Application of CAM to those wave fields was studied qualitatively and quantitatively. The behavior of the interaction term

was discussed relating to the distance of the array from the wall, the array size, the incident angle to the wall and so on. These discussions were confirmed by simulations.

i

INTRODUCTION

Estimating the directional distributioñ of short-crested seas has beén increasingly important for naval architects and civil engineers. Recently some methods based on sta-tistical models have been developed and getting popular like MLM', MEM2), MBM3 and so on. These methods are reported to have a better resolution than conventional methods with small number of sensors, but have some instability inherent in the modeling. In this report an array-dependent method, using wave gauges distributed on a circle and on the origin, is proposed. It is basically based on the cOnventional Fourier series expansion, but supplemented by MEM extrapolation if necessary. Application to the wave fields thoseare

contaminated by reflected waves has also been discussed.

'Ship Dynamics Division

PROPERTIES OF CROSS SPECTRUM

In general 3-D wave spectrum S(f, 9) is expressed as a product of power spectrum S0(f) and directional distribution function D(f, O).,

S(f,9)

=

S0(f)D(f,9),

(1)

where D(f, 9) is normalized like

£ D(f, 9) dO =1.

Cross spectrum f surface elevatiOn measured at two points is

(f, r) JT S(f, O) ejkrdû

=

S0(f)

£ D(f, O) e2rdO

(2)

where k: wave number vector

r: position vector between two measuring points.

Later frequeiicy f will be dropped for simplidty. Integral signs will mean integrals from ir to ir. It is a common way to express D(0) with acomplex Fourier series as follows because it must be a periodic function of 2ir,

D(8)

=

Pm ,m9 (3)

Pm =pmei#m _{= ß'-m} ßO= 1

where p and q5 are the absolute value and the phase of respectively. Then cross

spectrum which is nondimensionalized by S0(f) becomes

=

_JD(0)e"d9

=

moo

Jm(kR)'13m ejm) (4)

This equation means that contribution of the component ¡6m to the cross spectrum is proportional to Bessel functioñ Jm(kR) . When the argument is small the absolute value of Bessel function rapidly tends to zero with order in . So if kR value is small, i.e. two

wave gauges are too close compared to the wave length, it is not easy to evaluate higher order coefficients correctly because in that case cross spectrum includes little information on higher order coefficients, in other words cross spectrum is not sensitive to the detailed

shape of D(9).

3

CIRCULAR ARRAY METHOD

The idea of Circülar Array Method (CAM) is to evaluate ¡3m by inverse Fourier

transformation of cross spectrum expressed by eq.(4) with keeping R constant like -

-

_{f'(crR\e_jm(0+f)da}

Pm

- Jm(kR)2irJ

'

' /

When we only have N cross spectra with the same spacial distance R and equiangular distribution with the initial angle a0 like

2ir

= a0 + evaluated coefficient ßm becomes

1m(l1) N

'(cr; R) e_imf)

=

J,(kR) ¡3, e«°f }

Jm(kR) f(l - m) ßj where

1(r) =

Substituting eq.(l) to eq.(6) we get

- Jm+ZN(kR) EN fiNao -Pm -

1-

7 (L o\

j

e Pm+1N mI.'.'-J e»'° n=1

Ç e''°° p = O, ±N, ±2N,

O other values. (7) (8) (597)

so, evaluated coefficient /3m by eq.(6) includes contributions from other Fourier components

with the distance of order ¿N.

We know

((r) = cI"(o!lr;R)

=

=

m=oe

Jm(kR) ß e_jmf)

=

1)tmJm(kR) 13m ejmf). (9)

Using

'(air; R) instead of

'(a; R) in eq.(6) and taking an average with eq.(8) we get

00 i I i'tIN

-

'm+1NV) 1N fiNa,,

-Pm

¿

_J

_kR' e

2 Pm+ZN.

¡.

m J

If N is an odd nurkiber we can eliminate contributions from 15k 's, (k = in ± N, rn± 3N, ...)

like 00 Jm+21N(kR) iN 2iNà,, -Pm

= L

'

'kR'

( 1) e Pm+21N

ioo

'm J

ARR4Y DESIGN

We have designed the circular array like Fig.1. N wave gauges (N : odd number) are distributed on the circle and one on the origin. Let us call a group of N gauge pairs those have the same distance R "nest". There are (N + 1)/2 nests available.

