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WWKZ-84.S013KINEMATICS ANO OIRECTIONALITY OF WAVES IN THE SURF ZONE Sy J.van Heteren and M.J.F. Stive
May 1985
KINEMATICS AND DIRECTIONALITY OF WAVES IN THE SURF ZONE
J. van Heteren, Rijkswaterstaat M.J.F. Stive, Delft Hydraulics Laboratory
Rijkswaterstaat
Report WWKZ - 84.S013
May 1985
KINEMATICS AND DlRECTIONALITY OF WAVES IN THE SURF ZONE
CONTENTS
PAGE
1. INTRODUCTION 4
2. LOCATION FIELD EXPERIMENTS 6
3. MEASURING STATIONS 7 4. INSTRIJMENTATION 4.1. Introduction 4.2. Velocity meters 4.3. Wave gauge 4.4. Anemometer
4.5. Data acquisition system
8 8 8 11 12 12
5. DATA ANALYSIS PROCEDURE
5.1. Introduction
5.2. Estimation of auto- and cross spectra 5.3. Estimation of coherence spectra
5.4. Estimation of phase spectra 5.5. Estimation of gain spectra
5.6. Estimation of wave directional parameters
13 13 13 14 15 15 16 6. OBSERVATIONS 18 6.1. Data sequences
6.2. Wind and wave conditions
18 19
7.1. Introduction
7.2. Colnbrook and ASTM 7.3. Vector-akwa and Simrad
7.4. Vector-akwa and Marsh MeBirney 7.5. Discussion 21 22 22 22 23
8. WAVE KINEMATICS RESULTS 24
8.1. Introduction
8.2. spectra of a typical measurement series
8.3. Transfer functions at the peak frequency and its higher harmonies 8.4. The r.m.s. fluctuations 24 24 29 35
9. WAVE DlRECTIONAL RESULTS 36
10. RADIATION STRESS RESULTS 37
11.1. Introduction
11.2. Spectral transfer functions 11.3. Wave directions 11.4. Radiation stresses 39 39 39 45 45
11. SUMMARY AND a:>NCLUSIONS
ACKNOWLEDGEMENTS 47 REFERENCES 48 LIST OF TABLES 50 LIST OF FIGURES 51 APPENDIX A TABLES FIGURES
4
-1. Introduction
Realistic modelling of the internal kinematics of wind waves breaking on a beach is of great practical importance, e.g. for the modelling of coast-al processes and for the design of coastal structures. A fundarnental aspect in this modelling concerns. the relation between the surface elevation and the internal kinematics. In this respect it appears th at in water of interrnediate relative depth the linear Gaussian model of the wave motion performs satis-factorily (see Battjes and Van Heteren, 1983, 1984). Although the linearity assumption is expected to be violated near and in the surf zone, it is prac-tically useful to check the quantitative performance of the linear Gaussian model in this case. An important part of the present study is devoted to the
applicability of this theory, making use of field measurements which we re collected in the spring of 1981 on the beach near Egmond.
The quanti tati ve performance of linear theory in predicting the wave kinematics from the surface elevation is investigated on the basis of spec-tral transfer functions between surface elevation and velocity and of mea-sured and theoretical r.m.s. fluctuation of the velocity. As far as the spec-tral transfer functions are concerned the squared coherence, the gain and the
phase were investigated between surf ace elevation and both horizontal and vertical velocity.
Relevant earlier studies on the performance of linear theory in the surf zone are those of Thorn ton et al. (1976 ), of Mi tsuguchi et al. (1980 )
and of Guza and Thornton (1980). The general findings are that linear theory generally overestimates wave induced horizontal veloci ties by 10% to 30%. Mitsuguchi et al. merely state their conclusion without analyzing apparent trends in their data wi th e. g. increasing frequency. They only consider a lirnited set of horizontal velocity data. Guza and Thornton obtain their
over-all conclusion merely on the basis of the r.m.s. fluctuation of the horizon-tal velocity as measured and as predicted wi th the linear theory from the
surface elevation. The present study confirrns the conclusions of the above investigations by and large, but extends the analysis both in width by
inclu-ding the vertical velocities and in depth by investigating the role of
turbu-lent kinetic energy generated by breaking. Also special attention is given to
the phase relationship between the vertical velocity and the horizontal
The contents .of the present report is as follows. The chapters 2 and 3 give a description of the measurement location and of the measuring stations respectively. For the measurements different types of velocity meters were used, which are described in chapter 4. This chapter also gives a description of the wave gauges and the data acquisition system. Chapter 5 deals with the
data analysis procedure and chapter 6 with the observations made. Chapter 7
gives the results of an intercomparison between the different types of velo-city meters and chapter 8 the results of the wave kinematics and their rela-tion with dissiparela-tion. The results of the direcrela-tional spectrum parameters are presented in chapter 9 and those of the radiation stress in chapter 10.
