Fluid Accelerations Under Irregural Waves

ELSEVIER

Applied Ocean Research 17 (1995) 43-54 Elsevier Science Limited

Printed in Great Britain

0141-1187(94)00019-0 0141-1187/95/SO9.50

Fluid accelerations under irregular waves

Jeffrey A. Zelt,* Ove T. Gudmestad t & James E. Skjelbreia*

*Wave Te f

hnologies, Kjopmannsgata 59, 7011 Trondheim, Norway Statoil as, Box 300, 4001 Stavanger, Norway

(Received 13 May 1993; revised version received and accepted 11 September 1994)

With reference to theoretical models for the prediction of fluid accelerations under irregular waves, comparisons with measured data are reported.

Numerical techniques have been developed to filter and differentiate laboratory velocity data collected under irregular waves in order to compute estimates of the local fluid accelerations. Algorithms for evaluating several acceleration theories are furthermore presented. The algorithms described are thereafter applied to laboratory data. Conclusions and recommendations for future studies are presented in the final section of the paper

1 INTRODUCTION

Particular attention has been focused during recent years on estimating wave velocities under irregular waves in order to improve the input to the Morison formula for the calculation of wave loads on slender offshore structures. For such structures, which are dominated by the drag loading term, accurate estimation of inertia loading has been given less emphasis.

For large volume structures which are dominated by the inertia loading term proportional to the fluid acceleration under irregular waves, the most accurate values of the fluid acceleration should be used. For such structures (e.g. tension leg platforms and gravity base structures), dynamic transient response through reso- nances with higher order acceleration terms can be an important phenomena.

It is, however, very difficult to carry out measurements of water particle accelerations under irregular waves, and it is therefore suggested that measured velocity data be used for the calculation of acceleration values. Very accurate measurements of wave velocities and associated wave surface elevations in a wave flume using Laser Doppler Velocimetry equipment (LDV) have been carried out by Skjelbreia2 and Skjelbreia et LZ~.~‘~ These data have been used for the accurate estimation of fluid acceleration through the use of a newly developed algorithm. Particular emphasis has been put on a stretching model.5

It is believed that this research represents the first attempt to compare ‘measured’ acceleration values and theoretical predictions of fluid acceleration under irregular waves.

43

2 THEORETICAL MODELS FOR PREDICTION OF FLUID ACCELERATIONS UNDER IRREGULAR WAVES

2.1 General

The latest information about wave velocities under irregular waves”3’4 indicates that application of the Wheeler stretching method gives a good fit to measured wave velocities. There is furthermore a need to compare theoretical values of fluid accelerations with estimates of fluid acceleration based on measurements.

Before presenting various wave theories, it is necessary to define some nomenclature. Let the time history of the surface elevation recorded at a fixed location be denoted by n(t). Let the horizontal and vertical velocities, and their local time derivatives measured at an elevation z be denoted by u(z, t), w(z, t), ti(z, t) and +(z, t), respectively. The still water surface corresponds to z = 0 and the bottom corresponds to z = -d. Let the Fourier transforms of these time series be denoted by H(w), U(z,w), W(z,w), U(z, w) and W(z, w), respectively.

Each method evaluated for computing the local fluid accelerations represents different relations between H(w), the Fourier transform of the surface elevation, and the Fourier transforms for the velocities and accelerations. For simplicity, these relations will in this paper be presented as if the frequency w was continuous. In practice, all data evaluated are sampled and the FFT algorithm is used to compute the DFT approximation to the Fourier transform of the variable.

