15 SEP.A97
/'/
ZS
ARCHIEF
I
bIioheek van d
Ondera{dehns .P'-- - - SSI n;sche Hoqeschoo, DOCUMENTATEDATUM:
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
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.
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
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
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) If
H(w) S (; T1, H13) dw
where E[aI represents the 'average value of.. a, S() is the wavespectrum, .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 , .
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-termvariance of response is
E[X(T1, H113)]
.=(fX(T.i.
H13)
p(T1,
H13) dT1dH1,3
(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
SH13) 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 andthe 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
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., 1The use of this diagram will be illustrated below.
.5
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 26.84f
J
S(w)dw
0 H1134\/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)
23,360S1(
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
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 1percent 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
and = 22Tf S() d(&) T1(1), T1 - fc.i.Sp(w)d
2iri!S ()dc.,
T =°
(i)P 1S()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 frequencyband. 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
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
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.
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
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 transformJ2/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)
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
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) Dl4
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
1At 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
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 fundamentaldistance 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
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
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
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
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
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.
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.
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
Ewing, J. S., "Some Measurements of the Directional Wave Spectrum", Journal of Marine Research, Vol.
27,
No.2,
1969Pierson, 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, 1968Barber, 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 Coherencyin Vector Gaussian Processes Due' to Computational Procedures with Application to Ship Motions and Random Seas", College of
Engineering, ResearchDivision, New YorkUniversity, 1960
(-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
0.785FIGURE 1 - AVERAGE HINDCAST WAVE SPECTRA CALCULATED
P(10<T1
12, 4<H1136)
'.4
L
10 12 14 16 18 20T1 SEC
FIGURE 2 - PROBABILITY DIAGRAM OF T1 AND H113
BASED ON HANbCAST WAVE DATA
25 H1, FT
46
8 10 12 14 16 18 181614
12 10 8 22X = 0.4 C
10 12 14 16 18 22
T SEC
FIGURE 3 - PROBABILITY DISTRIBUTION OF THE IMAGINARY FAMILY
= 0.1 C
H113 FT
4 6
8 10
14 18X = 0.4 C
FIGURE 4 - PROBABILITY DISTRIBUTION OF THE IMAGINARY FAMILY
OF SHIP RESPONSES WITH T1*
= 15 SEC
27
II.
%% I68
10 12 14 16 18 T1 SEC 22 20m2-sec
AVERAGE MEASURED SPECTRUM T
= T1(-1) 1 AVERAGE IDEALIZED T1 T11, SPECTRUM = T1(2) p
000T1
05
10 15 C..) SEC112
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)gIw2I.
' I 130° 900 500 300 0 1..1
$ I 15°1
0° I 180° 165° 150° 19,
U)
0
LU2
'U C, LU>
WAVE GENERATORS
J 0 0 0 0 0 0 0I
I
I
2
100 FT
NORTH BANK K360 FT
3D -2D DENLARGED LINEAR ARRAY CONFIGURATION. D = 2.5 FT
WEST
ENLARGE PENTAGON CONFIGURATION
FIGURE 7 - WAVE-HEIGHT PROBE
CONFIGURATIONS
S
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 SECFIGURE 8 - MEASURED DIRECTIONAL SPECTRA OF 1.6 SEC PERIOD WAVES ONLY, WITH ARRAY PARALLEL TO WEST BANK
31
S
1.0
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 10FIGURE 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
12 10 8 6 0 -2
-4
N I I 1.0 -0.5 0.0 k1 = k1g/w2FIGURE 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 SECI
1.0 0.5S 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
.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
1.6SECFIGURE 12 - MEASURED DIRECTIONAL SPECTRUM OF IRREGULAR WAVES GENERATED AT NORTH BANK
N -4 -0.5 -1.0 S 1.0 0.5
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
;////
= k1glwA.
.11.
ii
-VA--
ra
flf1Ik__
I
__um:_
Liii
wrawA'
T" '
-:
1I
:1:
12 10CENTER 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,2FIGURE 14 - MEASURED DIRECTIONAL SPECTRUM OF AN IRREGULAR WAVE TRAIN WITH A LOW AMPLITUDE REGULAR WAVE TRAIN OF 2.0 SEC PERIOD
*
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 SEC4II1I__
4i 1111
4111_
411_Ii
t1FA
'war
N I W I S1.0
0.5
= 0.0 0.5 k1gIw2 1.012
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 SECFIGURE 16 - DIRECTIONAL SPECTRUM OF FIGURE 12 WITH A NARROW FREQUENCY BANDWIDTH