DIRECTIONAL WAVE GENERATION
by
Stig E. Sand 1)
Danish Hydraulic Institute Agern Alle 5, DK-2970 HOrsholm
Denmark
A B ST R A C T
The directional character of natural waves can be of vital impor-tance for the design of many offshore structures. Consequently, it
has become necessary to develop a directional wave generation
technique in the laboratory for model testing of fixed and float-ing offshore structures. The most common
type
of generator con-sists of a large number of independent elements. The basiC theory behind the control of such a wave generator is reviewed. Specialemphasis is placed on the sources of error
originating
from thewave kinematics itself, as well as from the boundaries of the fa-cility. In addition to this, the deficiencies of various synthesis
models are outlined. It is shown that the problems inherent in extending the traditional 2-D models to 3-D are rather complex,
and that more research is needed in future to reach a satisfactory solution to the synthesis problem.
1)
Member "Working Group on Wave Generation and Analysis", ternational Association for Hydraulic Research (IAHR). 4405/SES/VIB/APA
Lab. v. Scheepsbouvikuncle
29 JULI 1988ARCH1EF
Technische Hogeschool
Delft
1. INTRODUCTION
scription of natural wave fields, and (model and prototype) analyzed ? - i.e ration techniques and directional wave
The need for physical model testing has obviously arisen from an
insufficient understanding of natural phenomena, in this case
waves, wind and currents, and their interaction with structures. Although the numerical modelling techniques are in strong progress physical model tests are still needed, because of the complexity of wave-wave, wave-current, wave-structure interactions and of force-motion descriptions.
The ultimate goal of the physical models must be to produce an
exact scaled version of nature. This explains the tendency during
the last decade to extend the traditional 2-D (uni-directional)
wave generation methods with one additional parameter: the direc-tional spread. During the years, the 2-D reproduction techniques have been highly refined and new phenomena (often of higher-order character) are constantly under investigation by researches all over the world. However, notwithstanding the questions left open in the 2-D area, it is the desire of
including
all observed (al-though not all well described) natural phenomena in the physical model that pushes the 3-Dwave generation
methods.It is impossible to make isolated considerations of 3-D wave gene-ration in physical models. The subject is closely linked to two other important issues: 1) What should be modelled ? - i.e. a de-2) How are the wave fields the relation between gene-analyses.
In wave generation and analysis two basically different principles exist. These are based on the i) stochastic and the ii) determi-nistic philosophy. In the present context these concepts are used
state is the modelling of a time series with correct amplitudes, frequencies, directions and phases, whereas the stochastic princi-ple is associated with a (stochastic) reproduction of amplitudes, frequencies and directions (the spectral density), but with the phases chosen randomly. As per definition the point spectrum will be the same whether one or the other basic principle of reproduc-tion is used, but the actual time series recorded in the model may deviate because of the phase selections. The three key elements discussed above are shown in Fig. 1 in relation to the stochastic/-deterministic philosophies.
Description of Natural Wave Fields
3
Directional Wave Analysis
Fig. 1 Key concepts in relation to a) stochastic and b) determi-nistic wave generation.
In both cases a basid theory of the composition of
natural wave
fields is required. For the stochastic reproduction
statistical information is extracted from nature (e.g. from a specific
site) and used in connection with more general models. Thus, a specific spectral density, Sn(f), very often belongs to parameterized
stan-dard versions such as the Pierson-Moskowitz (PM) or the JONSWAP spectrum. This is supplemented by a directional spreading
func-tion, D(f,0), which may be selected as a cos2s0/2 or a N(a21e)
type; 0 being the direction, s(f) the spread, and a2(f)
the va-riance of the Normal distribution. With this standardized descrip-tion of natural sea states the direcdescrip-tional spectrum becomes:
Sl(f/e) = S (f) p(f e) (1)
When the directional spectrum is generated in the physical model the directional wave analysis becomes of vital importance for any use of the tests. This is also indicated in Fig. 1. Finally, the
obtained statistics may be compared to prototype conditions. An
example of a directional spectrum is seen in Fig. 2.
Fig. 2
f
A directional spectrum, Sn(f,e).
The deterministic wave modelling requires first of all site speci-fic records. To derive the necessary information for the physical model a suitable wave analysis method must exist. This should be able to resolve the directional energy distribution as well as the actual phase relations. Both must be detected without the intro-duction of parameterized
functions
or distributions. The next step is to'generate the decomposed sea state, and finally compare with the original prototype information.As Fig. 1 indicates the perception of natural seas, the analysis and the physical scale modelling are closely linked and the change of methods (or point of view) in one area affects the other.
Hen-ce, the wave generation techniques can be very different from one to the other (e.g. deterministic/stochastic)
or even
Within one ofthem.
In the following sections it will be attempted to give a review of the existing directional wave generation theories neglecting (to a certain extent) the links to the analyses discussed above. The
principles are mainly seen in relation to physical modelling al-though most of the basic ideas apply to numerical models as well.
2. WAVE GENERATOR SET-UP
In general the directional Wave basins are of a rectangular shape. Typical dimensions of existing facilities range from 10 by 15 m up to 50 by 80 m. The generators producing the directional wave input usually take up one side of the basin, whereas efficient wave
ab-sorbers are installed along one or all of the remaining three
sides.
A directional wave generator is composed of a number of
individu-ally controlled elements. It is the phase lagging between the
ele-ments that determines the direction of the wave components as
shown later. The typical width of each element is around 0.3 to 1.0 m.
In
general all the wave generator elements are arranged on onestraight line along the long side of the basin. However, in a sin-gle case it has been chosen to dispose 10 individual elements on the 1000 of an arc. The generators all focus at the central point of the basin. For this set-up, however, the number of directional components in the wave field is limited to the number of genera-tors
in
the arc, see Huntington and Thompson (1976). Such a limi-tation is not applicable to the straight-line system, andin
addi-tion it leaves more space in the basin for model installaaddi-tions as well as for transverse current generation.The wave generator systems can be either electrically or hydrauli-cally driven. In practice it seems that the hydraulic solution is
more powerful and is used to run larger individual elements (either wider or higher elements), whereas electrical motors are chosen for smaller elements producing lower total wave heights.