44 J. A. Zelt, 0. T. Gudmestad, J. E. Skjelbreia

2.2 Linear wave theory

Linear dispersive wave theory yields6

U(z, w) = f “““,“,ffk; d, H(w)

W(z, w) = -is sinfoifkl d, H(w) (1)

for the fluid velocities, and

o(z, w) = -igk

fi(z, w) = -gk SiZoffk;d) H(w) (2)

for the fluid accelerations, where

w2 = gk tanh kd

2.3 Wheeler stretching applied to linear acceleration equations

Wheeler’ described the motivation of replacing the vertical coordinate z in eqn (1) by the expression

z - rl(t)

1 + rl(t)ld (3)

If, instead, this expression is used with the formulae for the accelerations, eqn (2), the result is

ri(z,w) = -igk ‘Osh k( 1 :;;,d) H(w)

cash kd

ti(z, w) = -gk sinh k( 1 :;;,d) H(w) (4)

cash kd

This is difficult to compute in its present form since n is a function of t; Fourier techniques cannot be applied directly. However, with the change of variable

c= z - 44 1 + 44/d

(5)

then eqn (4) formally reduces to the linear expressions with z replaced by C:

i?(C, w) = -igk

sinh k(c + d) w(&w) = -gk cosh kd H(w)

(6)

Inverting these Fourier transforms yields zi([, t) and &(C, t). These are the accelerations evaluated at a time

varying z elevation (through r](t)). An example of such time varying z elevations is shown in Fig. 1. In order to obtain ti(z, t) and ti(z, t), i.e. the accelerations at a tied elevation (e.g. where LDV measurements are recorded), zi(<, t) and ti(C, t) can be evaluated for several values of C that bracket the z value of interest. A simple inter- polation between the nearest C values can then be performed to obtain zi and ti at the required z value.

2.4 Numerical differentiation of the Wbeeler velocities

The ‘Wheeler’ velocities, weeter, wmeeier are determined from the expression that is obtained by substituting eqn (3) for z into eqn (1):

u gk ‘Osh k(l ;;;,d) H(w) wlleeler(Z, 4 = -

W cash kd

W gk sinh k(l :;t;,d) H(w)

~h~i&w) = -i- W cash kd

(7) The same < algorithm as described above for accelerations can be applied to these expressions to obtain the velocities at a fixed elevation.

Analytical expressions for the time derivatives of

Wheelers WWheelet based on eqn (7) are difficult to evaluate. However, local acceleration estimates can be obtained by numerically differentiating weter, wwhceler with respect to time using the algorithm described in Section 3.1.

3 NUMERICAL ALGORITHM

3.1 Numerical differentiation

An algorithm was developed to compute the numerical derivative of the sampled data. It is used to locate wave crests and troughs, as described in Section 3.2. In

I I I I

tl \ -I

I I I I I

time Fig. 1. Time varying elevations C1 (t) and G(t).

Fluid accelerations under irregular waves 45

Section 4 it is used to compute local fluid accelerations under irregular waves from laboratory measurements of the fluid velocity. It is also used to differentiate theoretical values of the fluid velocity computed using the ‘Wheeler’ approximation, as described in Section 2.4.

The algorithm can be divided into seven steps. These steps are described first, and then illustrated below for two cases presented in Figs 2 and 3 and in Figs 4 and 5.

‘Detrend’ the data with a quadratic polynomial using a singular value decomposition (SVD) algorithm. This polynomial is then subtracted from the sampled data to leave a ‘residual’ signal. Extend the residual signal with a cubic polynomial into a fictitious padding region to render the signal and its derivative continuous at the boundaries. Without this step, the energy is falsely transferred into higher frequency Fourier components that are subsequently filtered in steps 4 and 5. Compute the discrete Fourier transform (DFT) of the detrended, padded data using a standard fast Fourier transform algorithm (FFT). Apply a window to remove the energy in the high frequency region associated with various noise sources (electrical, mechanical vibration, etc.) that represent a small fraction of the total energy, but which can contribute significantly to the first derivative of the data. The cumulative energy distribution is computed for the raw data and energy associated with the high frequency tail that accounts for a small percentage of the total energy is removed. For this investigation a fraction of 0.2% was used, but the results are quite insensitive to its choice. This step effectively locates the upper frequency bound associated with wave induced motion. Energy associated with frequencies above this value represents noise in the system. For the data treated in this investigation, this procedure also removed all the energy in the capillary wave region as well as all the energy associated with frequencies greater than that resolved by four data points. The cut off frequency using this algorithm

__~________~____________--____________----~-___ I 1 I 1 I I I I ~ I I

Fig. 2. Numerical differentiation test for ‘long’ data segments - data values.