6
-2. Location field experiments.
The field experiments were conducted on the Dutch coast near Egmond, see fig. 1. This location was selected because of its longshore uniformity in bottom topography and the absence of beach groynes.
The site selected is gently sloped and barred, see fig. 2. The beach consists of sand with a median grain diameter of approximately 200 IIm à
250 llm.
The location of the surf zone and its width vary with the tide and incident wave height. Another factor which influences the location is the presence of bars. At high tide the surf zone is often close to the beach, although high waves already break above the bars. At low tide breaking occurs mainly at the bars.
The orientation of the shore is such that the direction of a line nor-mal to the shore and directed seaward is 2750 relative to true North. This
direction is somewhat arbitrary, since it depends on the extension of the coast which is considered. Here the direction is defined as the line normal to an extended part of the coast (»surf zone width). In addition the direc-tion of the coast .varies from bottom contour to bottom contour. The varia-tions within the direct environment are estimated to be within 10°.
3. Measuring stations.
The measurernentswere made at five stations along a line normal to the shore, three of these were situated in the surf zone during high tide".The other two were located outside the surf zone as reference, one at the 4 rn depth contour and the other at the 10 rn depth contour, with M.S.L. as refer -ence level. The stations are nurnbered 1 through 5 from shore to sea (see fig. 3).
The stations in the surf zone (stations 1 through 3) consisted of a platform, a sensor pile and, with the exception of station 1, a wave gauge. The platforms rested on transparent space trusses with four .115 rn diameter tubes in the corners of a 3 rnx 3 rnsquare. The batter piles and longitudinal pipes consist of tubes with a diameter of .048 m, see fig. 4. The stations were rnutually connected by a foot jetty which was also used for the cable
connection with the shore.
The sensor piles were placed at the southwest corner of each of the three stations, spaeed m from the platforms. In order to change the eleva -tion of the sensors a measuring car was used consisting of a steel socket fitted round the pile and provided with wheels made of a synthetic material. This car could be moved with a winch along the pile in order to give the sensors the elevation desired. On both sides of the socket two longitudinal pipes were welded in a direct line parallel to the coast. At the end of each of these pipes a sensor could be fixed for intercomparison and correlation. The ends of these pipes were spaeed 1.10 m from the centre of the pile so that the horizontal distance between the two sensors was 2.20 m, see fig. 5. At station 2 and 3 wave gauges were located at the northeast side of the stations, spaeed 1 m from the platforms and 3 m from the sensor piles.
The station at the 4 m depth contour (station 4) consisted of a fixed pile with a diameter of 0.60 m. This station was used for rneasuring the tide elevation and the wind waves. The data were transmitted to the shore using a radio link. At the 10 m depth contour (station 5) a waverider and a tidal
current meter were situated. The current meter recorded the data automatical-ly (NBA current meter).
8
-4. Instrumentation.
4.1 Introduction
The principal instruments used in this study were two-dimensional velo-city meters, three-dimensional velovelo-city meters and wave gauges. These instru-ments are described below, together with the calibration procedures.
4.2 Velocity meters.
Velocities were measured at each of the three locations with three dif-ferent types of velocity meters. The following types were used:
a) electro-magnetic current meters, one manufactured by Marsh MCBirney Inc., and one manufactured by Colnbrook Instrument Development Ltd., further on indicated by the name of the manufacturers;
b) velocity meters based on the principle of travel time of acoustic pulses, one of the Simrad firm and a newly developed three-dimensio-nal velocity meter: the so-called vector-akwa;
c) one acoustic doppIer shift meter, also newly developed, with which sediment concentrations can also be measured: ASTM (Acoustic Sedi-ment Transport Meter).
Each velocity meter was calibrated before and after the experiment.
4.2.1 Marsh MCBirney
The Marsh MCBirney is an electro-magnetic velocity meter based on Faraday's law. with a coil an electro-magnetic field is generated. Water flowing through this field generates a potential difference, which is directly proportional to the flow velocity. By measuring the potential differences the water velocities can be measured.