A common wave generator set-up for reproduction of directional seas is shown in Fig. S.
A large number of individual elements disposed on a straight line for directional wave generation.
OBLIQUE REGULAR WAVES
If the wave generation system described above should produce a
long crested regular wave in a zero degree angle (front parallel to the line of generators) then all the elements had to be in phase and work as one solid unit. The wave parameters would be determined by the:
amplitude frequency phase
of the wave generation system. ,As in case of traditional wave
flumes the three parameters are transferred to the travelling wave
so that the frequency is the same as that of the generator, the
phase is shifted 90 degrees and the paddle stroke is converted to
a wave amplitude by means of a transfer function a/Xa = F2(f)',
where a is the wave amplitude and Xa the paddle amplitude. In the linear case this function is given by for instance Biesel (1951) for ,a piston and a hinged type wave board.
In producing a regular oblique wave component the parameters are essentially the same.
To obtain an angle between
the line of gene-rators and the travelling wave, a phase shift has to be introducedbetween each of the individual paddle elements. In addition the
transfer function for amplitudes is altered, i.e. a/Xa = F3(f,9). The relation between wave angle and phase shifting of the elements is illustrated in Fig. 4, and the amplitude transfer function will be discussed below. From the figure the following relation is
sin =
sin
NB
L
Fig. 4, Oblique wave produCed.
by box
shaped eleMents of width B.
easily derived:
in which e is the wave direction, L the wave length, N the number of elements of width B in one oscillation in the y-direction. From one element to the next the progressive phase shift 0 can be writ-ten as:
If this is inserted in (2) then the wave direction becomes:
2 B
(2)
(3)
(4)
Since the wave length is a function of the frequency by virtue of the dispersion relation, it appears that the
direction of the
oblique wave is determined by the progressiVe phase shift of the generator elements and their frequency of oscillation.
As illustrated in Fig. 4 the wave direction can also be expressed
in terms of the wave number vector t and its projection on the y-axis
k
.The applicable transfer function for wave amplitude can be derived from Fig. 4 and Huygens principle. Thus, in the 3-D case it can be shown that
F3(f'e)
= F2(f)/cose, i.e.for zero degrees the 2-D case appears.With all parameters given it is possible to precalculate the gene-rator movements for the reproduction of a specificed wave
compo-nent. Say that it is desired to generate a wave of angle e with the following time variation of the wave elevation:
n(xr,yr,t) = a cos (. t) (5)
in which the coordinates (xr,yr)
are a
specified reproductionpoint, for instance the centre of the basin. Based on the above
relations the required wave generator displacements appear as:
X(n,t) = a tin( t+kx
cosek(nB-yr)sine) (6)' 3(f,e
where n is the element number counted from the one located in
(0,0). The phase shift between the generator elements is expressed in terms of the projection of the wave number vector I.
It is intuitively clear from Fig. 4 and from (6) that a perfect oblique wave front is obtained only with an infinitely small width
B of the paddle elements. Hence, the total paddle front should
form a perfect sinusoid. Any deviation from that, because of
fin-ite width of the elements, will contribute to the generation of
spurious waves. The larger the elements the more imperfect will the produced oblique wave appear. The theory describing the rela-tion between paddle width, wave direcrela-tion and spurious wave ampli-tude will not be derived here, but reference is made to e.g. Sand (1979). The result can be expressed as an alteration of (4), i.e. the wave direction relation becomes:
sin 01) 0 - Zwp) L
2 w B
where p takes the integer values 0,+1,+2,... The zero value cor-responds to the main wave as described by the formulas (2) to (6).
However, on top of this it may be possible to observe spurious
wave components. The applicable p interval is:
0 B
2 w P < + L
Thus, for a given wave direction and wave length, (8) gives the usable p interval for spurious waves. When these p values are in-serted in (7) the actual wave angles of all possible
components
can be found. For example, if it is desired to generate a wave in e = 200 with a wave length to paddle width ratio of L/B = 1.0 then the phase shift can be found by means of (4):L/B 2w
= 2.92 = => 0 = 2.15
sin 0 0
When the L/B ratio and the value of the progressive phase shift 0 are inserted in Eq. (8) it appears that the valid p values are only p = 0, +1. With these values of P. Eq.
(7) indicates that oblique waves will be generated in the directions 200 (as desired) and -410 (a spurious
component).
The equations discussed above can be set up in the diagramme shown in Fig. 5. It illustrates the number of possible spurious compo-nents as a function of L/B and the desired direction of the main. wave. It is seen that as long as L/B is larger than 2.0 it is
ac-tually possible to generate waves up to 90° without any spurious wave content. It is also seen that the limit for reasonable wave generation is characterized by a phase shift between the
indivi-dual elements of 0 = r. For larger values the desired wave can hardly be detected sinae the amplitude of the spurious components are dominating.
4))1T
spurious waves
higher than
main wave
Imaginary, exponentially attenuated waves
2it B
0,
= ke
Fig. 5 Number of spurious waves as a function of the main Wave parameters, 1,,e and the paddle width B.
In the design of a directional wave facility the spurious wave phenomenon is usually avoided by choosing the width of the ele-ments so small that the L/B ratio is about 2.0 for the whole range
of realistic wave lengths. This range is again dependent on the
planned range of wave spectra and the test scale of the facility. HoWever, there are other sources of error which cannot easily be avoided. These are, as described below, mostly related to the
phy-sical limits of wave basins.
A sine 1.0 7.90 .9
-60
8 .7 45 6.5 30
2.0 2.5Lie
13
-4. SOURCES OF ERROR
On the basis of the formulas presented in the preceding section it is possible to generate an oblique regular wave with prescribed amplitude, frequency and phase. However, the characteristics of the wave cannot be expected to hold all over the test basin. In fact, even the most favourable area in the vicinity of the wave generator is often contaminated by undesirable effects originating from the physical boundaries of the facility. The sources of error that influence the quality of the generated waves are:
diffraction reflection
local disburbances servocontrol system.
The diffraction process can significantly affect the amplitude of the generated wave. The phenomenon is schematically illustrated in Fig. 6. On both sides of the imaginary boundary indicated by the broken line, the. desired wave height is influenced by diffraction.
