• Nie Znaleziono Wyników

Ocean wave spectra

N/A
N/A
Protected

Academic year: 2021

Share "Ocean wave spectra"

Copied!
41
0
0

Pełen tekst

(1)

15 SEP.A97

/'/

ZS

ARCHIEF

I

bIioheek van d

Ondera{dehns .P'-- - - SSI n;sche Hoqeschoo, DOCUMENTATE

DATUM:

OCEAN WAVE SPECTRA

by

Ming-Shun Chang

ONt2

I: ,4'ij2/2f

9

Nava' Ship Research and Development Center Bethesda) Mary'and

To be Presented at the Ninth Symposium on Naval Hydrodynamics

Lab.

v. Scheepsbouwkunde

Technische Hogeschool

(2)

ABSTRACT

Thi.s paper presents the results of two studies: one dealing. with the analytical representation of unidirectional wave spectra, and the other dealing with experimental determination of directional wave spectra.

A two-parameter wave-spectrum formulation for determining the seakeeping qualities of ships was evaluated by application to hindcasted data for the North Atlantic Ocean. Computations indicate that the two-parameter representation does not properly distribute wave energy over the full range of wave frequencies.

An experiment was conducted in a large seakeeping basin to assess techniques for determining directional wave spectra from wave elevation measurements obtained with sonic probe arrays. The

measurements were found to be sufficiently accurate for analysis of the wave directions, when the directional spectra are approximated by a ninth-order Fourier series.

(3)

INTRODUCTION

In order to describe the properties of ocean waves, one

con-siders the seaway as a random process having a spectral. representation. The sepctrum of ocean waves is two-dimensional, and thus it is a

function of both wave frequency and wave direction. It is difficult to obtain a directional wave spectrum; and in many applications the spectrum is considered to be a function of wave frequency only with its direction arbitrarily specified. .A commonly used

representa-tion of ocean waves is the Pierson-Moskowitz spectrum1. This

spectrum is a special case of a fOrm suggested ear+lier by

Bretschneider2; The international Towing Tank Conference (IT.TC)

in l969 recommended a two-parameter ideal ized spectrum of the

Bretschneider form whenever statistical information on the

characteristic wave period and height was available and recommended the Pierson-Moskowitz spectrum whenever such information was not available. Because of the.lack of confidence. in wave period and height data, the Pierson-Moskowitz spectrum has been widely employed as the basis for evaluating ship performance in a seaway and ih

the design of marine structures.

For the study of long-term ship performance, Cummins suggested

the use. of the two-parameter ideal ized spectrum of the Bretschneider form. This spectrum recommended by the ITTC has been.recently applied

to North Atlantic hindcast wave data5, and the results obtained were considered to be less than completely satisfactory. These

(4)

2

results s';owed :iat in the idealized spectra the wave energy was

not properly distributed with respect to frequency. This could result in serious errors in the prediction of ship motions. The same conclusion is substantiated by more recent measurements of

ocean wave spectra6.

In modern ocean engineering, the need for knowledge of directional wave spectra is especially important. Several techniques have been developed to determine the wave directions in the ocean. Examples are the stereo-photographic method developed at New York University7,

the floating buoy method developed by Longuet-Higgins at the

NatiOnal Institute of Oceanography8, and the array method suggested by Barber and Pierson9. Despite these efforts only a few measured'' ,10

directional-wave spectra for the ocean are available. Moreovcr, the accuracies of the measurement.s are not known.

n an attempt to better assess the problem of determining directional wave spectra, it. was decided to measure and analyze wave data under controlled conditions. An experiment was conducted

in the Naval Ship Research and Development Center's seakeeping basin to study the angular resolution associated with measured wave spectra. The water surface elevation was measured by an array of sonic probes and the directions of the waves were then estimated from those

measurements. The results indicate that the measurements obtained by the probes are indeed suitable for analyzing directional spectra. However, the technique used in estimating the cross spectra between

(5)

different probes was not sufficiently accurate for determining the wave directions in' thecase where a regular wave train and

an irregular wave train are propagating at 90 degrees to each other. This paper reports on the, experimentally determined directional wave spectra and a Study of the application of the two-parameter wave spectrum model.

UNIDIRECTIONAL WAVE SPECTRA

One is aware of the variety of the ocean spectra, yet one must establish some order in this chaos fo,r practical application. For:

estimating the seakeeping qualities of ships, Cumins proposed a,

technique which makes use of a two-parameter wave spectrum of the general Bretschneider form. This two-parameter spectral formulation was studied by applying it to North Atlantic hindcast wave data5. 'The analysis procedure and results are given below.