This principle is used in two mutually orthogonal directions 50
that two velocity components in an arbitrary plan can be measured. The type used during the experiment consisted of a spherical sensor. It was mounted in such a way that the two horizontal veloci -ties could be measured.
The calibration factor was determined experimentally in a labo-ratory basin. For checking the directional cosine response eight orientations were used at 45° intervals.
From the calibrations it appeared that the maximum deviation from the theoretical cosine response is 10\ for velocities lower than 0.5
mis,
decreasing to 5% for velocities up to 1mis.
4.2.2 Colnbrook
The Colnbrook is based on the same principle as the Marsh McBirney. It also measures two velocity components. The shape of the sensors differs from that of the Marsh McBirney: it has.a disc form. This results in a better eosine response. However, due to this design,
measurements of the two orthogonal velocity components are influenced if a velocity component normal to the plane of the sensor is present.
From calibration tests in a laboratory basin it appeared that inaccu-rate results are obtained if the angle between the plane of the sensor and the velocity component is greater than 7°. Therefore, this sensor could only be used for measurements of the two horizontal velocity
components near the bottom, were the vertical velocity component is small.
4.2.3 Simrad
The Simrad is one of the velocity meters based on the principle of travel time of acoustic pulses. This principle is used in two orthogonal directions, allowing measurements of two orthogonal veloci-ty components. The electronics container is mounted 20 cm above the survey lines, 50 that a velocity component normal to the plane of the
survey line may disturb the measurements.
From the calibration it appeared that the relation between velo-city and output voltage is linear and that deviation of the cosine
- 10
-4.2.4 Vector-akwa
The Vector-akwa is also based on the principle of travel time of acoustic pulses. A meter of this type, designed for use in physical models, has been described by Botma (1978). He also designed another meter of this type, allowing measurements of three orthogonal velocity components for use in relatively shallow water. A description of this meter and the results of measurements made with this three-dimensional meter can be found in Battjes and Van Heteren, 1983, 1984. The survey lines had a length of 1.71 m each, so that this instrument could not be used in the surf zone.
Therefore, Botma designed another instrument, based on the same principle, with survey lines of only 0.23 m. This instrument is highly transparant. Only the survey line parallel to the velocity is influ-enced by the presence of the transducers. To eliminate this effect Botma added a fourth survey line 50 that the line most affected by the
transducers Can be rejected for the calculation of the three orthogo-nal velocity components, see fig. 6.
The calibration factor was calculated using the nominal speed of sound in sea water. Temperature and salinity variations are automati-cally corrected for. The calculated calibration factor was checked ex-perimentally by oscillating the velocity meter in a laboratory basin. The calibration factor calculated beforehand turned out to underesti-mate the velocity by about 10%. The calibration factors were corrected for this deviation. The eosine response showed deviations up to + 5%, see fig. 7.
Since the survey lines are very short compared to the length scales of the waves being studied, the influence of the averaging pro-cess caused by the finite lengths of the survey lines can be neglect-ed.
4.2.5 ASTM
The acoustic sediment transport meter can measure both the sedi-ment concentration and the horizontal velocity components. This meter is based on the measurement of the strenght and the Doppiershift of an
acoustic signal scattered by inhomogeneities (sand, silt, etc.). The reflected signal is detected by a receiver. The sediment concentration is deduced from the amplitude of the reflected signal, the water velo-city from the DoppIershift. However, there are several uncertainties, such as the representativity of the sediment velocity for the water velocity, the variations in composition of the inhomogeneities, etc. This instrument was also calibrated experimentally in a laboratory basin.
4.3 Wave gauge
The wave gauges used for the measurements .of the surface elevation in the surf zone were Plessey gauges. The sensor consists of a resistance wir~,
which is wound round akernel, consisting of a 6 m or 9 m long steelwire, covered with polyethylene. The resistance of the wave gauge section above the water level varies linearly with the changing surface elevation. To protect the gauges from damage due to floating objects the gauges were mounted in a tube, diameter .15 m, provided with slits (0.015 m x .300 m) to allow the water surface within the tube to follow the changing surface elevation
out-side the tube with sufficient accuracy, see fig. 8.
To investigate the influence of the tube, experiments were carried out
in a laboratory basin, dimensions: width 5 m x depth 7 m, length 233 m, In this basin the waves were measured at the same cross section inside and out-side the tube. Both regular and irregular waves were used. This experiment showed a maximal reduction of 6%, depending on the steepness of the waves, see fig. 9. Since in the laboratory basin the circumstances near Egmond could not be reproduced exactly (i.e. breaking waves, etc.) the maximal reduction is estimated to be 10%. If corrections are applied for this reduction a nomi-nal average value of 5% is used for all frequencies, accepting a random error
of ± 5%.