Nest-O consists of gauge pairs O-1, O-2, O-3,.. For this nest initial angle a is zero

and distance R is equal to the radius Of the circle A. Nest-i includes the pairs 1-2, 2-3,

3-4, ... , nest-2 consists of 1-3, 2-4, 3-5, ... , and so on. For the nests other than O initial angles and distances can be expressed with nest number

a0 =

R = 2i48fl(fle).

(12)

Substituting a0 to eq.(11) ¡5m can be expressed using Kronecker's delta function ö(n)

jm+2lN(1R)

( 1)1'ßm+2iN. Jm(kR)

removed.

In principle we can calculate coefficients of any order as long as the number. of nests is large enough to remove significant effect from other coefficients in eq.(12). But there is

a problem in order in = 2N. Using nests other than O we. can have cross spectra of 2N

equiangular directions. Directions of gauge pairs of nest-O are located at the middles of those of other nests. So we can measure cross spectra of 4N equiangular directions. All N(Ñ + 1)/2 cross spectra measuied.with this array only have a part of information of wave field which is produced by order m = ±2N. it is similar to the Nyquist frequency problem in the auto spectrum analysis. So in practice we have to stop calculation at m = 2N - 1. In conventional methods with Fourier series expansion4 maximum order is equal to the number of cross spectra using N(N + 1)/2 simultaneous equations or is less than that

using least-square technique. But evaluated coefficients are not independent with each other, that can lead to an unstable result if some error is included in the cross spectra.

Maximum order of CAM is smaller than that but results should be more stable because each coefficients are calculated independently and because we can select nests those include much contribution from the order we are interested in by considenng the magnitudes of

Bessel functions.

As mentioned before relating to eq.(4), little information of higher order coefficients will be included in the cross spectra if the diameter of the circle is small compared to the wave length we are interested in, that leads to low signal/noise ratio. So the array which

has a larger diameter can give more stable and better estimates. The simulation results

with statistical noise say the evaluated higher order Fourier coefficients become more stable when the wave length gets shorter. For example if the diameter is close to the wave length we can get a good estimate of the coefficients till about order 10.

Increasiñg the number of wave gauges not only enables us to go to higher orders

but alsO leads to more stable results because we can selet more proper nest(s) which has large IJ,(kR)I value. But considering vulnerability of higher order coefficients to noise and costs, practical number of gauges will be 6, 8 or to.

5

MEM EXTRAPOLATION

If evaluated maximum order of Fouriér coefficients is rather small because of noise or for other reasons D(9) will have a serious truncation effect. One sOlution to avoid that is to extrapolate higher order coefflcjents using the similar procedure to extrapolating auto correlation function by MEM in the auto spectrum analysis5).

If we can assume that Fourier coefficients have the the similar property to auto correlation of wave elevation like

where E{} : expected value

f

: random Gaussian.variable

E{f,}=O

then we can caicülate from ßo, , , ß by

1 ß

...M+1

po

PI

PiM ...

í3m=E{fif_m} (13)

Coefficients higher thän M + 1 can be calculated by repeating this process. Evaluated

distribution D(9) is nonnegative because

D(9)

21+1

.i

uu1m)

-L.ljm

_'

(i eJme)

=

_i-

0 (15)

where we assume I> M, ßM+1 = ßM+2 =

=

=

0.

Eq.(13) requires that the determinants of any order of Toeplitz matrixes must be nonnegative,

=0.

(14)

(601) The program not only extrapoÏates higher order coefficients but modifies lóweÈ order ones to make eq.(16) valid. That insures no serious negative lobe of D(9).

This MEM procedure is trying to maximize the information entropy of virtual randôm variable f,. There seems to be other ways, for example maximizing the entropy of the variable which has thé probability density function of D(9)2).

But that has not been

examined yet.