In the central area tests and
measurements will usually takeplace. However, in this area the wave height will be alternating from point to point dependent on the diffractional effects. Only
far from the imaginary (broken) line the fluctuations will be rather modest. The actual wave height fluctuations can be assessed by means of the diffraction diagrammes commonly used in connection with harbour layout. The obtained values will be a function of the wave frequency and the direction of propagation. However, in the central area, discrepancies of 10 to 30% can easily be found.
Fig. 6 Illustration of some sources of error in directional wave generation: "1" diffraction, "2" reflection and "3" local disturbance.
The wave quality in
the central
test area will to a certain extent also be affected by reflections from the boundaries of the basin. Often one side is equipped with a rather effective absorber, and the other two will be simple vertical concrete walls. In this case only a small amount of the incoming wave energy will be reflected from the rear end (with the absorber). Typically 5 to 15% of the wave height will be reflected by a reasonably effective installa-tion. However, the vertical side walls will in general contribute considerably more to the disturbing effects in the test area. In the left hand side of Fig. 6 the wave fronts will reflect directly on the wall and propagate - and diffract - into the main area. The magnitude of the reflection effect is difficult to assess. How-ever, again diffraction diagrammes (and assumption of 100%re-,
WAVE ABSORBERS
.,- -, -_ `..
...
it
..
N..1
III
111111111 111111111i 111111111 HllfllIl M1111111
III
15
-flecting walls) can be used for a rough estimate. The typical in-fluence of the walls on the test area is ±5 to. 20% of the wave height dependent on the width of the basin. The sign of the dis-turbance depends on the relative phase between the incoming and reflected wave in the point of consideration.
The third source of error is related to the wave generator itself.
It is a matter of the shape of the wave paddle. In deep water waves - which are the typical waves in an offshore basin - the
vertical velocity profile is exponential. However, a wave paddle with such a profile does not exist. The most practical approxima-tion in that situaapproxima-tion is the hinged straight wave board, whereas for shallow water waves the translatory wave board seems most suit-able. The deviation from the actual wave, profile creates distur-bances on the real progressive wave. These disturdistur-bances are often referred to as local disturbances, since they decay exponentially with the distance from the wave generator. Obviously, the magni-tude of this effect depends on the wave frequency, the water depth
and the shape of the wave paddle. In deep water cases the wave
amplitude added close to the generator may amount to some 10% of the progressive one. In a few metres from the wave board the ef-fect will be reduced to only a couple of percent.
Finally, it should be mentioned that also the setvocontrol system
may belong to the list of sources of error in directional wave
generation. For a good quality of the wave reproduction the number of control characters per second has to be reasonable high, i.e. the
information
transmitted from the computerto the servo system. In addition, the amplification of the servo loop must not be too low, since this would introduce phase lags and reduced amplitudes relative to the desired input signal. Alternatively, the
characte-ristics of the servo system should be included in the
overall transfer function for the wave generation.
5. BOUNDARIES OF TEST AREA
The oblique regular waves discussed in the preceding sections are important elements in directional wave generation. Hence, in prin-ciple a short crested sea is simply a linear superposition of
re-gular oblique wave
components.
Therefore the boundaries of the optimal testing area can also be determined from considerationsbased on a few oblique components. This will be shown below.
Consider the directional distribution D(0) at a given frequency f. This could be the one obtained from a cross section of Fig. 2 at
the frequency f. For wave reproduction purposes such a distribu-tion would normally be symmetrical around zero degrees (the direc-tion perpendicular to the line of wave generators). For the physi-cal reasons outlined above the distribution has to be cut (truncat-ed) at an angle of less than 900. Dependent on the actual spread
of the given directional wave generation case, various maximum
wave angles can be determined. The bounds of the distribution may be denoted +emax and -0max, as shown in Fig. 7 a). If these pre-scribed maximum angles are represented by two regular oblique wave
components
then the boundaries of the test area can be estimated as illustrated in Fig. 7 b). Again the broken lines indicate thelimits of the two wave fronts generated in the directions +0 max and
-emax. The useful test area can now be taken as the circular area limited by the broken lines. Within this, all the wave
com-ponents
in the distribution limited by+emax and -emax (cf. Fig. 7 a)) will be represented.
If the total width of the wave basin is denoted
LB then the dia-meter d of the circle can be expressed as:
b)
7 17 7
N't).rnax
Fig. 7 a) Maximum wave angles for a satisfactory representation of a directional distribution, and b) the resulting limitations in the available test area.
For example, if the desired maximum angle for a given directional spectrum is °max = ± 600, and the total width of the test basin is 30 m, then the useful area for installation of models corresponds to the circle with 'a diameter of 8.0 m. However, if the maximum
wave angle for a satisfactory reproduction of the directional spectrum can be reduced to, say, 450 then the test area increases to 12.4 m.
The above considerations give a simple ihdication of the spacial limitations imposed by the maximum angles of the directional
di-stribution. Thus, according to the preceding section additional
on-ginates from, especially, diffraction. With these further limita-tions on the available model area it appears that a good quality of the reproduction of a directional spectrum requires a reason-ably wide test basin as well as rather small individual elements of the wave generator.
6.
DIRECTIONAL WAVE_SYNTHESIB_MODELSThe regular oblique waves and the sources of error have been
tho-roughly discussed above because an irregular directional wave field is essentially composed of regular components. Thus, the control signal in (6) can be extended to:
J a4. X(n,t)
=:E:
F sin(w.t
+k.xri
cose.j-k.(nB-Yr)sine..) (11)3(f'e)
i=1 j=1
where summation is made over I frequencies and J directions. There is no doubt that (11) will produce an irregular sea, but the major problem is to exercise control over the output, i.e. how to deter-mine the amplitudes a. , the I frequencies, the J direction and to
ij
Select the phases so that the desired wave field it obtained.