Another factor which is influenced by the tube is the phase of the sig-nal. The laboratory experiments showed that this phase shift is not consider-able «12°), but difficult to determine since it is neither linear nor con-stant. To investigate this phase shift a mathematical model was used. This
model learns that the phase shift depends on the shape and the height of the waves, the mean water level and the dimensions of the slits, see appendix A.
- 12
-More problematic is the fluctuation of the reference level of the gauge. The frequency of these fluctuations is so small (order of period: 1-2 days) that they do not atfect the wave measurements, but the results of the wave gauge could not be used for a very accurate determining of the mean water level, nèeded for the investigation of the wave set-up.
4.4 Anemometer
Measurements of wind speed and direction were made at the location with a Thies-anemometer. This meter is provided with a cupmill and a vane. It was mounted on a high mast at an elevation of approximately 15 m above M.S.L.
(see fig.2).
4.5 Data aCquisition system
During the experiment at Egmond a pulse code modulation system (P.e.M.) was used. with this system 112 channels can be sampled and recorded simulta-neously. The data were recorded with a 14 track tape recorder (4010 Bell and Howell). The system consists of 14 units which can sample 8 channels each. Every unit can be connected with one track of the tape recorder. The sample frequency of the system at the lowest record speed (1 7/8 i.p.s.) is 133 sam-ples per second per channel. The system is synchronized because one of the units functions as "master".
5. Data analysis procedure
5.1 Introduction
As a standard procedure mean values, variances and auto spectra were calculated of the surface elevation and wave velocity signais. In addition cross spectra 1 calculations were carried out between offshore wave velocity components of two velocity meters fixed at the same sensor pile for the pur-pose of intercomparison. To investigate the transfer functions (gain- and phase functions) and to estimate the directional spectra, cross spectral cal-culations were also carried out between the surface elevation (ç ) and the three orthogonal velocity components (u,v,w) measured in approximately the same vertical. Due to shor-ccrescedneas of the waves spectral analysis cannot be used for the calculation of gain functions between
ç
and the resultant velocity in the horizontal plane (u). For the calculation of these gain func-tions the auto spectra were used, assuming absence of noise.5.2 Estimation of auto- and cross spectra
The analyses were performed on 30-minute long records, digitized with
Is t = 0.25 s . No dilution of the data was applied, so that the Nyquist fre-quency is at f = 2.0 Hz. This was deemed high enough to prevent significant aliasing.
Standard spectral analysis procedures as described e.g. by Bendat and piersol (1972) and Jenkins and Watts (1968) were used. These included correc-tions for missing data or unaccountable values and removal of sample mean and linear trend. Singleton's (1969) FFT algorithm was applied for the spectral calculations.
The records were divided into 30 segments of 60 s each , No data or spectral window was applied, so that the resulting spectral estimates each had 60 degrees of freedom, except those at zero frequency and at the Nyquist frequency, which had 30. Using the X 2-distribution, the lower and upper bounds of the 90%-confidence bands of the estimates of the auto-spectra are 0.72 and 1.32 times the samples estimate in case of 60 degrees of freedom.
- 14
-The frequency resolution 6,f = 1/60 Hz. Each estimated spectrum is here
deno-'"
ted as SxyCf), with subscripts indicating the processes of which it is the auto- or cross spectrum. If SxyCf) is a cross spectrum Sxy(f) is complex and can be written as:
A
'" ~ (f) S (f)
=
H (f). e xyxy xy
A
in which Hxy(f) represents the amplitude of y per unit amplitude of x, and '"
~xy(f) represents the phase lead of y with respect to x, both as function of
'" A
frequency. Hxy(f) is the gain spectrum and ~xy(f) is the phase spectrum. The circumflex above the symbols indicate that they are estimated from the data. If no circumflex has been used the corresponding quantities according to the linear theory are indicated. The auto spectra have been made one-sided, so that their~integral over all positive frequencies equals the variance of the process.
5.3 Estimation of coherence spectra
The cross spectra have been normalized to spectra of coherence. In case of [,(t)and wet) this is written as:
.y
2
r,w(f)90% confidence bands for the coherence were calculated using the expressions presented by Bendat and Piersol (1972). These authors limit the validity of their expressions for the confidence interval of
y2
to the range:A
0.35 < y2 < 0.95
Outside this range no confidence bands were calculated.