Let x (t; 11, H113) be the response of a ship in a seaway which has average wave period 11 and significant wave height H,113, where

t is time. If x. is linear, its variance X(T1, H13) is given by

X(T1, H113) = E

[2

(t; T1, H13) I

f

H(w) S (; T1, H13) dw

where E[aI represents the 'average value of.. a, S() is the wave

spectrum, .H is the frequency response function of the ship and w

is the wave frequency. By the use of the wave spectrum form recomende,d

by ITTC3, that is , .

(6)

where S(w) is the idealized two-parameter spectrum. The statistics

of the. response,X(T1, H,3) are compete1y determined by the statistics

of T and H1113

if His a deterministic function.

The long-term

variance of response is

E[X(T1, H113)]

.=(fX(T.i.

H13)

p(T1,

H13) dT1

dH1,3

(1.3)

where p(T1, H113) is the joint probability density function of T'1 and H1,3.

Substituting equations (1.1) and (1.2) into (1.3), one has

E [X(T1,

H13)

=f

H(w)

S*(w)

dw

(1L)

where

S(w)

=ff

S

H13) p

(T1, H13) dT1dH13

(1.5)

S*is the averaged idealized spectrum by definition.

It is seen from equation (1.1) that the long-term averaged

variance of the response X(T1, H13) is the integral with respect

to

frequency of the product of the frequency response function and

the averaged idealized spectrum, Si,'. Knowing the averaged idealized spectrum of the environment determines the long-'term averaged variance

of the respone for agiven frequency response function of a ship.

In addition to the convenience in calculating the averaged long-term

173 H2

S(CiJ) = S'(w)

1/3 e91(Ti)14

(7)

ship responses, the two-parameter spectrum approach provides a rapid method for estimation of the probability of short term average ship response and its higher moments. If a probability diagram of and 'K113 is constructed such that

z = P(T1 a) and

y = P(H113 b; T1)

where zand y are the two coordinates of the diagram and the P's

are the probability fiincti.ons, then from (1.1) the probability

distribution of short-term ship response,P(X ca), is given by

g(T1,H13) = a P(X a)

=fj

p(T1, H113) dH1,3

=ffg(T1.H113)

= a p(H1,3; T1) p 1 pg(T1 (z).H113(zy) )

=J J

0 0 where

)=

H()S,(c..;

T1, H13)dc..

P(X .a)ts simply enclosed by the two coordinate axes and the curve

of

g(T1,

1

H(w)

SI.(

; T., 1

The use of this diagram will be illustrated below.

.5

(8)

North Atlantic hindcast wave data which contains spectra hind

casted at 519 point in the.North Atlantic every sixhours for a period

of one year, was selected to study thetwo-parameterspectral model.

Spectra at 16 grid pointswere selected for this study. A total of

16x365x1+ = 23,360 spectra were analyzed. T1 and H1113 were calculated for each spectrum by the use of the following formulas, which have been proposed by the ITTC3: t i

J

S()d

24.17 o '1 26.84

f

J

S(w)dw

0 H113

4\/jS(w)dw

'

(1.7)

The statistics of T and H1113, p(Ti, H113), are then constructed from these calculated values with equal weights. The averaged idealized spectrum was calculated from equations (1.5) (1.2), and p(T1, H1,3).

The resulting averaged idealized spectrum was compared with the

averaged hindcast spectrum SA() of those 23,360 spectra; that is

1

23360

SA(w)

23,360

S1(

j=1

where S1(w) are the hindcast spectra. As seen in Figure 1, the

comparisons do not agree very well. In comparison to the averaged hindcast spectrum, the idealized spectrum does not contain enough energy over both the high frequency and very low frequency range, and

is high for the middle frequency band. Figure 2 shows the probability diagram constructed from the statistics of T1 and H1113 of the hindcast

(9)

wave data. For illustration purposes, Figures 3 and 1 show the corresponding purely imaginary family of responses which result from assuming that

g(T1, H113)

= H,3 (T1/T112 exp' (-2T1/T1)

where C and T1are constants. The curves of response for aà O.lC

and 0.4C are shown in the figures. With T10

= 10

sec the probability that the' response would exceed O.4C and 0.IC was estimated 'as 1

percent and' 1.7 percent, respectively, by measuring the areas. The

probabilities were higher' for, the case of T= 15 sec. , These were

1.5 percent and '21 percent for a,= O.4C and 0.lC, respectively.

Recently, Webb Institute analyzed a groupof wave records

measured at weather station '16. Preliminary unpublished results

from these newly analyzed data indicate that the idealized two-parameter spectral form does not agree well with the measured spectra.

Figure 5 illustrates how the wave energy is' improperly distributed

over the frequencies in the idealized spectrum when it is compared

against the measured wave spectrum. The,sol'id curve in the figure is the average measured spectrum calculated from some of the recently analyzed spectra6. The other curves are the corresponding average

idealized spectra calculated from three different definitions ofT

(10)

and = 22Tf S() d(&) T1(1), T1 - fc.i.Sp(w)d

2iri!S ()dc.,

T =°

(i)P 1

S()dw

24.171 26.847 T1(-1)

I rs (w)da.'