If the stochastic approach is considered (as briefly discussed in the Introduction), then the target would be defined in terms of a directional spectrum such as that in Fig. 2. Both the spectrum and the spreading function might actually be described by parameter-ized functions as mentioned in connection with (1). Commonly used Parameterized spreading functions are:
That is, a normal distribution characterized by a mean value, em, and a corresponding spread, a2, both as a function of frequency, and a cosine to the power of 2s, where the mean value and spread-ing are determined by the variables Om and s. Thus, a very narrow
D(f,0) and =
N(0,
02) = 1/(a7i) exp(-(0-Om)2/202) (12) - 2s D(f,0) =221
F2(s+1)/(7a(2s+1))cos((0- m)/2)
(13)distribution (small spread) is described by a small a2 or a large s. For high frequencies often characterized by a considerable spread the value of a2 is large (for the normal distribution), while the value of s has to be quite low (for the cosine
func-tion).
Note that for any spreading function the following must apply:
D(f
e)
de = 1 (14)This is used together with (1) to form the basis for the wave ge-neration. In order to take the step from here to a surface eleva-tion time series and to the required control signals for the wave generator a synthesis model is needed. As briefly outlined below
several different models have been presented in the literature over the past decade.
6.1 Double Summation Model
In synthesis of waves there are a number of general problems,
which arise no matter if the target spectrum is 2-D or 3-D. One of the basic decisions is whether to apply the probabilistic or the deterministic approach to synthesis. In addition, equidistant or non-equidistant frequency spacing, repetitive or non-repetitive
time series, etc, have to be decided upon. All these problems have been reviewed by Huntington (1986) and Funke (1986). Therefore the general aspects of wave synthesis will not be dealt with here, but only those directly related to the 3-D nature of the spectra will be discussed.
In one of the earliest synthesis models (presented by Borgman (1969)) the surface elevations are described as:
21
-n(x,Yft)
=Ii..cos(w
.t-k.xcose. -k.ysine .+jieij
)3.
a. i3
i=1 j=1where the amplitudes a.. are determined by:
a,, =
I2Sn(f-, -)A01Af I/2S(f)D(f-,0-)A0Af
(16)
1 3
The double summation in (15) is apparently carried out for I fre-quencies each of which has J directional components. The associ-ated amplitudes are, according to (16), determined as the square root of two times the area of the vertical column shown in Figure 2. The phase angles e.. are randomly selected from a uniform di-stribution U(0 1.).
Instead of (16), which is often referred to as the random phase method, Tucker et al (1984) has advocated for the use of a random
coefficient principle. In this the aij is considered a Gaussian distributed random variable with zero mean and variance S (f ).
TI
j
AfA0. The variance of the variance of the synthesized time series is claimed to be too small with the random phase method in (16), and more realistic (close to the theoretical value for a Gaussian process) with the random coefficient method.
The double summation model seems to be a rather obvious choice for representation of a natural directional sea. However, for practic-al applications it is important to note that only limited length of time series containing a limited number of frequencies are rea-listic due to computer capacity, testing and analysis costs, etc. Therefore, the implications of these facts on the results of the model are discussed below.
Consider the directional wave field in the point (x,y) = (0,0). Then (15) becomes
41(t)
=Ea.
.cos(w.t+e.,)- 1 1
i=1 j=1
Goda (1977) and (1981) showed that this can more efficiently be written as
n(t)
A. cos(wit) + g sin(wt) i (18) i=1 with A,.E1
- lj j=1 ]=1and B.
-Ea..
sine..1, 13
1]
The formulations in (18) and (19) are computationally less time
consuming than .(15).
For any model it must be required that the area, mo, under the spectral
density in Fig. 2
equals the variance of the wave eleva-tions. This fundamental identity can be written as:2
=E
(f-;,(9_)
A0Afn
i=1 j=1
Using (18) and (19) this turns into
I J n2 = 13 ik 13 i=1 j=1 k=1 (17) (19)
23
-This shows that when the double summation model is used for synthe-sis of a directional wave field the actually produced spectrum as well as the area
m0 under it will depend on the randotly selected phase angles. That is, each realization of the process will
pro-duce a new spectrum and mo. The reason for the phase dependent
spectral density is that J (directional) wave components are added up at the same frequency. Thus, the total energy is in this case not simply the sum of the squared contributions, but it depends on the relative phases.
The difficulties encountered with the double summation model have been reported by Forristall (1981), Pinkster (1984) and Lambrakos (1982). The latter pointed out that a simulation of surface eleva-tions in connection with the design of a pipeline showed signifi-cant variations in mo (and thus in signifisignifi-cant wave height, Hs) along the trace. An example of the possible variations in the mean energy over 1 km2 is shown in Fig. 8. It is representing a cos2e directional sea (with 36 components) at the period T = 10 s. Appa-rently the mean energy can vary with a factor of four within this limited area. 3.75 3.25 2.75 2.25 1.75 1.25 .75 .25 1.0 -.75 3.25 2.75 2..25 1.75 .75 .25 1.0
Fig. 8 Variation of mean energy in 1 kit? of a
cos2e
seaThe phase locking phenomenon giving the fluctuating rms of the
double summation model can appear not only as a result of the
syn-thesis. Hence, it is important to realize that the same effect emerge from reflecting walls in the basin. That is, an incoming
and a reflected wave at the same frequency will create a phase
.locked pattern with rms oscillations. This is what will happen
when the side walls are used as reflectors with the purpose of
extending the area of the directional wave field.
For comparison with (21) it is of interest to calculate the ensem-ble mean of a series of realizations produced by the douensem-ble summa-tion model:
I J
E {
n2) =
:E:
iaZ.ij
i=1 j=1
Since this expression does not equal the result obtained in (21) it can be concluded that the double summation model is not ergo-dic.
One way of
eliminating
the phase term in (21) is by averaging the realization, i.e. either over space or over neighbouring frequen-cies. Pinkster (1984) calculated the variance of the variance,i.e. Var{n2}, of records produced by the double summation model
and found that for N+03 it approached zero. Hence in this limit the model apparantly becomes ergodic.
The averaging over several realizations or over space might not be feasible in typical model tests with directional wave generation. Therefore the question is how many frequency components are needed in the synthesis to give a reasonable approximation of N-.. Based on actual simulations it has been shown that a few hundred is de-finitely insufficient. Stansberg (1986) found from numerical simu-lations that with 100.000 components the variability of the direc-tional spectrum was reduced to a reasonable level.