In the following, coherence values are used only for the vertical velo-city in relation to the surface elevation, in which case its theoretical value in absence of noise is 1 (assuming linearity). The coherences between a horizontal velocity component (u or v) and surface elevation (r,) or vertical velocity (w) depend on the directional wave energy distribution. In fact, their values are used in estimating that distribution.
5.4 Estimation of phase spectra
Phase spectra were derived from cross spectra accordinq to:
~ (f)= arg S (f)
xy xy
Confidence bands on this estimate were calculated usinq the equations presen-ted in Bendat and Piersol (1972).
5.5 Estimation of qain spectra
In order to eliminate possible influences of noise, the qain function
A
Hçw(f) was estimated accordinq to:
H (f) Çw
A
Confidence bands for Hçw(f) were calculated, usinq the procedure described by
,Bendat and Piersol (1972).
The horizontal velocity components require a separate treatment, becau-se of the directional distribution of the wave enerqy. Gain functions were calculated for the resultant velocity in the horizontal plane (u) accordinq
to: '" H _ (f) Çu S (f) + S (f) ~
= {
uu vv} Sçç(f)This procedure does not permit the estimation of confidence bands. In point
of fact, this procedure to calculate the qain spectra can only be used if the siqnals are virtually free of noise.
- 16
-5.6 Estimation of wave directional parameters
The two-dimensional wave spectrum Sçç(f,8) is often written as:
in which S (f) is the variance density spectrum of the surface elevation and
çç
Df(8)is the directional spreading function, normalized so that Df(8) is a periodical function with a period of 2TI. This means that Df(8) can be written as an unlimited Fourier series:
Assuming a linear relation between the spectra of the surface elevation and those of the velocity components the first four Fourier coefficients can be calculated using the auto- and cross spectra of these orthogonal velocity components, see Longuet-Higgins et al., 1963, and Borgman, 1979:
j_ TI sww(f) 1 H _ (f) wu
I
s
(f) I 1 vw ! TI S (f) ww Hwu- (f)s
(f) - S (f) 1 Az (f)=
j_ { uu vv }.
" TI S (f) H- -(f) ww wu.z.
[Suv(f)i
1 S (f) + S (f) B2 (f) TI " { uu vv}
~
S (f) W _(f) with H -w,.,r wu wu S (f) wwFor the first two Fourier coefficients measurements of the three orthogonal velocity components are required while for the calculation of the second two
coefficients measurements of the two horizontal velocity components are
The parameters of the cos2S(0/2) model, defined as:
A(s) cos2s {0 -00} 2
71'
in which A(s) is a normalizing factor so that
f
Of(0) d0:::: 1, can be estima--71'ted from the first two Fourier coefficients as weIl as from the second two, see Mitsuyasu et al., 1975:
mod 1800
Cl
with: (A12 ~l 2) ~ fAo
501 =-- Cl = + 0 i-Cl 1 + 3 C
z
+ (C 2 + 14 C2 + 1) ~ B g) ~fA 2 (A 2 502 with: C2 = + 2-
2 C2 2 2 0in which 00 is the main wave direction of the cos2S(0/2) model and So the spreading parameter of this model.
- 18
-6. 'Observations
6.1 Data sequences
Five measurement series were collected around high tide with durations of 4 to 5 hours. The dates of these series of measurements and an indication of the significant wave height (Hs) just outside the surf zone on these days are listed in table A.
Measurement series Date Hs{m}
'-, T1 14 May 1981 0.30 - 0.60 T2 19 May 1981 0.15 - 0.20 T3 22 May 1981 0.40 - 0.60 T4 26 May 1981 0.70 - 1.50 T5 5 June 1981 1.00 - 1.50
Table A: Dates and offshore significant wave heights of measurement series.
Since the low significant wave heights of series T2 result in an
inac-ceptably low signal/noise ratio, this measurement series has not been taken
in consideration.
The measurements were made at each of the five stations. In each mea-surement series the instruments were kept at the same station and mainly at a constant elevation. This elevation was 0.10 m above the bottom at station 1,
0.80 m above the bottom at station 2 and 1.00 m above the bottom at station
3. During TS the elevations at station 2 and 3 were changed to 0.20 m above
the bottom.
Figure 10 shows the weather conditions in April, May and June. This
figure also presents the significant wave height at deep water (permanent
wave measuring station of IJmuiden, see fig. 1, mean local water depth 25 m).