6

FILTERING

In general calculated Fourier coefficients have some errpr. Moreover MEM

extrap-olation tends to stop in lower order than expected because of numerical problem and

sometimes extrapolated coefficients do not converge smoothly. So we need ifitering to get stable estimation of D(9). Instead of filtering in 9.domain we multiply a Gaussian shape window W(m) to the absolute values of evaluated Fourier coefficients themselves,

ImI = W(m) l/51!zI (17)

whe

W(m)=e.

(18)

The value of the standard deviation o is decided to make D(G) smooth and not to make peaks too low. The. window is selected to have a Gaussian shape because if D(9) can be expressed with the power of cosiüe function, we call it "únimodal distribution" hereafter, the absolute values of Fourier coefficients can be approximated with eq.(18).

In the case of ummodal distribution, ifitered coefficients can be interpreted as a

extended ones to m direction along the shape of eq.(18)6),

D(9) = gJfllSID

(19)

m-M

where

im =

W(m)ßme(»m.

(20)

From eqs.(18) and (20) the ratio of extension K

(21)

K should. be the same value for any order of. m. D(9) is assumed to be zero for the range out of the new expansion. This unimodal filter works to make the distribution very smooth and to make it sharp when almost long-creste4 sea comes. But it can be a too strong filter

when we, expect à big main peak and srnall sub peak(s). We shoUld decide if this filter should be used òr not by what we eìpect.

7

APPLICATION TO THE WAVE FIELD NEAR.

A REFLECTION WALL

Near a reflection wall wave field is not stationary in space because of the interaction between the incient and réflected waves. That can be a serious problem for experiments in mädel basins and for field measurements near breakwaters and so on With a wall-sided reflector of öeffiient r, cross spectrum of surface elevation measured at Xj and x2 is,

-

=

JD(){e<(21) + r2ejk<c2r_x

)

+

reik2r_x1) }de (22)

where XT means the mirror image of x, fór the reflector. The first and second terms corne

from the incident and reflected waves respectively The third and forth ternis mean the interaction effect.

Expressing spacial lags with the angles and the distances of the gauge pairs like

(cosa

x2xi = R.

I

cosa

X2. - ir = R

fli(co3a

X2Xlr

_-

_sina

-

I cosa'

R

_sina'

(see Fig2), then eq.(22) leads to

- X1)=

{ Jm(kR) (ßm +

raß*) gi"*+)

m=oo

-I- r Jrn(IR') (i3m + i) eim(0'+F)} (23)

Note that this-equation is. not a Fourier expansion because R' anda', the distance and the angle between the real and imaginaty gauges, are the functiojas of a as follows,

a =

f2yo (i - 2a cos'y cosa + a2cosa2)

i2yo(1+asina+a2/4)

2

Iacosycosa)

{ ir

(

asinycosa

I a cosa

--

2+asina)

where a is the nondimensionalized radius of the circle A/yo, and y is related to nest number

ir 'Y= Tie

(see Fig..2). if a is small, i.e. the array is located far from the reflector compared to the radius of the array, R' and a' will be reduced to

R'

2yo{1+aC(a;n)}

+ ai(a;n).

where

Substituting eq.(23) to eq.(5) and taking an average with ' just like eq.(1O) without paying any attention to the interaction terms we get

(28)

=

J Jj(JcR') cos(la' ) e_imada.

So even if we have so many gauges on the circle the evaluated coefficients by CAM will be contaminated by the real parts of m+2Z 's of the inci4ent wave.

It is not so easy to estimate the behavior of ßm from eq.(28). So let us assume

that kR' > i and a '

1, i.e.e. the array is located far from the wall compared to thethe array is located far from the wall compared to the

- -

2-i. (603) (603) where

I cos'ycosa

(n=1,2,...)

((a;n) =

siria (n.= O)

(.2

I

sin-y cosa

cosa

(n=1,2,...)

(rie = O). 2

(n=1,2,..)

₍₂₄₎ (ne = O)

(n=1,2,-..)