11/2 24.171 = T1(2) I Jo P x.2z248

[J2s(w)dwJ

For an idealized spectrum, T1(l), T(l), and T1(2) are the same by

their definition. However, for a measured ocean spectrum the various

T '.s can be different. The differences reflect the departure of a

measured spectrum from an ideal ized spectrum. By use of the different

definitions of T one will thus obtain the different corresponding

idealized. spectra. The use of T1(l) is recommended by ITTC3

Fi.gure 5 indicates that all of the ideal izedspectra under

estimate the wave energy for the high and low frequency

band bya

factor of 1/2 and overpredict the wave energy for the middle frequency

band. The use of different definitions of T1 has not altered the

result significantly; calculation based on T(-l) is slightly better than that of the others, but is not. significantly better. The

probability diagram was not constructed for measured spectra because of the small number of the samples.

These differences in the energy distribution of the ideal ized

spectra and that of the measured spectra are consistent with the

differences found between the energy distribution of the idealized

spectra and the average hindcast spectrum shown in Figure 1. In both

(11)

by the idealized spectra, and in the middle frequency range it is

overestimated. This leads one to suspect the usefulness of the two-parameter spectrum for approximating the ocean environment, especially in studying the motions of platforms and buoys.

In spite of the bias described above, the parameterized spectrum approach has simplified the procedured in calculating the statistical properties of the ship responses. The probability diagram

representa-tion isa good tool to engineers, if theidalized spectrum form is

improved and the. statistical bias is tolerable..

DIRECTIONAL WAVE SPECTRA

The motion of a ship in a seaway depends not only on the

frequencies of the waves, .but also on the directions of the component

waves. The unidirectional spectrum discussed previously has little use when the ocean waves do not propagate in a dominant direction. Unlike the unidirectional wave spectrum, a directional wave spectrum

can nOt be obtained from a continuous record of wave elevation áta

single location on the sea surface. It requires knowledge of wave elevations over an area of the sea. Due to this requirement.only a few measured directional ocean wave spectra are available7'8'

The rapid growth of techniques for seakeeping analysis requires a more accurate description of a ship's environment. In order to niake meaningful comparisons between the analytic .results, basin

experiments, and full scale trials one has to establish the capability of measuring the directions of the ocean waves and generating directional

9

(12)

waves in a seakeeping basin. In this section the result of pre

liminary work. on basin-generated directional wave spectra is presented. The elevation of the sea surface is considered to be a

stationary and homogeneous random process in time t and space respectively. in the linear thiory the spectral representation of

T) is given by

(t,) =Rffe1(k

wt)d()

(:2. 1)

where (x1, x2) is the position vector, with x1 and x2 the two surface

coordtnates; i = (kcos 0, k sin 0) = (k1, k2) is the wave number vector,

with k = (k12 + k22)1"2. and 0 = tan1 k21k1 the wave number and wave

direction, respectively; k1 and k2 are respectively the wave-number

components in the x, and x2 directions; and dE(k) is the random

iariable. According to linear theory the wave frequency, is given by

= g(k + k22)112 in deep water.

+

In applications, one assumes that dE(k) satisfies the following expected value condition

1

+ -+

IS()difi=i'

-

F 1d(k) d(k' 1 =

2

-where a bar denotes the cOmple cOnjugate.

In the above expression S(i) is the directional wave spectrum. can be interpreted as the mean-square value of arising from wave

elements which lie in the infinitesimal range of wave number components

(k1, k1 + dk1) and (k2, k2 + dk2). Knowing the directional wave spectrum,

S(), of the sea determines the composition of waves in all directions.

(13)

In applications it is usually necessary to parameterize S(k)

A general representation of S() is given by its Fourier series. By

decomposing the directional wave spectrum S(; w) at a given frequency

into a Fourier series with respect to direction 0, Longuet-Higgins8

was able to relate the Fouriercoefficients to the cross spectra of

the surface elevation and its space derivatives. The float mg buoy built at the National Lnstftute of Oceanography was designed for this approach. Barber8 related the Fourier coefficients to the cross spectra of the surface elevations measured at several points in

space. However, in the latter case the relations become more. complicated and the lower order harmonic coefficients can not be determined without assumptions regarding the higher order harmonic coefficients.