1.5, I I I I I I II

1.0 i -1

0.5

b-+ 'S s:

0.0

-0.5

-1.0

t

t/ ‘i/

-1.51

I I I I I I I I [

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 3. Numerical differentiation test for ‘long’ data segments - derivatives.

is not fixed; however, for the data from case i 18.1 6,2-4 the cut off frequency was approximately 2-l Hz. The sample time step was At = 0.025 s. Apply a window to the DFT to differentiate the data. Let i$ represent the discretized Fourier components of a functionS(t), i.e.

f(t) = C qe’q’ (8) i

Hence, the appropriate window is simply iwj. Invert the DFT to obtain the filtered, differentiated and detrended data. Compute the analytical derivative of the detrend- ing function and add it to the results of step 6 to obtain the final estimate for the derivative of the raw sampled data series.

Examples are presented for both a ‘long’ data series and a ‘short’ data series to illustrate the effectiveness of the algorithm regardless of the number of points in the data segment. A ‘long’ data series with noise was generated artificially by taking a segment of a sine wave (not an integral number of periods) and adding to this a random noise signal. The noise is represented by the open circles in Fig. 2, and the composite noisy signal (raw data) is represented by the filled circles in the same figure. The parabolic trend is represented by the short-dashed line. The residual is obtained by subtracting this trend from

-O.~~o-----d~~_~~_-~~b-----r~s-----2~~-----~~S-----!/ 3.0

Fig. 4. Numerical differentiation test for ‘short’ data segments - data values.


-0.2. I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 5. Numerical differentiation test for ‘short’ data segments - derivatives.

the raw data. It is represented by the long-dashed line. The padding region is also shown and it can be seen that the periodic padded residual signal and its derivative are continuous at the boundaries. It is the DFT of this signal that is low-pass filtered and differentiated as described in steps 4 and 5 above. The results from the differentiation algorithm are presented in Fig. 3. The derivative of the segment of the sine wave should be another sine wave with an amplitude of exactly 1. This is reproduced very closely by the algorithm, with only small errors at the boundaries.

Figures 4 and 5 present similar results for a ‘short’ data segment. For this case, a cubic polynomial was evaluated at five points to provide the raw data. The ‘actual’ derivative in Fig. 5 is the analytic derivative of the cubic polynomial used to generate the raw data. It can be seen here that the residual signal is much smaller relative to the original raw data than for the previous case.

In general, step 1 provides much of the low-pass filtering and step 7 provides the largest contribution to the derivative for very short data segments. In that case the treatment in Fourier space may contribute very little to the overall result. However, for longer data segments with an oscillatory nature, the treatment in Fourier space provides the main contribution to the final result.

3.2 Locating wave crests and troughs

In Section 4, results from the various wave theories are compared with the accelerations obtained from LDV data.* It is useful for engineering purposes to make these comparisons only at certain ‘extreme’ points. Here, four types of points were chosen: wave crests, troughs, leading edge slope extrema, and trailing edge slope extrema. Although the concept of a wave crest and trough is simple enough to imagine, their precise identification is not straightforward from wave elevation time series such as that used in this study. Crests and troughs for uni-directional waves are generally associated with points on the wave surface with a zero

surface slope. However, a fixed wave gauge provides a time series of the surface elevation at a fixed location, and the precise identification of such crests and troughs is not strictly possible in theory. They do not correspond to points where the time derivative of the wave gauge elevation is zero. To illustrate this problem, consider a tini-directional wave:

(9)

then

aq . ax=-” c k.A .,i(wjt-+)

J J j

(10)

Crests and troughs in the wave gauge time records correspond to times when an/ax = 0. At these times it will not generally be true that dq/dt = 0 unless k is proportional to w (non-dispersive waves) or the wave has a single frequency component.