From each measurement series several 30 minute records were selected
for the present investigation. This selection was based on several factors, such as the variation of the conditions, the quality of the data, etc. This resulted in 14 half hour records of each instrument per station.
6.2 Wind and wave conditions
Values of the principal parameters of tide and wind of the records selected are tabulated in table 10 The following quantities pertaining to tide and wind are listed in this tabIe: record number, date and start time of record, time of high-tide, all in M.E.To, mean water elevation measured posi-tive upward from M.SoL. (Ç),mean velocity component along the shore (u) and
offshore (v) (the last three quantities obtained from station 3; the bar
above the symbol denotes a 30-minutes average value of the quantity),
10-minutes average wind velocity (vw) and wind direction with respect to true
North (Bw). Table 2 summarizes the principal parameters of the waves at the
stations 2, 3 and 4: the significant wave height, calculated as 4
lmo,
inwhich
me
is the variance ofç
,
the peak frequency of the var-Lance densityspectrum of
ç:
fp' while table 3 contains the wave directional parameters atfp: the main wave direction with respect to true North and the directional
width of the waves (00' for definition: see par. 9) as weIl as the values of
the breaking index (6) and the wave steepness (0) at the stations 2 and 3.
The wave directional parameters are also given for station 1, since these
parameters were calculated using the two horizontal velocity components. The breaking index and wave steepness are defined as:
H. 'S
B
= h and <5=
H s L P L P c P g {_2TIh} = - - tanh f - 2TIfL L P P Pin which h local mean water depth = d <I-
ç
d local depth below M.SoL.
g acceleration of the gravity
cp
=
phase speed at the peak frequencyLp
=
length of the wave component at the peak frequencyFigo 11a - 11d show sketches of the situations at the measuring site of
breaker bar and channel system and of the position of the breakers at low
tide, while fig. 12a - 12d give cross sections for each measurement series.
Fig. 13a _ 13c give the variance density spectra of the surface elevation at
each station of Tj-1, T4-1 and T5-1. The spectra of T1-1 are not given since
20
-The measurements were all made during south-westerly wind. The wind speed ranged from 4 mis to 10 mis. Since the wave direction is about the same
as the wind direction the waves can be characterized as sea. The peak fre-quencies ranged from 0.15 Hz up to 0.30 Hz. Although the sketches of the fig.
11a - 11d indicate that during T1 and T3 breaking occurred at the breaker
bar, the variance density spectra of the waves at the stations 3 and 4 indi-cate that only little energy was lost at this bar so that for these measure-ments significant breaking mainly occurred at the beach. The sketches and the spectra of T4 and TS show clearly that during these measurements the stations 1, 2 and 3 were located within the surf zone so that T1 and T3 are more ly describing nearshore but non-breaking waves and T4 and T5 describe near-shore, breaking waves.
The difference between T1 and T3 on the one hand and T4 and T5 on the
other hand is aLso illustrated by a different value of Hslh, w,hich may be
considered as a breaking index. This ratio ranges from 0.14 up to 0.27 for T1 and T3 and from 0.38 up to 0.45 for T4 and T5 (see table 3).
7. Intercomparison of velocity meters
7.1 Introduction
One of the objectives of the field campaign of 1981 was to investiqate the performance of several types of velocity meters in a natural surf zone by intercomparison. To this end the velocity meters were closely placed at the same location and heiqht above the bottom. Cross spectral analysis was per-formed between the meters mutually. The meters were compared on basis of the measured r.m.s. values of the magnitude of the fluctuatinq part of the hori-zontal velocity:
in which
u
is the horizontal velocity in the propaqation direction and u and v are the two orthoqonal velocity components in the horizontal plane. In ad-dition a comparison was made on basis of.the qain and squared coherence at fp of the offshore velocity between them.Due to the lack of an absolute measurement result and of insiqht into the
spatial variability of the velocity field these comparisons give no definite
answer to the question which meter is the better one. The only quality
assessment of the meters may be based on the squared coherence between the
offshore velocity and the surface elevation. This function was derived from
cross spectra estimated from the measured offshore velocity and surface
ele-vation. Table 4 summarizes the positions of the velocity meters used, while
the mean water depth (d+Ç) and elevation of the meters (z) are given in table 5 (Ç and z are measured positive upward from M.S.L.). The variances of u, v
and
u
are given in table 6, the values of the squared coherence and gain ofv, the offshore velocity component, of the different velocity meters located
at the same station at fp and the squared coherence of surface elevation and
offshore velocity component at this frequency in table 7. The results qiven
in table 6 are presented graphically in fig. 14 and those of table 7 in fiq. 15a and 15b. All above mentioned results are discussed below. A more detailed consideration of these intercomparisons is given by Derks and stive, 1984.