From the Fourier representation of S(i; w0) with respect to the

wave-number component k1 , Barber and Pierson9 have shown that S(i; co0) can be approximated directly from an array of probes which lies in a direction parallel to the x1 coordinate. The Fourier coefficients obtained from this approach have a one-to-one correspondence with the cross spectra of measurements having space separations of

+ + .

r = (nD, 0), n = 1, 2, . . . N; where r is the separation vector and D

is the fundamental Separation of the probes. Thus, from an array of probes which has spearations D, 2D, . .. ND, one is able to approximate the directional spectrum up to the Nth harmonic. The derivation of

this i.s given below.

By multiplying equation (2.1) by the corresponding equation for

(14)

t(t+r,+fl

and taking the expected value, one has E [t7(t, ) i (t+r, +?)J = ..

ffJf

e1(t'

(1

-(J(t4r)]

(2.3)

'E[dE()d(l')]

=ffei(wT_kr)(i)di

where r is the time lag. Since the wave field is assumed stationary as well as homogeneous, the correlation function

E[(t,)i?(t+r,

+) I

in the above equation is a function of r and t only; it is independent

of t and . If the correlation function is denoted by R(r, ?)

equation (2.3) can be written as

=1! (wr-()d

=

where S(k1, w) dk1dw = () d . The dependence of S on k1 and k2

is replaced here by dependence on k1 and w, making use of the dispersion relation between k1, k2,. and c. Equation (2.Le) relates the directional

wave spectrum to the correlation function R(r,) . In the above

pO.2/g

equation

I

S(k; w)e_rdk1

is the Fourier transform

J2/g

of R(rj) with respect to rand is called the cross spectrum, which

is denoted by o(w)-iQ( . Thus,

whe.re

Pw2/g

+4

JS(k;

w)erdk1

R(r,) =J [Co(w;) -iQ(w;)] e"dw (2.5)

Thus, if one represents the directional wave spectrum S(k1,w) in a

Fourier series with. respect to k1D, where D satisfies -. D

9

(2.4)

(15)

one can obtain the Fourier coefficients directly from equation (2.6) by their definitions. That is

S(k1, w) A1 +

(Ar, cos nk1 D + B sin nk1 D)

(2.7) where A0 = - co(w; ? = (0,0)) An = Co (w; = (nD,0)) n 1, 2, (2.8) B,, = P. (w; ? = (nD,0))

By arranging the probes to measure wave elevations such that one can calculate the Co's and Q's up to = (ND,0) , the coefficients

A0, A1, B1, ...AN, BN can be readily obtained from (2.8). The, relations

in equation (2.8) are simple and they form the basis for the experiments described below.

The accuracy of the approximation of equation (2.7) is dependent on both the order N and the non-dimensional parameter w2D/g . For

given N and D the angular resolution,

0r , is defined as o

=

r

\ND w2/g

Physically this is a measure of the width of the angle over which

a narrow band wave-spectrum is spread. °r increases with decreasing w; that is the angluar resolution increases with increasing wave length. On the other hand if N and w are given, then

°r increases with decreasing

probe separation, D. For a specific experimental setup, D can be adjusted for optimum results.

'As an example consider a narrow band directional wave spectrum

(16)

which satisfies:

r

+k

J

k1

-k1

S(k; w) dk1 =

= 0 Otherwise

where C(w) is an arbitrary function of frequency w and 2k1 is the

band width of the wave number k1. The directional spectrum can then

be represented by a delta function, 5, and its Fourier representation is given by

S(k1; w) =C(w)D5(k1D - k0D)

C(w)D[1

+ cos n (k kO)D] (2.lD)

where n = 1, 2, . . The Nth order approximation gives

S(k1 ; w)

C(w)D

[ + cos n(k - k0) D

l4

fl=i N n=1 if k1 =k0 (2.9) (2.11)

C(w)D'l

).

At k1 k0 the approximation S(k1; w) has its maximum value of N

ir 2

C(w)D

1

At k1 -k0 one has S(k1;w)

(-.-It is a factor of -l/(1+2N) smaller than the maximum value. The

approximation of equation (2.8) to the narrow band directional spectrum, equation (2.9) of k0 = 0 , is shown in Figure 6. From this figure, one

sees how a narrow-band directional wave-spectrum is directionally spread out as a function of D in this approach. The loss of accuracy, i.e.,

g

increase of spreading, with decreasing - D , which has been discussed

9

(17)

EXPERIMENT

The basic appraoch described in the previous sect ion was applied to the measurement of the directional spectrum of waves generated in NSRDC's seakeeping basin. All wave measurements were taken with sonic probes, using three separate array configurations, which are shown in Figure

7.