(25) (ne = O)

wave length and the radius. Using the asymptotic expression of the Bess1 functionand

dropping unimportant terms,

'm+21.m

₌

J Jm+2i(1R')08{(in + 21)a'} e1da

ft-Jcos(2kA+ß_?rn)

coa{(rn+ 21)a17 + (29)

where

ß=2kyo.

r exchanging the, summation and the integration,

-

2s+

r

nui

Pm+" Pm

Jm(kR)V7y0

._L td f

Te(a) cos(2kA( + ß) e"'

(in : even) 2ir J

a

T,(a) sin(2kA( + ß) e_'"

(m : odd)

where

Pm

Tfri) i

1- -.

'

1ii' cos{(rn+2l)ar}

Ta(a) f -

Pm+2i + Pm+21) ( ) sin{(m + 21)arj}.

(30)

From eqs.(28), (29) and (30) we can know söme properties of ßm as follows.

Approximately the interaction effect is proportional to l//.

Even number cóefficients are easier to be contaminated than odd number 'ones,

be-cause terms of order ±(m + 21) in T(a) tend to cancel each other by the work of

(-1)' for odd orders.

Usually the. nest which has th biggest number includes the least interaction effect

because (, which is proportional to cos y, should be very small (see eq.(26)). We

should use the information of this nest as long as IjmVR)I is not so small.

Too large array leads to bad estimates because of large kA value in eq.(30). On the other hand a small array makes the absolute value of Jm(kR) in the denominator too small. The radius of the same order as the wave length should be recommended. When the incident wave comes almost parallel to the wall the interaction effect will

be small because real parts of all ßm+ii 's decay smoothly and those are expected to

On the other hand when the incident wave comes with the right angle to the wall we

will have senous interaction effect because /3m 's aie proportional to 3m7 winch means all Pm+21(1)' 's have the same signs. But if m is an odd number we can have an exact estimate because /3m+21 's have no real parts.

As long as D(8) has broad and smooth shape, i.e. higher order coefficients are negligible, the interaction term is not so sensitive to the parameter a.

8

SOME SIMULATION RESULTS

Simulations are ¿arried out for cos25G distributions and superpositions of thera. All

the résults in this report are of cross spectra without noise. The radius of the circUlar

array is 1 meter and 8 wave gauges are used (N=7) fòr all cases. So the maximum order of Founer coefficients is 13 MEM extrapolation is only used when calculated higher order coefficients are not reliable or when negative lobe(s) cannot be removed by filtering.

Fig.3 shows the results of unimodal distributions using the unimodal filter. In the broadest case the peak is a little lower than the theory because of the filter, but the results are almost satisfactory. When narrów and broad distribUtions are superimposed like Fig.4 the peak value gets lower because unimodal filter.is not used.

The resolution is studied in Fig.5. Two peaks of 25 degrees ápart (Fig.5(d)) cannot be distinguished because the resolution of the maximum order, 13, is 27 7 degrees

Fig.6 is the demonstrations of the. MEM extrapolation. In both examples the

maid-mum order of Fourier coefficients before MEM process is 6. Two peaks are clearly separated

after MEM because the maximum orders became 19 (Fig.6(a)) and 9 (Fig.6(b)). But in (a) the. distribution shapes are a little distorted, and iñ (b) extrapolation stops at order 9 by a numerical problem. Moreover the resultant directional distributions sometimes have

serious fluctuation because extrapolated higher order coefficients do not converge smoothly.

There seems to be a room for improvement in MEM extrapolation process.

Calculations on wave fields with a wall reflector of the coefficients r = i and r = 0 3 are shown in Figs 7 and 8 respectively When yo/A 10 the estimated directional

distribution can be almost expressed as the superposition of the incident and reflected waves. But when the array gets closer to the wall we can see not only spurious lobes

but also distorted main peaks caused by the standing wave effect. According to another simulation result the reflection wall of the ratio of 0.1 causes no serious fluctuation. So the ability of usual wave absorbers seems to be high enough to get reasonable experimental result, but attention should be paid on the other no-absorbing walls.

The éffect of the mean directioú of incident wave is also studied in Fig.9. When it is 90 degrees the interaction effect is serious as discussed in the previous section. It is

interesting to see that the distribution is the periodic function of ir. The. réason is all odd

number of coefficiénts axe zero, which is mentioned in item 6. In other directions no-serious

interaction effect càn be Seen.