25
-6.2 Single Summation Model
To avoid the effects appearing in short time series generated by the double summation model it is important not to add up compo-nents with different directions at the same frequency. An obvious alternative is therefore
n
(xfIrft)
= a..cos(w.t-k.xcoseji.
i
-k ysine.j )i
i=1
+ bijsin(wit-kixcoseij-kiysineij)
At each frequency only one direction of propagatidin is chosen. As
opposed to the double summation model it takes a small band of
frequencies to describe a whole direCtional distribution. It is assumed that the single summation model works with rather narrow
frequency intervals so that a small band of frequencies fully co-' vers a directional distribution (defined by the average frequen-cy). The angular boxes
e,
may be drawn at random such that the full distribution will be statistically covered after a certain number of selections. With the single summation model the ampli-tudes (coefficients) are chosen asa! + = 2S (f)Af (24)
The phase angles (ratio of b. /a. ) should belong to a uniform
ij random distribution.
Again the alternative to the random phase method above (i.e. (24)) may be formed by the random coefficient scheme suggested by Tucker et al. (1984).
The selection of direction of propagation of a given wave
compo-nent
(frequency) can be made in several ways. One (implemented at PHI) is described below. As a basis for a random choice of thedirection en of the frequency fn the cumulative directional di-stribution for fn is established, cf. Fig. 9. The total
distribu-tion from eto
e will then be represented within the abcissamin max
values 0 to 1. Hence the direction of propagation of a given
b)
Fig. 9 Random selection of wave directions, given distribution at frequency fn cumulative function
quency
fn can be selected from a uniform random distribution
U(0.1), cf. Fig. 9 b). For a narrow directional distribution the cumulative function becomes rather steep around the 0.5 crossing
implying that more random numbers will draw values close to the mean, em.
In the process of assigning random directions to the individual
frequencies it should be noted that the spread (distribution) from which the angles are drawn (e.g. Fig. 9) should vary with the fre-quency, cf. the examples in (12) and (13). The trend confirmed by
many publications is that the spread seems to increase with the
frequency. The results of prototype measurements performed by DHI in the Baltic and the North Sea were also very identical in terms of spreading. On that basis the following relation has become part of the DHI synthesis software
2 2 = 1 2.5
(f/f )
a (f) s(f) +
These frequency dependent distributions have to be combined with (14) and a given maximum
angle ()max (cf. Fig. 7) for the
particu-lar test basin.
Usingemax = 600 and random number selection from
distributions corresponding to (25) lead to a scheme as shown in Table 1.
(25)
This gives the directions of propagation, en, for given s-parame-ter (horizontally) and given random number. Only the positive an-gles corresponding to probability 0.5 to 1 are shown; the negative half is identical (symmetrical around U = 0.5). For s = 1 - a very
wide distribution - it is seen that random numbers from 0.5 to
0.72 correspond to angles less than 250, whereas for s = 20 - a narrow distribution - angles less than 250 are drawn from 0.5 right up to 0.91. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.1 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 2.2 2.0 1.9 1.7 1.6 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1.0 1.0 1.0 1.0 0.9 0.9 3.3 3.0 2.8 2.6 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.7 1.6 1.6 1.5 1.5 1.4 1.4 1.4 4.4 4.0 3.7 3.5 3.3 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.1 2.0 2.0 1.9 1.9 1.8 5.5 5.0 4.7 4.4 4.1 3.8 3.6 3.4 3.3 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.4 2.3 2.2 6.6 6.1 5.6 5.2 4.9 4.6 4.4 4.1 3.9 3.2 3.6 3.5 3.3 3.2 3.1 3.0 2.9 2.9 2.8 2.7 7.7 7.1 6.6 6.1 5.7 5.4 5.1 4.8 4.6 4.4 42 4.0 3.9 3.3 3.6 3.5 3.4 3.3 3.3 3.2 8.8 8.1 7.5 7.0 6.5 6.2 5.8 5.5 5.3 5.0 4.8 4.6 4.5 4.3 4.2 4.0 3.9 3.8 3.7 3.6 9.9 9.1 8.4 7.9 7.4 6.9 6.6 6.2 5.9 5.7 5.4 5.2 5.0 4.9 4.7 4.6 4.4 4.3 4.2 4.1 11.0 