For the linear arrays, the fundamental

distance between the probes was 2.5 feet and the total array.length

waS 32.5 feet.. This arrangement enables. one to approximate the Fourier series of the directional wave spectrum up to the ninth

harmonic. This configuration was suggested by Pierson11. The reason this arrangernent was used rather than the optimum array suggested by Barber12'13 is that in Barber's corif iguration the total array length would have been only 22 5 feet, a greater length was preferred The

linear array was arranged in two different orientatiOns relaflve to the wave generators. In the first case the array was mounted paraile.l to the West bank of the basin and at a distance of 100 feet from the

bank, as shown in Figure 3. In the second case the same array was rotated L15 degrees clock-wise to the North. The third array consisted of a pentagon arrangement, and employed six probes: one in the center and five Outside. forming an equilateral pentagon. The sides were designed to be 10 feet long. The orientation s shown in Figure

3.

The seakeeping'basin has wave generators along both the West and North banks and the wave generators on the two banks are operated

independently. During the study three kinds of directional wave fields

were generated. These were: waves coming from West bank, waves coming

(18)

from the North bank and waves coming simultaneously from both banks Both regular and irregular waves were generated. The periods of the regular waves were 1.6, 2.0, 2.5, and 3.0 seconds. These wave periods correspond to waves with angular resolution of less than 10 degrees up to 90 degrees. The irregular waves were generated from available

random seaway tapes. These wave trains had average wave periods

ranging from 1.6 to 3.0 seconds.

Sonic probes operating at a frequency of 200 kc were mounted approximately 20 inches above the still water surface. These devices can measure the iristaritanious water surface elevations with great

accuracy. From the digitized records, cross spectra were calculated

for all irregular waves. Wave ampi itudes and wave phases were calculated for the regular waves by means of Four.ier transforms. The directional wave spectra were then obtained by the use of equation (2.8).

Some resulting directional wave spectra measured from the linear

array are shown in Figures 8 through 16. Figures 8 through 11 are for regular waves and Figures 12 through 16 are for irregular waves.

The curves for regular waves represent the directional spectra obtained

under several different conditions such as different wave-maker dome

air pressures, which are indicated in terms of blower rpm, different

directions, which are designated by N or W for waves generated at the

North and West banks, respectively, and different wave combinations.

The coordinates of the figures are the normalized wave vector components

in the direction parallel to the array, and the normalized spectrum S(k1;w)

density S*(k1;c) = where SA(.)) is the average

(19)

of the one-dimensional spectrum densities obtained by the five, probes. Since S(k1;w) is approximated by a ninth order Fourier series, the normalized spectrum density S(k1;w) shou!d be less than 9. as

previously discussed. The theoretical maximum value of S(k1;w)

depends on the wave conditions. If a wave of a given frequency were generated at only one bank, S*(k1;w) should have a maximum value of

9.5 in the direction inwhich it was generated. For other directions,

it will be less than 9.5. The actual valu'e depend.s on, the combination

of the wave' ampl itudes generated at the two banks. The regular wave

result's a shown in the figure.s agree with this theoretical value

very well regardless of the wave frequencies, wave amplitudes and the presence of waves coming from other directions. Figure 11 'shows ow the angular resolutions varied with the frequencies of the waves. As,

dicussed previously, the angular resolution increases with increasing

wave period and the theoretical value of S*(ki;w) i.s -0.5 at direction

, (k -k0) . The measured angular resolution 'agrees fairly

well with the theoretical value! However, away from the peaks, S*(k1;w) 'oscillated with a much higher value than one would expect, especially

for the wave periodsof 2.5 and 3 seconds. The irrgular wave results

do not 'agree as well, with the theoretical value. Figure 12 shows the resulting diretio,nal 'Wave spectra of irregular wave trains generated at the North bank it indicateS that the waves were all coming from the North bank and the' peak value's of the spectra are between 65 atd 8. In

comparison With the regular wave measurements of Figure 7, the pak

value is decreased. However, as the' tail value also decreases, one

(20)

concludes that the mixture of the frequencies in the same direction does not affect the angular spreading significantly.

Figure 13 shows the measured directional spectra under a different wave condition. From it is concluded that there were long waves of period 2. to 3 seconds coming from the West and short waves of period

1.6 seconds propagating to the South. The directional distribution of the 2.0 second period wave is meaningless. However, the actual wave field was different from the one pictured above. An irregular

wave train was generated at the North bank and a regular wave of

period 2.5 seconds was generated at the West bank. The loss of

accuracies of the wave directions for the waves with wave frequencies

near that of the, regular wave is clear. Figure IL+ illustrates.the

same phenomena. For this case the period of the regular wave was 2.0 seconds and the amplitude was smaller in comparison with the previous

case. The loss of information on the wave directions in this case

was not as serious as in the previous case. The presence of the 2.0 second period wave reduced the peak value of the 2.4 second period wave but increased that of the 1.6 second period wave. Figure 15 shows

the measured directional wave spectra for the case of two low-amplitude irregular wave trains propagating at 90 degrees to each other. The directional distribution of the wave is reasonable.