Locating crests and troughs for bi-directional waves is even more difficult. For the laboratory data treated in this report, waves were generated in a wave channel with a wave absorber located at the opposite end from the generator. All wave absorbers must reflect at least some energy and therefore the waves observed consisted of two oppositely travelling wave systems. Let:

= C aRjei(wjf-k,x) + C aLjei(ujf+k,)

i i (11)

Then

2 = i C wjaR,ei(wPk,x) + i C WjaLjei(qf+$x)

1 J

aq (12)

ax=--I c kjaRj&“i’-kP) + i c kjaL,ei(uj’+kjX)

i i

For this case (e.g. consider a standing wave), the distinction between dq/dx = 0 and dq/dt = 0 is even more obvious.

The importance of this distinction was investigated. The decomposition algorithm of Zelt & Skjelbreia7 was applied to wave gauge data collected for the present investigation and the reflected wave was confirmed to be very small. Therefore, the simpler case of eqns (9) and (10) can be assumed instead of the more complex case of eqns (11) and (12).

Moreover, the approximation that aq/ax = 0 at times when dq/dt = 0 is reasonable here. To show this, wave crests and troughs were located by both methods. In


addition, the extreme points of dq/dx and aq/at were also located. In order to compute aq/ax using eqn (10) the linear dispersion relation U* = gk tanh kd was used. The results for a representative segment of a wave gauge record are presented in Figs 6 and 7.

As can be seen, the results are very similar for both cases. The method based on surface slope located an extra crest and trough not seen by the method based on the surface elevation time derivative, but the differences are generally very small. Therefore, all extreme point locations were obtained for this investigation from the surface elevation time derivative.

4 RESULTS FROM LABORATORY DATA

4.1 General

The algorithms described in Section 2 have been applied to selected data segments described by Skjelbreia* (see also Skjelbreia et aL314). These results from case i18 (Hs = 0.21 m, Tp = 1.8 s) are presented to document the feasibility of computing local accelerations from wave gauge and/or LDV data and to present preliminary results rather than representing an exhaustive study of accelerations induced by irregular waves.

The velocities and acceleration values for each theory were extracted at the extreme points described in Section 3.2. These values are plotted against the quantities obtained from the LDV data at the same times to obtain a sort of ‘scatter’ plot. Good agreement between the theoretical value and the LDV-based value results in a point plotted near a line of unity slope. Although this technique provides valuable data for engineering purposes, it must be noted that it suffers from one difficulty in its present form. The extreme points are determined from the surface elevation record, and therefore each extreme point is associated with a unique time value, i.e. each point plotted on a scatter plot represents one point from a theoretical time record and one point at the same time from an LDV time record.

0.2, I I I I ,;0.6

-0.1 - ,Y - io.4

- iO.6

-0.2 I 1 I I LO.6 14 16 16 20 22 24

time [s]

Fig. 6. Wave extreme points based on surface elevation time derivative.

-0.2 1 I I I I LO.2 14 16 18 20 22 24

time [s]

Fig. 7. Wave extreme points based on surface elevation spatial derivative.

However, there are, of course, slight phase differences between the various theories and the LDV-based data records. These phase shifts are of little importance for engineering purposes since they do not influence the peak values nor the overall shape of the acceleration time records. However, these phase differences are responsible for apparent discrepancies between the two quantities being compared, especially if they are changing rapidly at the time they are compared. A better technique to avoid this problem is discussed in the conclusions and recommendations presented in Section 5.

In Section 4.4, short segments of sample time records are also presented to give an indication of the accuracy of the various techniques in the time domain.