10.1 9.4 8.8 8.2 7.7 7.3 6.9 6.6 6.3 6.0 5.8 5.6 5.4 5.2 5.1 4.9 4.8 4.7 4.6 12.1 11.2 10.3 9.6 9.0 8.5 8.0 7.6 7.2 7.0 6.7 6.4 6.2 6.0 5.8 5.6 5.4 5.5 5.2 5.0 13.2 12.2 11.3 10.5 9.9 9.3 8.8 8.4 8.0 7.6 7.3 7.0 6.8 6.5 6.3 6.1 5.9 5.8 5.6 5.5 14.3 13.2 12.3 11.4 10.7 10.1 9.6 9.1 8.6 3.3 7.9 7.6 7.3 7.1 6.9 6.7 6.5 6.3 6.1 6.0 15.1 14.3 13.2 12.4 11.6 10.9 10.3 9.8 9.3 2.9 8.6 2.2 7.9 7.7 7.4 7.2 7.0 6.8 6.6 6.4 16.6 15.3 14.2 13.3 12.4 11.7 11.1 10.5 10.0 9.6 9.2 8.8 8.5 8.2 8.0 7.7 7.5 7.3 7.1 6.9 11.7 16.4 15.2 14.2 13.3 12.5 11.9 11.3 10.7 10.3 9.8 9.0 9.1 8.8 8.0 8.3 8.0 7.8 7.6 7.4 18.8 17.4 16.2 15.1 14.2 13.4 12.6 12.0 11.4 10.9 10.5 10.1 9.7 9.4 9.1 8.8 8.6 8.2 8.1 7.9 19.9 18.5 17.2 16.1 15.1 14.2 13.4 12.8 12.2 11.6 11.2 10.7 10.3 10.0 9.7 9.4 9.1 8.9 8.6 8.4 21.1 19.5 18.2 17.0 16.0 15.0 14.2 13.5 12.9 12.3 11.8 11.4 11.0 10.6 10.2 9.3 9.7 9.4 9.1 8.9 22.2 20.6 19.2 17.9 16.9 15.9 15.0 14.3 13.6 13.0 12.5 12.0 11.6 11.2 10.8 10.5 10.; 9.9 9.7 9.4 23.3 21.7 20.2 18.9 17.3 16.8 15.9 15.1 14.4 13.8 13.; 12.7 12.; 11.8 11.4 11.1 10.8 10.5 10.2 10.0 24.5 22.8 21.2 19.9 18.7 17.6 16.7 15.9 15.1 14.5 13.9 13.4 12.9 12.4 12.0 11.7 11.3 11.0 10.7 10.5 25.6 23.9 22.3 20.9 19.6 18.5 17.6 16.7 15.9 15.2 14.6 14.0 13.5 13.1 12.7 12.3 11.9 11.6 11.3 11.0 26.8 25.0 23.3 21.9 20.6 19.4 18.4 17.5 16.7 16.0 15.3 14.7 14.2 12.7 13.3 12.9 12.5 12.2 11.9 11.6 28.0 26.1 24.4 22.9 21.5 20.4 19.3 18.3 17.5 16.7 16.1 15.4 14.9 14.4 13.9 13.5 13.1 12.2 12.4 12.1 29.1 27.2 25.5 23.9 22.5 21.3 20.2 19.2 18.3 17.5 16.8 16.2 15.6 15.1 14.6 14.1 13.7 13.4 13.0 12.7 30.3 28.3 26.6 25.0 23.5 22.2 21.1 20.1 19.1 18.3 17.6 16.9 16.3 15.8 15.3 14.8 14.4 14.0 13.6 13.3 31.5 29.5 27.7 26.0 24.5 23.2 22.0 20.9 20.0 19.1 18.4 17.7 17.0 16.5 15.9 15.5 15.0 14.6 14.2 13.9 32.7 30.6 28.8 27.1 25.6 24.2 23.0 21.9 20.9 20.0 19.2 18.4 17.8 17.2 16.6 16.1 15.7 15.3 14.9 14.5 33.9 31.8 29.9 28.2 266232 23.9 22.8 21.8 20.8 20.0 19.2 18.6 17.9 17.4 16.8 16.4 15.9 15.5151 35.1 33.0 31.1 29.3 27.7 26.; 24.9 23.7 22.7 21.7 20.8 20.1 19.4 18.7 18.1 17.6 17.1 16.6 16.2 15.8 36.3 34.2 32.3 30.5 28.8 27.3 26.0 24.7 23.6 22.6 21.7 20.9 20.2 19.5 18.9 18.3 17.8 17.2 16.9 16.4 37.5 35.4 33.5 31.6 29.9 28.4 27.0 25.7 24.6 23.6 22.6 21.8 21.0 20.3 19.7 19.1 18.5 18.0 17.6 17.1 38.7 16.7 34.7 32.8 31.1 29.5 23.1 26.8 25.6 24.5 23.6 22.7 21.9 21.2 20.5 13.9 15.3 18.8 13.3 17.9 39.9 37.9 35.9 34.1 32.3 30.7 29.2 27.9 26.7 25.6 24.G 23.6 22.8 22.0 21.4 20.7 20.1 1961q1 18.6 41.2 39.2 37.2 35.3 33.6 31.9 30.4 29.0 27.8 26.6 25.6 24.6 23.8 23.0 22.2 21.6 21.0 20.4 19.9 19.4 42.4 40.5 38.5 36.6 34.8 33.2 31.6 30.2 28.9 27.7 25.6 25.7 24.8 23.9 23.2 22.5 21.9 21.3 20.7 20.2 43.7 41.8 39.8 38.0 36.2 34.5 32.9 31.4 31.1 28.9 27.8 26.7 25.8 24.9 24.2 23.5 22.8 22.2 21.6 21.1 45.0 43.1 41.2 39.3 37.5 35.8 34.2 32.7 31.3 30.1 28.9 27.9 26.9 26.1 25.2 24.5 22.8 23.1 22.5 22.0 46.3 44.5 42.6 40.8 39.0 37.; 25.6 34.1 32.7 31.4 30.2 29.1 29.1 27.2 26.3 25.6 24.8 24.2 23.6 23.0 47.6 45.9 44.1 42.3 40.5 38.7 37.1 35.5 34.1 32.8 31.5 30.4 29.4 28.4 27.5 26.7 26.0 25.3 24.6 24.1 48.9 47.3 45.6 43.8 42.1 40.3 38.7 37.1 35.6 34.3 33.0 31.8 30.7 297 23.3 28.0 27.2 26.5 25.2 25.2 50.2 48.7 47.1 45.4 43.7 42.0 40.4 28.3 37.3 35.9 34.633.4 32.3 31.2 30.3 29.4 22.5 27.3 27.1 26.4 51.6 50.2 48.7 47.1 45.5 43.8 42.2 40.6 39.1 37.7 36.3 25.133.9 32.8 31.8 10.9 20.1 29.2 28.5 27.9 52.9 51.7 50.4 48.9 47.4 45.8 44.2 42.6 41.1 39.6 23.3 37.0 35.8 34.7 33.6 32.7 21.7 23.9 30.2 29.4 54.3 53.3 52.1 50.8 49.4 47.9 46.4 44.9 43.4 41.9 40.5 39.2 17.9 36.8 25.7 24.7 33.3 32.9 22.1 31.2 55.7 54.9 53.9 52.2 51.6 50.3 49.9 47.5 46.1 44.6 43.3 41.9 40.6 39.4 23.3 30.2 36.2 25.3 24.4 33.6 57.1 56.5 55.8 55.1 54.1 53.1 51.9 50.6 49.3 47.9 46.6 45.3 44.0 42.7 41.6 40.5 39.4 30.4 37.5 36.6 58.6 58.2 57.9 57.4 56.8 56.2 55.5 54.6 53.6 52.5 51.4 50.3 49./ 47.3 46.6 45.5 44.4 43.4 42.4 41.4 60.0 60.0 60.0 60.0 c0.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 63.0 60.0 61.0 60.r, A0.n 0.86 0.87 0.83 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.'48 0.59 1.00 27 -9 10 11 12 11 14 lc 16 17 12 19 20
Table 1 Example of random selection of directions of propagation with 0max = 60o and 1 < s(f) < 20.