By' examining the calculated cross-spectra we found that the method used in estimating the cross spectrum was responsible for the errors which appear in Figures 13 and 14. Special care is necessary when, analyzing the directional wave fields in these cases. By the use of a

(21)

narrower frequency band-width, the result was improved; it Is shown

in Figure 16.

The directional spectra obtained from the pentagonal arrangement were not good and are not shown here. The measured phase lags between the probes had the same accuracies as those obtained by the linear

array. However, as previously mentioned, the relations between the Fourier coefficients and cross spectra are more complicated and thus, the results were not as good as those obtained from the linear array methOd.

DISCUSS tON

The result of the study On the idealized unidirectional spectrum indicates the need for improvements in recommended spectral forms in order to obtain a better prediction of long term ship motions. The two-parameter spectral form underestimates the wave energy for both high and low frequencies and overestimates the wave energy over the

wave frequency range of 0.1 cycle/sec to 0.R cycle/sec. This has

been illustrated in Figures 1 and 5.

For applications, one has not only to be aware of this limitation associated with the idealized spectrum but also the varieties of ocean wave spectra. For a better representation of the ocean environment, one needs to know not only the averaged wave period and wave height but also other parameters. With more measured spectra a data bank of wave spectra

can be established on a digital computer and stored on tapes for dirct

access. Such a data bank would eventually make idealized spectra

(22)

obsolete. It would certainly be more accurate than the idealized spectra and would contain samples of all of the variOus ocean wave

spectra.

The experiment on the directional waves suggests that the accuracy of a measured directional spectrum depends more on the

directional compositions of a wave field than on frequency compositions of the waves. This is demonstrated in Figures 11, 12 and 13. In

order to accurately measure the spectrum of a swell 'and wind waves

cOmbined sea, one has to use a technique which can estimate a sharp

peaked cross-spectrum accurately, such as the narrow band process or th time shift process'. However, it will require much longer

records of surface elevation.

The linear array is a better: probe arrangement than a pentagon arrangement for the wave fields generated for this paper. However,

a linear array does not allow one to separate the Waves propagating

in the direction left of the array from those of the right. Two linear arrays may be needed for the measuring of an actual wave field in which waves propagate in an angle of more than 90' degrees.

The angular resolution illustrated previously can be improved

i:f one applies a weighting function to the Fourier coefficients. The

choice of the weighting function depends on one's taste, as has been

discussed by Longuet-Higgins8.

The separation D was 2.5 feet in the present experiment. This separation cah be adjusted to improve the long-wave angular resolution.

(23)

is interested. If one is only interested in the long waves then

2

one can chose a suitably large D such that D for

g

all frequencies in which one is interested. But in this case, the total array length will certainly be increased, and the increase in the array length can complicate the operations of the detectors. This has to be considered in the choice of an optimum 0. By moving the probes opposite to the direction of the waves, one can also improve the angular resolution significantly11. It is applicable in an open

sea where the waves are considerably homogenious over a large area. It may not be suitable for a model basin unless the basin is large and the waves are very homogenious with respect to space.

The length of the wave records used for the above irregular wave calculations are approximately one minute long. The accuracy might have been improved if longer wave records had been used.

ACKNOWLEDGMENTS

The author is indebt to Dr. Wm. E. Cummins for his suggestion of the subject and the guidance given during the course of the work. Also, the author wishes to thank Mr. Daniel Huminik for his assistance with the experiments and many others who helped on the manuscript.

(24)

REFERENCES

Pierson, J. W. and Moskowitz. L., "A Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S. S. Kitaigorodskii", J. Geophys. Res., Vol.

69,

No.

2k, 196k

Bretschneider, "Wave Variability and Wave Spectra for Wave Generated Gravity Waves", Technical Memorandum, No. 118, Corps of Engineers, Department of the Army, 1959

Proceedings of the 12th International Towing Tank Conference, .1969 Cummins, W. E., "A Proposal on the Use of Multipararneter Stand Spectra", Proceedings of the 12th ITTC, 1969

5.

Pierson, W. J.. and Tick, L. J., "Wave Spectra Hindca.sts and Forecasts and Their Potential Uses in Military Oceanography", .

The U. S. Navy Symposium on Military Oceanography, 1965

6, Miles, .M., "Wave Spectra Estimated from a Stratified Sample of.

23 NOrth, Atlantic Wave Records", NRC Report LTR-SH-118, October 1971,

(Unpubl ished)

7. Chase., J., et al, "The Directional Spectrum of a Wind-Generated

Sea as Determined from Data Obtained by the Stereo Wave Observation Project", New York University, College of Engineering Report, July 1957

.8. "Ocean Wave Spectra, Proceedings. of a Conference", Prentice

Hall, New York, 1963

9. Barber, W. F. and Pierson, W. J., "Review of Methods of Finding

the Directional Spectra in a Towing Tank", New York University, College of Engineering Report, 1963

(25)

Ewing, J. S., "Some Measurements of the Directional Wave Spectrum", Journal of Marine Research, Vol.