4.2 Accelerations below the still water line

A segment of data from case i18.162-4 was processed to provide data for the case where the LDV measurements were below the still water line. For this case the water depth was d = 1.302m, the LDV measurement location was z = -0.25m, and the sample time step was At = 0.025 s.

Figures 8- 11 contain the scatter plots that compare the various theories with the LDV measurements. Figure 8 presents data values at times associated with wave crests. As expected, the horizontal velocity values are large, but the horizontal acceleration values are small. The linear theory tends to over-predict the horizontal velocity, whereas Wheeler theory under- predicts these values. As expected, the vertical accelerations are negative. It appears that the differentiated Wheeler velocity theory gives slightly better results than the ‘Wheeler acceleration’ theory for the vertical accelerations. The vertical velocity values are small, although not negligible. This may be due to the phase difference problem mentioned above, or to small errors in locating the crest times themselves.

Figure 9 presents data values at times associated with wave troughs. The data here are roughly a mirror image of those seen at the wave crests, except that the


-0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2

&Bt [gl- LDV dw/at [g] - LDV

Fig. g. Measurements of velocities and accelerations under wave crests. Case i18.16. LDV positioned at .Z = -0.25 m.

linear theory under-predicts the horizontal velocity here, results than the ‘Wheeler acceleration’ theory for the whereas Wheeler theory over-predicts these values. The vertical accelerations. vertical velocity is consistently small relative to the Figure 10 presents data values at times associated horizontal velocity. Again, it appears that the differ- with leading edge slope extrema (maximum values of entiated Wheeler velocity theory gives slightly better l@/atl associated with the leading edge of the

-0.4 -0.2 0.0 0.2 0.4

-0.2 -0.1 0.0 0.1 0.2

-0.4 -0.2 0.0 0.2 0.4

WZt [g] - LDV aWlat (g] - LDV

Fig. 9. Measurements of velocities and accelerations under wave troughs. Case i18.16. LDV positioned at z = -0.25 m.


-0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2

au/at [g] - LDV aWBt [g] - LDV

Fig. 10. Measurements of velocities and accelerations under leading edge slope extrema. Case i18.16. LDV positioned at z = -0.25m.

approaching waves, assuming that the waves are moving passing waves). As expected, the vertical velocities are to the right) and Fig. 11 presents data values at times large and positive for the leading edge, and large and associated with trailing edge slope extrema (maximum negative for the trailing edge. It might be expected that values of l&j/&l associated with the trailing edge of the vertical accelerations should be small for both cases;

-0.2 -0.1 0.0 0.1 0.2

-0.4 -0.2 0.0 0.2 0.4

-0.2 -0.1 0.0 0.1 0.2

adt [g]-LDV &t/at [gj - LDV

Fig. 11. Measurements of velocities and accelerations under trailing edge slope extrema. Case i18.16. LDV positioned at z = -0.25m.


however, they appear to have significant values and are negative for both cases. This can be attributed to the asymmetric nature of the surface elevation wave pattern for nonlinear waves. As the wave steepness increases for narrow band spectra, the individual wave crests become more pointed and the troughs more flattened. As a result, the extreme points of lLFlq/atl are located above the still water line where the vertical acceleration is negative. This can be seen easily for the third, fifth, and last crests in Fig. 6. As expected, the horizontal accelerations are positive at the leading edge and negative at the trailing edge, although the horizontal velocity values themselves are not particularly small relative to the vertical velocities, as might be expected. In fact, they are generally positive for both cases; this is in agreement with the observation that the extreme points are generally located above the still water surface where the horizontal velocities are generally positive.

Apart from discrepancies due to the location of the extreme points chosen to compare the various theories, overall agreement between the various theories is very good.

4.3 Accelerations at the still water line

A segment of data from case i18.262 was processed to provide data for the case where the LDV measurements were located at the still water line. For this case the water depth was d = 1.302m, the LDV measurement

location was z = 0 m, and the sample time step was At = 0.025 s.