1 2 1 4 5 6 0.50 0.51 0.52 0.52 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.6, 0.64 P. 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0./8 0.79 0.80 0.8] 0.82 0.83 0.84 0.85
If calculations similar to those in (21) and (22) are carried out
for the single summation model it readily appears that this is ergodic.
For practical use of any of the synthesis models it is important
to include a reasonably large number of frequencies (Long time series). This is to ensure that a given response function (e.g. of
a floating structure) is well covered by wave components in the
directional as well as the frequency domain. In the case of very narrow response functions an optimal coverage of the frequency and directional intervals in the synthesis can be obtained by a sub-series technique.
Say that the total length of the synthesized time series has to be 1 hour. This may be composed of 10 subseries each 6 min. long. In
the first subseries the frequencies
fn, fm, . and associated directions en, em, ... are determined according to the above pro-cedure. In the following subseries, the frequencies and directions are allowed to change slightly in order to cover "in between" va-lues, cf. Fig. 10. That is, if the frequency spacing is pf = fn
fm' then the new frequency f' is chosen as
f' = fn + U(-f/2, Af/2) (26)
and the associated direction of travel is
et
=e
+ U(-61cn n
in which ec is chosen to e.g. a 50 interval.
- 29
Fig. 10. Illustration of area covered by varying frequencies and directions in a number of subseries.
In this way frequencies and directions are determined for all of the remaining subseries. The concatenated 1 hour record will then form a rather smooth directional spectral density instead of the traditional comb picture.
Obviously discontinuities between one subseries and the next can-not be allowed. The change of frequency and direction is therefore
handled smoothly by a phase shift in the transition between the
subseries. That is, the following phase relation is established for the time of the transition tt
2wf t -knxcosen - k- ysine + e =
n t n n n
2wf't -k xcose'n t - knysine' + el
n n
From this, the new phase e
of
the component f, 0;1 in thesubse-quent subSeries is derived.
The expense of applying the subseries technique with varying
fre-quencies and directions is that the FFT pattern is broken up so
that conventional (slower) sine/cosine computations have to be
made.
The comparison of the single and the double summation models shows that the basic differences lie in the variance of the variance of the synthesized records. In addition, the synthesis technique
as-sociated with either the random phase or the random coefficient principle also produces a different variance of the variance.
Which level to choose is a basic 2-D synthesis question as well.
It seems that for determination of the response functions of
floating structures by means of physical model tests a well-de-fined spectral input is important - thus emphasizing the single
summation and the random phase method. In other cases where mainly statistical properties of the waves and the responses have to be
examined the double summation model and the random coefficient
method may be preferable. However, all aspects of the two catego-ries of synthesis and their relation to practical applications are
31
-6.3 Diffraction Model
The synthesis .models discussed above are all based on the socalled snake' principle. This implies that all wave generator elements contribute to every individual oblique wave component, cf. Fig. 4.
These are then linearly added up to form the short crested sea. However, it is possible to use the individual generator elements
in an alternative way for the reproduction of a given directional spectrum. Naeser (1979) suggested to use the natural diffraction of waves from sources formed by the small generator elements.
If just one of the elements in the straight line of generators is moving then diffraction diagrammes (traditionally used for break-water gaps) give the following approximate wave height distribu-tion in the basin:
H B H 7:5-c coss/ 2e ; x>2B1 > L in which 3 s = 2 (T, + 0.6) ; and
H0 is the wave height close to the paddle (also the height that would exist all through a wave flume of width B). When using this diffractional principle of wave generation it appears that
the spread of the waves will follow a cosine distribution. The
narrowness of this distribution is controlled by the ratio B/L.
The typical tendency in natural short crested seas is that the
directional spread seems to increase with the frequency. Hence, for shorter wave lengths the parameter s should decrease. For a constant B in (30) the exact opposite is the case. Therefore, the number of paddles 'playing' together has to be varied as a func-tion of frequency.
For practical use of the diffraction method it should be noted
that reflecting side walls, for instance, will disturb the expect-ed diffraction pattern significantly. Furthermore, there are a
number of restrictions on the applicable B/L ratio for which the wave-conditions will be sufficiently stable. For these reasons the method is rather Complicated to work with, and it seems only
33
-7. DETERMINISTIC WAVE FIELD REPRODUCTION
The word 'deterministic' is used here in the sense that not only statistical properties (i.e. directional wave spectra) are repro-duced in the model but also a set of specified phases. Hence typi-cally the task is to reproduce time series recorded in nature, and
here it is aimed at modelling a natural wave field in the test
basin. This has also been discussed in the Introduction.
Deterministic 2-D reproductions appeared in early 1970'ies as a consequence of insufficient understanding of natural waves. The basic philosophy was that instead of assumed statistical models it seemed more 'safe' to reproduce as accurately as possible a mea-sured time series from nature. Wave characteristics not yet fully. understood may in this way have been successfully transferred to
the models At todays level of 3-D wave research many directional properties are far from understood. To represent special phase and directional relations in a wave field a deterministic reproduction may in some situations be desirable.
As mentioned in the Introduction the reproduction of a certain
measured wave field requires a very close link to the directional analysis, cf. Fig. 1. One deterministic package of analysis and reproduction has been introduced by Sand (1979) for a set of three simultaneous time series, viz, elevations and two components of horizontal orbital velocities. The theory has also shown to be applicable to pitch-and-roll buoys, see Sand (1984).
The method of analysis is related to.a single summation assumption of the structure of the wave field. It basically works in Fourier coefficients of the recorded time series. It preserves the phase information, and furthermore it does not introduce any assumptions of the shape and boundaries of the directional distributions.
The theory has been tested at the Danish Hydraulic Institute. In a central point of the basin a wave gauge and a two component cur-rent meter were set up. The u-velocity was orientated along the
mean direction of the waves, whereas the v-velocity was parallel
to the line of generators. During a storm in the North Sea the
same three quantities had been measured. The quality of the repro-duction with the linear deterministic theory is shown in Fig. 11.