27,

No.

2,

1969

Pierson, W. J., "The Estimation of Vector Wave Number Spectra

by Means of Data Obtained from a Rapidly Moving Hydrofoil Vehic1",

Technical Report No.

68-56,

Oceanics, 1968

Barber, N; F., "Optimum Arrays for Direction Finding", N.Z.J. Science 1 (1), 1958

Cummins, W. E., "The Determination of Directional Wave Spectra

in the TMB ManeuveringSeakeeping Basin", DTMB Report

1362, 1959

]L. Pierson, W. .J. and Dal:zell, J. F., "The Apparent Loss of Coherency

in Vector Gaussian Processes Due' to Computational Procedures with Application to Ship Motions and Random Seas", College of

Engineering, ResearchDivision, New YorkUniversity, 1960

(26)

(-I U E 3.0 2.0 C,, -w 1.0

/

I

I

/

- I - I I 0.157 0.314 0.471 w SEC-1 AVERAGE IDEALIZED

/

\

SPECTRUM 0.628 AVERAGE HANDCAST. SPECTRUM

\

N

N

0.785

FIGURE 1 - AVERAGE HINDCAST WAVE SPECTRA CALCULATED

(27)

P(10<T1

12, 4<H1136)

'.4

L

10 12 14 16 18 20

T1 SEC

FIGURE 2 - PROBABILITY DIAGRAM OF T1 AND H113

BASED ON HANbCAST WAVE DATA

25 H1, FT

46

8 10 12 14 16 18 18

1614

12 10 8 22

(28)

X = 0.4 C

10 12 14 16 18 22

T SEC

FIGURE 3 - PROBABILITY DISTRIBUTION OF THE IMAGINARY FAMILY

(29)

= 0.1 C

H113 FT

4 6

8 10

14 18

X = 0.4 C

FIGURE 4 - PROBABILITY DISTRIBUTION OF THE IMAGINARY FAMILY

OF SHIP RESPONSES WITH T1*

= 15 SEC

27

II.

%% I

68

10 12 14 16 18 T1 SEC 22 20

(30)

m2-sec

AVERAGE MEASURED SPECTRUM T

= T1(-1) 1 AVERAGE IDEALIZED T1 T11, SPECTRUM = T1(2) p

000T1

05

10 15 C..) SEC1

(31)

12

10

6

(

FIGURE 6 - THE NINTH ORDER HARMONIC REPRESENTATION OF THE NARROW BAND DIRECTIONAL SPECTRA

29 I- I = ('h+ cos(k1-k0)D1 IT 1 . (.)2Jg D = 0.33 D = 0.49 D = 1.2

2/gD0.75

---- w2Ig

S(k1;w) .

-w2/g

vail'-will____

ilhIiILIlL

_s_I.

Illu

E1WIAWI

.1Mui_i1I1

/

.11

. (k1-k0)gIw2

I.

' I 130° 900 500 300 0 1.

.1

$ I 15°

1

0° I 180° 165° 150° 1

9,

(32)

U)

0

LU

2

'U C, LU

>

WAVE GENERATORS

J 0 0 0 0 0 0 0

I

I

I

2

100 FT

NORTH BANK K

360 FT

3D

-2D D

ENLARGED LINEAR ARRAY CONFIGURATION. D = 2.5 FT

WEST

ENLARGE PENTAGON CONFIGURATION

FIGURE 7 - WAVE-HEIGHT PROBE

CONFIGURATIONS

S

(33)

12

10

N w

-0.0 k1* = k1g/w2

BLOWER RPM DIRECTION PERIOD

700 RPM N 1.6 SEC 700 RPM N 1.6 SEC 600 RPM VV .3.0 SEC 1200 RPM N 1.6 SEC 600 RPM W 2.5 SEC

000 600 RPM

W 1.6 SEC

FIGURE 8 - MEASURED DIRECTIONAL SPECTRA OF 1.6 SEC PERIOD WAVES ONLY, WITH ARRAY PARALLEL TO WEST BANK

31

S

1.0

(34)

BLOWER RPM DIRECTION PERIOD 1000 RPM N 20 SEC 700 RPM N 2.0 SEC 700RPM N 2.0 SEC 600 RPM W 3.0 SEC

000

700 RPM N 2.0 SEC 1000 RPM W 2.5 SEC 700 RPM N 2.0 SEC 1000 RPM W 2.0 SEC 3 * 12 10

FIGURE 9 - MEASURED DIRECTIONAL SPECTRA OF 2.0 SEC PERIOD WAVES ONLY. WITH ARRAY PARALLEL TO WEST BANK

N

W S

-4

I I I I I

-1.0 0.5 0.0 0.5 1.0

(35)

12 10 8 6 0 -2

-4

N I I 1.0 -0.5 0.0 k1 = k1g/w2

FIGURE 10 - MEASURED DIRECTIONAL SPECTRA OF 2.5 SEC PERIOD WAVESONLY. WITH ARRAY 450 TO WEST BANK.