The crest scatter plots are presented in Fig. 12. The linear theory gives poor results but the agreement between the other theories and the LDV data is good.

The leading and trailing edge slope extrema scatter plots are presented in Figs 13 and 14, respectively. Agreement between the nonlinear theories and the LDV data is good, although there is some scatter in the acceleration data, particularly for the wave trailing edge values.

4.4 Accelerations above the still water line

A segment of data from case il8.1 l2 was processed to provide data for the case where the LDV measurements were above the still water line. For this case the water depth was d = 1.302m, the LDV measurement location was z = +0*05m, and the sample time step was At = 0,025 s.

For this case, scatter plot data are presented only for wave crests, since the LDV measurement location was not submerged for wave troughs and generally was not submerged during the occurrence of lc3q/dtl extrema.

Short segments of the time records are first presented in Figs 15-18. It is from the full time records that the scatter plot data are obtained. Segments of the horizontal velocity records are presented in Fig. 15.

-0.4 0.0 0.4

W3t [g] - LDV aW/at [g] - LDV

Fig. 12. Measurements of velocities and accelerations under wave crests. Case i18.26. LDV positioned at z = Om.

51 Fluid accelerations under irregular waves

-0.5 0.0 0.5 1.0

au/at [g] - LDV &v/at IsI - LDV

Fig. 13. Measurements of velocities and accelerations under leading edge slope extrema. Case i18.26. LDV positioned at z = Om.

Both the raw LDV velocities as well as the Wheeler velocity are shown. Linear theory is not valid at this elevation and cannot be applied. The surface elevation time record is also presented for comparison. Notice

-1 .o -0.5 0.0 0.5 1.0 1.01 I 1 I A

0.5 _____.._..__.__~ ._________.___._._t_.~..._....._________.. j _...............

g ty=Y#Yj _A 0.0 _..._.,.______._ i . . . . . . . . . . . . . . . . . . . i j

how valid velocity data are present only for elevations greater than 5cm.

The vertical velocity records are presented in Fig. 16. From these plots it can be appreciated how non-zero

-1 .o -0.5 0.0 0.5 1.0

:-:i’

PW3t [g] -LDV aWlat [gl - LDV

Fig. 14. Measurements of velocities and accelerations under trailing edge slope extrema. Case i18.26. LDV positioned at z = Om.

52 J, A. Zelt, 0. T. Gudmestad, J. E. Skjelbreia

time [s]

Fig. 15. Horizontal velocity time series. Case i18.11. LDV positioned at z = +0.05 m.

vertical velocities can be associated with wave crests if there are slight errors in locating the crest times.

The horizontal acceleration records are compared in Fig. 17. In order to provide more detail the range of time values has been restricted to include data only from the third crest displayed in Figs 15 and 16. The curve labelled ‘Wheeler acceleration theory’ represents data from the theory presented in Section 2.3. The curve labelled ‘Wheeler velocity theory (differentiated)’ represents data from the theory presented in Section 2.4. The curve labelled ‘LDV’ represents numerically differentiated raw LDV velocity data using the differentiation algorithm presented in Section 3.1.

The vertical acceleration records are compared in Fig. 18. Although this plot represents only a single point in the scatter plots, it appears that the differentiated

,~_~____~~~~~~~~’

. I 1 !

I + wLov

I __

Wbul8okf I

-7\ o.5j_ _.___.___________............. + . I

Wheeler velocity theory agrees somewhat better than the ‘Wheeler acceleration’ theory.

The crest scatter plots are presented in Fig. 19. Agreement between theory and the LDV data is good. Small horizontal accelerations and small vertical velocities are observed, as should be expected. The vertical velocities are large and slightly under-predicted by the Wheeler theory. The vertical accelerations are negative, with some scatter that might be attributable to the phase difference problem described earlier.