22
-7.2 Colnbrook and ASTM
The Colnbrook and ASTM were mounted at station 1 at an elevation of about 0.10 m above the bottorn, see table 5. The varianees of u, v and the resultant velocity in the horizontal plane (~) are surnmarized in table 6.
Table 7 gives the gain and squared coherence between the two offshore velocity components at fp' The rnutual differences are considerable. The gain values at the peak frequency show deviations up to 20% in spite of the high coherence values at fp:
y
2vv > 0.9.Rernemberingthe uncertainties of the ASTM, mentioned in par. 4.2.5, the authors have rnainly relied for the present investigation on the results of the Colnbrook.
7.3 Vector-akwa and Simrad
The intercomparison of one of the Vector-akwa's (coded Vector-akwa I, see table 4) and the Simrad occurred at station 2. The elevation of the two velocity meters was 0.80 m above the bottom with the exception of T5-3, T5-4 and T5-5, during which the elevation was 0.20 m above the bottom (see table
5). Fig. 14 shows the scatter diagram of the r.m.s. value of ~ of this
Vector-akwa and the Simrad (see also table 6). These results show a substan-tial discrepancy between the varianees mentioned, i.e. the regression is cal-culated to have a mean of -16 % with a standard deviation of 11 %. Fig. 15a shows that the squared coherence between
ç
and v of the hydrodynamically well designed Vector-akwa is always higher than that of the Simrad (see also table 7), indicating that we prefer the output of this veloeity meter over theSimrad's.
7.4 Vector-akwa and Marsh MCBirney
The Vector-akwa used for the intercomparison with the Marsh MCBirney is coded Vector-akwa II, see table 4. They were intercompared at station 3. The elevation was 1.00 m above the bottom, with the exception of TS-4 and T5-5, were the elevation was 0.20 m above the bottom (see table 5). The scatter diagram of the r.m.s. values of
u
of these two velocity meters can be seen inFor the measurements made during T3 dd.sczepanc.ies are found of the order of 30%. For the other measurements deviations occur of about 8%.
From fig. 15b it can be seen that for T3, y2~v at fp of the Marsh MCBirney is very low compared with that of the Vector-akwa (see also table 7). This im-plies that during T3 the Marsh MCBirney was malfunctioning. For the same reason as given in par. 7.3 the Vector-akwa is used for further investiga-tions of the measurements.
7.5 Discussion
In the above only qualitative conclusions about the performance of the-current meters could be drawn based on the squared coherence between the off-shore velocity and the surface elevation. An extension of the intercomparison
~was realized in a second field measurement campaign. The results, summarized
in Derks and stive (1984), indicate that the vector-akwa current meter
per-forms satisfactorily, i.e. the random measurement errors in the velocity
24
-8. Wave kinematics results
8.1 Introduction
The quantitative performance of the linear theory in predicting the internal wave kinematics from the surface elevation is investigated in sever-al ways. Firstly, results of one typical measurement series, i.e. the first measurement series of station 2, are described extensively. The variance den-sity spectra of the surface elevation of these examples are shown in the fig. 16a - 16d. Secondly, all the results are involved in a comparison of measured and theoretical gain values at the peak frequency, fp' and its sec-ond and third harmonie. Finally, all measurements of the r.m.s. fluctuations of the horizontal and vertical wave velocity are compared to predictibns ac-cording to the linear theory. In the latter two cases the results are inves-tigated in relation to the local turbulent energy due to wave breaking which
is shown to help explain the nature of the discrepancies between theory and
measurements.
The figures 16a - 16d (varianee density spectra surface elevation),
19a - 22d (gain spectra) and 28 - 33 (scatter diagrams and correlations with
the turbulent energy) have not been corrected for the wave height reduction
due to the tube of the wave gauge. The same holds for the tables numbered 1
to 12. All other gain values given in this paragraph and the results given in the tables numhered with capitals have been corrected for this reduction.
8.2 Spectra of a typical measurement series
8.2.1 Squared coherence spectra
As mentioned before, the coherence between surface elevation and
the offshore velocity component is quite high. This is also shown in
the examples of the squared coherence functions given in the fig.