9

CONCLUDING REMARKS

Circulai Array Method (CAM) is an unique array-dependent method introduced by C.T.Stansberg6 and improved by the author. The Fourier coefficients of directional

distribution function calculated by CAM should be more stable than conventional methods as mentioned in séction 4.

CAM is basically a linear and simple calculation, so it is easier than other (statistical)

methods to estimate the reliability of the result and to find the reason when we get an

unexpected distribution. For example if ßm 's in eq.( 12) of all nests do not agreewell when Jm+2lN(ICR) 's axe negligible small, that means cross spectra themselves include significant

noise, but if those Fourier coefficients agree well that means the calculatedcoeffiçients ß,,.

's are reliable even if it the directional distribution is not the expected one.

For unimodal wave the resolution is satisfactory using a skillful filter without any extrapolation. But for complicated distributions we should depend on statistical process like MEM, whith has some room for future improvement.

When a reflection wall exists the result will be contaminated by the interacLión

be-tween the incident and reflected waves like other methods. The behavior of it was discussed

theoretically. But in practice it is common to take an average with neighboring frequencies, which rmght cancel some interaction effects if the degree of freedom is large enough This problem should also be studied further..

lo

.

ACKNOWLEDGEMENTS

I have a regárd for the research work of C.T.Stansberg (MARINTEK) on the direc-tional sea problem, and am grateful to him for his advice and encouragement when I was staying in Norwegian Institute of Technology, Trondhéim, Norway from 1988 to1989.

REFERENCES

O.H.Oakley and J.B.Lozow : Directional Spectrum Measurement by Small Arrays, 0TC2745, (1977)

N.Hashimoto and K.Kobune: Estimation of Directional Spectra from the Maximum Entropy Principle, Report of the Port and Harbour Research Institute, Vol.24, No.3,

(1985)

N.Hashimoto and K.Kobune : Estimation of Directional Spectra from a Bayesian

Approach in Incident and Reflected Wave Fidd, Report of the Port and Harbour

Research Institute, Vol.26, No.4, (1987)

L.E.Borgman: Directional Spectra Models for Design lise, 0TC1069, (1969) T.J.Ulrych and T.N.Bishop: Maximum Entropy Spectral Analysis and

Autoregres-sive Decomposition, Review of Geophysics and Space Physics, Vol.13, No.1, (1975)

C.T.Stansberg: Numerical Study on Method for Directional Spectrum Simulation and Estimation, Symposium on Description and Modeling of Directional Seas, (1984) C.T.Stansberg : Statistical Properties of Directional Sea Measurements, Journal of Offshore Mechanics and Arctic Engineering, (1987)

C.T.Stansberg and S.Ishida: Directional Spectrum Estimation by Means of a Circu-lar Wave Gauge Array, Proc. of 23rd Congress of lAHR, (1989)

Abstract

For estimating the directional spectrum of short-crested seas Circulár Array ,Method (CAM). is proposed. CAM is an array-dependent method, distributing an odd number of wave gauges on a circle and one on the centeì of the circle. Through this array arrangement the Fourier coefficients of the directional distribution fünction can be evaluated by a simple calculation, inverse. Fourier transformation of cross spectrum. But fòr higher orders MEM extrapolation is used.

Fourier coefficients are calculated directly and distinctly with simple equation, so it

is easier than other methods (e.g. MLM) to estimate the reliability óf the result and to

find the reasón when we get an. unexpected directional distribution. For a single-peak distnbution the resolution of CAM is satisfactory with a skillful ifiter, but for complicated distribution therè is a room for improvement of MEM extrapolation process.

Standing wave effect caused by a reflection wail can be a serious problem for exper-iments in modul basins and for field measurements. Application of CA.M to those wave fields was studied qualitatively and qûantitatively. The. behavior of the interaçtion term was discussed relating to the distance of the array from the wall, the array size, the mcident angle to the wall and so on, These discussions were. confirmed by simulations.