33

W

BLOWER RPM DIRECTION PERIl)!) 1000 RPM N 2.5 SEC 600 RPM W 2.5 SEC 700 RPM N 2.5 SEC N 600 RPM W 2.5 SEC 700 RPM N 2.5 SEC

xl

000 600 RPM

W 2.5 SEC

I

1.0 0.5

(36)

S WAVE PERIOD 3.0 SEC 2.5 SEC 2.0 SEC

pop

1.6sEc

FIGURE 11 DEPENDENCE OF ANGULAR RESOLUTION ON WAVE PERIOD FbR WAVE

(37)

.3 S U, 12 10 35 W 0.0 k1 = k1g/w2

CENTER PERIOD OF FREQUENCY BAND

3.0 SEC 2.7 SEC

- - 2.0 SEC

000

1.6SEC

FIGURE 12 - MEASURED DIRECTIONAL SPECTRUM OF IRREGULAR WAVES GENERATED AT NORTH BANK

N -4 -0.5 -1.0 S 1.0 0.5

(38)

12 10 8 6 * C,, 2 2

FIGURE 13 - MEASURED DIRECTIONAL SPECTRUM OF AN IRREGULAR WAVE TRAIN WITH A HIGH AMPLITUDE REGULAR WAVE TRAIN OF 2!5 SEC PERIOD

-CENTER PERIOD OF FREQUENCY

3.0 SEC 2.5 SEC 2.OSEC 1.6SEC BAND

----.

--000

\

:1

11.4/

/

Ir

;////

= k1glw

(39)

A.

.11.

ii

-VA--

ra

flf1Ik__

I

__um:_

Liii

wrawA'

T" '

-:

1I

:1:

12 10

CENTER PERIOD OF FREQUENCY BAND

.0 SEC 2.7 SEC

- 2.0 SEC

000

1.6SEC W S I .

I-.

-

--'

I -1.0 -o.5 - 0.0 0.5 1.0 k1 = k1gk,2

FIGURE 14 - MEASURED DIRECTIONAL SPECTRUM OF AN IRREGULAR WAVE TRAIN WITH A LOW AMPLITUDE REGULAR WAVE TRAIN OF 2.0 SEC PERIOD

(40)

*

cn

10

6

2

FIGURE 15 - MEASURED DIRECTIONAL SPECTRUM OF TWO IRREGULAR WAVE TRAINS

- CENTER PERIOD OF FREQUENCY

-

3.0 SEC BAND 2.7 SEC 2.0 SEC

-00 0

1.6 SEC

4II1I__

4i 1111

4111_

411_Ii

t1FA

'war

N I W I S

1.0

0.5

= 0.0 0.5 k1gIw2 1.0

(41)

12

10

-1.0 -0.5 0.0 0.5 1.0

= k1g/w2

CENTER PERIOD OF FREQUENCY BAND

3.0 SEC 2.5 SEC

- 2.0 SEC

- 000

1.6 SEC

FIGURE 16 - DIRECTIONAL SPECTRUM OF FIGURE 12 WITH A NARROW FREQUENCY BANDWIDTH

Cytaty

Powiązane dokumenty

Przykładowo, w opracowaniu (Kaczmarczyk 2017) pomimo szeroko omawianej tematyki niskiej emisji i zmniejszania negatywnego oddziaływania na środowisko systemów zaopatrzenia

Zestawiając stereotypowy obraz krakowianina z wizerunkiem warszawiaka stwierdzamy między nimi ogromną przepaść, zarówno w aspekcie deskryptywnym (odmienne treści), jak i

Wystarczy zajrzeć do spisu treści, by zauważyć, że cały materiał książki dzieli się na trzy Części, poświęcone kolejno: Liturgii Godzin, problematyce

Od instynktownej niechęci do człowieka używającego obcego języ­ ka i przynoszącego ze sobą obce zwyczaje, poprzez wrogość w stosunku do obcego intruza,

Significant support on the political scene has been obtained by groups who even call for direct democracy to play a more important role than before in the political system

[r]

Niedojrzałe postawy wobec rodzicielstwa częściej wiążą się z takimi właściwościami indywidualnymi młodych dorosłych, jak niski poziom oceny wartości religijnych

stitutive relations depend on gradients of the strain rate, (ii) particle migration modifies the predictions of local models, but cannot account for the observed stress profiles,