6
423
0 see
Fig. 11 Deterministic reproduction of a North Sea record compris-ing elevations and two components of horizontal veloci-ties (Aage and Sand (1984)).
Ai
-Tiffiy!
,ILI
i
ATr
I A i i
ty wr!
11
itiA
Y!
;
4ii
.iikL
i
i
ii
1, 1 4 14i
LA.
A ir7
;T!
1 Y ! To v ! ... A 41 A .i.V
T '
vv
iv 4, city, Aw. AI,360 420 480 5.40 - BE
0 480 540 600 sec
yELOCITy. u ( ). MEASURED IN CENTRAL POINT OF BASIN ( 3 - 0)
0E3114E0 PROTOTyPe vELOCITy u ( t )
480 540 600 sec
yELOC:Ty. v( t) MEASuREO IN CENTRAL POINT OF BASIN ( 3-0)
DESIRED PROTOTYPE yELOciry. ( t )
- 3-0 ELEVATIONS MEASURED IN SELECTED POINT OF BASIN
rn 3-0 DESIRED PROTOTYPE ELEVATIONS
6 1(t) 0 rris 2
u(t)
2
m Is 2 v(t) 02
35
-So far the method has proved very useful for comparative 2-D and
3-D reproductions of rather short records. This has formed the
basis for investigations of the pure directional effects of waves on the response of floating structures.
8. OTHER SYNTHESIS METHODS
For the synthesis of directional spectra several other methods
exist. Some are more or less based on extensions of similar 2-D principles. An example is the white noise technique. A band of
white noise is filtered by a function formed by the shape of the spectral density. In the 3-D version this technique is, of course, rather complex.
The use of filters has
been suggested by Bryden and Greated(1984). For an input white noise source and a desired directional spectrum they ,derived a two dimensional digital filter, one axis being time and the other space so as to give the individual wave generator displacements along the straight line. However, the amount of computer time needed for this approach is considerable.
37
-9. CONCLUSIONS
The problems of directional wave generation an be divided into
two groups. One is related to the physical set up of the genera-tors and to the boundaries of the test facility. The other is di rectly associated with the method of synthesis of the spectrum.
It has been shown that the errors introduced in the generation of a simple regular oblique wave can be considerable. For instance, the diffraction effect often has a significant influence on the resulting wave height. Other effects to be taken into account in
the planning of tests and measurements have been outlined. A
simple method for determination of the optimal test area has been
presented. It has been shown that to obtain a reasonable large area and to reduce the various disturbances a rather wide test facility is required.
As regards the synthesis models much research is still needed to reach a satisfactory reproduction technique. Many of the models presently used possess evident deficiencies.
10. REFERENCES
Aage, C. and Sand, S.E. (1984)."Design and
construction
of the DUI3-D wave basin". Proc. Symp. Description and Modelling of
Directional Seas, Copenhagen, paper B-2, 2Opp.
Biesel, F. (1951)."Etude Th6orique d'un Type d'Appareil a Houle". La Houille Blanche, 6°
ann.,
No.2, pp 152-165.Borgman, L.E. (1969)."Ocean wave simulation for engineering de-sign". Journ. Waterways and Harbours Div., Proc. ASCE, Vol. 95, No. WW4, pp. 557-583.
Bryden, I.G. and Greated (1984)."Generation of three-dimensional random waves". Journ. Applied Physics, Vol. 17, pp. 2351-2366.
Forristall, G.Z. (1981). Kinematics of directionally spread
waves". Proc. Conf. Directional Wave Spectra Applications, Univ. of Berkeley, pp. 129-146.
Funke, E. (1986)."Deterministic approach to wave generation". IAHR working group on wave generation and analysis.
Goda, Y. (1977)."Numerical experiments on statistical variability Of ocean waves". Rept. Port and Harbour Res. Inst., Vol. 16, No. 2, pp. 3-26.
Goda, Y. (1981). Simulation in
examination
of directionalresolu-tion". Proc. Conf. Directional Wave Spectra Applications,
Univ. of Berkeley, pp. 387-407.
Huntington,
S.M. (1986)."Probabilistic approach to wave genera-tion". IAHR working group on wave generation and analysis.39
-Huntington, S.M. and Thompson (1976)."Forces on a large vertical cylinder
in
multi-directional random waves". Proc. OffshOre Techn. Conf..; 0TC2539, Dallas, pp. 169=.183.Jefferys, E.R. (1986)."Directional seas should be ergodic". Proc.
Conf. Offshore Mechanics and Arctic Engng., °MAE, Tokyo,
llpp.
Lambrakos, K.F. (1982)."Marine Pipeline Dynamic Response to Waves froMDIrectional Wave Spectra". Journ. ()dean Eng04., VOl. 9, No. 4, pp. 385-405.
Naeser, H. (1979)."Generation of uniform directional spectra in a
wave basin using the natural diffraction of waves". Proc.
Port and Ocean Engng. under Arctic Conditions, POAC, Norway, pp. 621-632.
Pinkster, J.A. (1984)."Numerical Modelling of Directional Seas". Proc. Symp. on Description. and Modelling of Directional
Seas, Techn. Univ. of Denmark, Copenhagen.
Salter, H. (1984)."Physical modelling of directional seas". Proc. Symp. Description and Modelling of Directional Seas, Copen-hagen, paper B-1, 9pp.
Sand, S.E. (1979)."Three-dimensional
deterministic Structure of Ocean Waves". Inst. Hydrodyn. and Hydraulic Engng., Techn. Univ. of Denmark, Series paper No. 24., 189 pp.
Sand, S.E. (1984)."Deterministic Decomposition of Pitch-and-Roll Buoy Measurements". Journ. Coastal Engng., Vol. 8, pp. 243-263.
Stansberg, C.T. (1986) ."Statistical properties of directional sea measurements". Proc. Conf. Offshore Mechanics and
Arctic Engng., OMAE,'Tokyo, 24pp.
Tucker, M.J., Challenor, P.G., and Carter, D.J.T. (1984).
"Numeri-cal simulation of 'a random sea: a common error and its