Fig. 1 Geometry of Circular Array

Fig. 2 Mirror Image, of Circular Array with a Reflection Mall

0.05 (I/dog) 0.05 0.0 D(0) n.n3 lI.nI 0.05 -IO) -120 60 9 (b)

S=20

120 lO) (dog) -60 ' - ESTIMATED INPUT 00

--ISO -ro o 68 120 lOO

9 (deg) (a)

S=2

(1/dog) 0.20 0.20 D(8) 0.10 -'20 -60 60 (20 IO) (dog) (c)

S500

Fig. 3 Simulation of Unimodal Distribution,

o o D(6 o. o o o

(1/deg)

0.02s 0.020

D(9)

0.11W n .no 11.01111 -60 120 180

(deg)

Fig. 4 Simulation of Bimodal Distribution,

D (e).cc cos4 (e) +3 cos4° (e)

, f = 0.96 Hz

(611)

ES1.I MATEO INPUT

/

"Y.,!

(1/deg) 0. D(O o o o - loo -120 O O 60 (20 I (deg) 0.03 0.020 D(9) n-OIS 11.11'S liii -(20 -60 GO 120 lOi (deg) (20 (80 (dog) 03 020 -- ESIIMATEO INPUT ('i OIS

-LL1..

010

J

lt (a) 00 (b) (1/deg) (i/dcg) 0.03 0.020 -O(0) (1.015 0.025 0.020 O(0) 0.0(5 0.010 0.005 0.0110 1.0 IO 11.11'S 0.101 -(20 lOi (deg) i 50 lOi e -(80 -l'O -GO O (c) Oo (d)

(1/deg) (1/deg) - 0.025 0. 60 120 180 (deg) 0.020 0.0 IO 0.005 o oca -180 (613) 020

---- £STIIIAT(O ClIN 8011 -IIUT ,'! 000 (a) Oo (b) eo =

Fig. 6 Behavior of MEM Extrapolation,

D (0) cos4° (ee)+ cos4° (e+e), f = 0.33 Hz

-120 -60 0 60 120 180 9 (deg) - 180 -120 -60 O 9 0. 0. 0. 0.

(1/deg) (1/deg) o 0, o O o 0 60 120 180 (deg) (1/deg) 0. (1/deg) 180 - -13 (1111

f't

JLJ\

uyJ ESTIMATED flPUT

-028 r.. 020

J

h

OIS 010

:ti

loo. -320 -Iç - . 050 028 '

A

(a) Yo

/Aco

(b) y0

/.A=10

(c) Yo

/A5

(d) Yo

/A2.5

-120 -80 0 60 120 e . (deg) -120 -60 0 60 120 180 e (deg) 18) -120 -60 e 0 60 120 (deg) -ro -60 e o. o. D( fi. o. fi II 0. 0. D( 0. 0. 0. O. o. D(6 0.1 o.

(1/deg) 0.05 0.01 D(9) 0.03 0.02 0.0' 0.03 (1/deg) 0.05 0.01 0.01 -loo ESTIMATED IUT p

fJl

-60 0 9 Yo

/A=co

60 120 103 (deg) O O o Yo

/A5

60 120 101) (deg) (1/dog) 0.05 (1.02 11.101 - Ib1) (1/deg) 0.05 0.01 Yo

/A10

CO l'O lIJO (deg)

Figé 8 Directional Distribution with a Reflection Wall,

D ()

cos4°

(O)

, r0.3 , f = 0.96 Hz (615) -ro (b) JA _... -bD O o -120 (d) -60 O GO l'O 100 O (deg) Yo

/A=2.5

(1/deg) 0.01 0.06 0.05 D(0) 0.OE -0.02 0.01 O .t1) (1/deg) O.030 0.025 0.020 D(0) 0.015 0.0 IO 0.805 -lOO I20 -60 (a)

=0

-120 -60 60 (c)

e0 =

- ESTIMATED INPUT I) fo) (deg) 120 ISO (deg) (1/dég) 0.030 --0.025 0.1120 D(e) filiO) - tod (1/deg) 0. 0. o o n n (b) (d) Ll l'O IOU (deg) o 60 l'O ISO e (deg) 030 026 . 020 0I5 -4t

) LA

n

i \/\

:

60 o -60 -l'o e

r

L.

Wall Function Method with Conservation tiow Property

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it26.fl

f9 8

I:'+7 (CPU,

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Model Tests oÉ Stepless and Stepped Planing Boats with Deep Vee Hull

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