5 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDIES

Overall good agreement between the Wheeler stretching- based theories and LDV velocity data shows that local

_----_--___-------------~-~--- 3 99 90 91

time [s]

0.20

0.15

3 0.10 -

.z

0.05

0.00

Fig. 16. Vertical velocity time series. Case i18.11. LDV positioned at z = +0.05 m.


10 I I ! ! 0.20

--- Wheelef~theory -.-.- Wheeler veMty theory (dilferenttated) ____- tJ-J”

i

5 _............?I ..___._............. ;‘...__... ._._________.__.....

time [s]

Fig. 17. Horizontal acceleration time series. Case i18.11. LDV positioned at z = +O.OS m.

accelerations can be reliably computed using the algorithms presented here with the data collected by Skjelbreia.*

Although the results obtained using the ‘Wheeler acceleration’ theory are comparable to those obtained by numerically differentiating the Wheeler velocities, the latter technique may yield slightly better results and should be used for future studies. In particular, the data of Skjelbreia’ should be processed to generate a more extensive data bank of acceleration measurements under controlled conditions.

The identification of ‘extreme’ points used to generate data for the ‘scatter’ plots could be improved. The current simple technique described in Section 3.2 suffers from slight phase variations as described in Section 4. An improved technique would be to first locate the

extreme points times as described in these investigations; these can be used as a first approximation for the extreme point locations. Then by searching each velocity or acceleration time record for extreme points in the vicinity of these locations, it should be possible to reduce the phase difference problem. For example, the maximum horizontal acceleration values for each time record observed near a slope extremum should be compared rather than comparing the acceleration values at exactly the same time for each time record. This technique could not be used to compare horizontal accelerations at wave crests or troughs since the horizontal acceleration does not exhibit an extremum in these regions, but that is not important because the horizontal accelerations are of little engineering interest in these regions.

- - Wheeler a&Aeration theory . -. -. Wheeler velocity theory (dittersntiated)

____. D”

--

90.4 90.6 9

time [s]

Fig. 18. Vertical acceleration time series. Case i 18.11. LDV positioned at z = +O-05 m.

J. A. Zeit, 0. T. Gudmestad, J. E. Skjelbreia

-0.5 0.0

-0.4 0.0 0.4 -0.4 0.0 0.4

iW3i [g] - LDV dw/at [g] - LDV

Fig. 1% Measurements of velocities and accelerations under wave crests. Case i18.11. LDV positioned at z = +@05 m.

REFERENCES

Gudmestad, 0. T., Measured and predicted deep water wave kinematics in regular and irregular seas. Marine Struct., 6 (1993) l-73. Skjelbreia, J. E., Kinematics in Irregular and Regular Waves. NHL-Marintek report, 1991. Skjelbreia, J. E., Torum, A., Berek, E., Gudmestad, 0. T., Heideman, J. & Spidsoe, N., Laboratory measurements of regular wave and irregular wave kinematics. In Wave and Current Kinematics and Loading. E&P Forum report 3.121 156, IFP, Paris, 1989. Skjelbreia, J. E., Berek, E., Bolen, J. K., Gudmestad, 0. T.,

Heideman, J., Ohmart, R. D., Spidscae, N. & Torum, A., Wave kinematics in irregular waves. In Proc. Offshore Mechanics and Arctic Engineering, OMAE, Stavanger, ASME, New York, 1991, pp. 223-8. Wheeler, J. D., Method for calculating forces produced by irregular waves. J. Petroleum Technoi. (March, 1970) 119-37. Mei, C. C., The Applied Dynamics of Ocean Surface Waves. John Wiley, New York, 1983, 740 pp. Zelt, J. A. & Skjelbreia, J. E., Estimating incident and reflected wave fields using an arbitrary number of wave gauges. Proc. 23rd fnf. Conf: Coastal Engng (in press).

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Fluid Accelerations Under Irregural Waves