17a - 17d. This conclusion holds for all records analyzed, regardless whether the measurements were made in non-breaking waves (T1 and T3)
This high coherence implies a nearly linear relationship between 1; and v, and a relatively small contribution to the variance due to turbulence. A further consequence is that the sampling variability in
A
<p (f)is low: 5° to 10°.
z;v
The squared coherence of 1; and w is much lower than the one of
:Z; and v, see fig. 18a - 18d. This results in high sampling variability
in the estimates of phase and gain spectra. The total widths of the
90% confidence interval for these quantities in the energetic
frequen-cy band are 15° for the phases and 30% of the theoretical values for
the gains.
8.2.2 Gain spectra
Fig. 19a - 19d and 20a - 20d show examples of the gain spectra
'" A
Hçu(f) and Hsw(f) respectively and of the corresponding gain spectra
calculated with linear theory.
The examples of
H
s u(f) show an overestimation by the lineartheory for f = 0.10 Hz, which overestimation decreases towards f=0.40
Hz and changes into an underestimation for higher frequencies. This
trend can more clearly be seen in the fig. 21a - 21d where the ratio
between the measured to the theoretical gain, lndicated by R, has been
plotted versus frequency. Since these spectra have been calculated
using the auto spectra of
S,
u and v it is likely that this trend iscaused by increase of turbulence at the higher frequency range. This
is confirmed by the squared coherence spectra which show a decrease
with increasing frequency.
The overestimation of these gain spectra by the linear theory
for the frequency range of 0.10 Hz to 0.40 Hz is about 15%. Corrected for the reduction of the surface elevation caused by the tube of the
wave gauge, see par. 4.3, the overprediction of the gain spectra by
the linear theory will be 20%, with a random error of 5%. In addition
an uncertainty of the veloeity meters must be taken into account,
which was estimated not to exceed 5%, see par. 4.2.4.
The examples of Hçw(f), show a greater discrepancy with linear
theory. The vertical velocity is overpredicted by the linear theory
26
-if a correction of 5% has been applied for the reduction of the sur-face elevation due to the tube of the gauge. For higher frequencies this overprediction increases slightly.
The gain spectra of ~ and w have not been influenced by the pre-sence of turbulence, since these spectra were calculated using the cross spectra of these quant ities. This may be the reason why the
A A
trend found for H
~;:i(
f) is not present in the spectra of H~w( f), seealso fig. 22a - 22d.
A _
Results of Hwu(f) have not been given here, since these spectra had to be calculated using the auto spectrum of w. These spectra are composed of the variance correlated with the waves and the variance due to turbulence. In the spectra of w the variance caused by turbu-lence is substantial, see par. 8.3.1.
8.2.3 Phase spectra
Time delays
For the interpretation of the phase spectra a scrutinous investigation of the entire system of measurement and data handling is necessary, as is shown by Battjes and Van Heteren, 1983, 1984. A detailed investiga-tion of possible time delays for the present study was carried out separately (Botma, '983).
I The wave gauge output signal lags behind the velocity signals due to counting and waiting.time, which results in a total ef-fective delay of the gauge, 6t" of 142.2 ms. This was checked by simulation using sinusoidal test signals with different fre-quencies. The output signalof the velocity meter lags behind the instantaneous value, due to slowness of the electronics • However, this delay is only , ms. The delays between u, v and w mutually are less than , ms and have been neglected. These time delays were taken into account at the spectral calculations.
11 In addition there is a lag of
ç
behind the velocity signals due to the tube of the wave gauge. To estimate this lag a mathemati-cal model was used, see appendix A. This model learns that this lag, 6t2' depends on the shape of the waves, indicated by X, the wave height, the mean water level and the width of the slits in the tubes. The value of X is uncertain, therefore two values were used: X = 0.25 and X = 0.75. For the wave height Hs was used. The differences between the calculated lag À t 21 for X = 0.25 and that for X = 0.75 were maximal 20 ms. Therefore, the mean values of the two calculations we re used, accepting a maximal inaccuracy of=
10 ms, which corresponds with an inaccu-racy of 401Hz in the phase spectra.111 Cue to the~scanning system of the data transmission system addi-tional lags are introduced. Counting from the last sample"the
following delays ( /:,.t3) are found:
wave gauge at station 2 0.95 ms ::: 1 ms
wave gauge at station 3
.
1.88 ms e! 2 ms wave gauge at station 2 5.16 ms ::: 5 mswave gauge at station 3 1.41 ms ::: ms
IV In addition delays are caused by the positions of the sensors.
This delay can be calculated as:
in which /:,.X distance between Vector-akwa and wave
gauge along the shore
/:,.Y distance between the two sensors normal
to the shore
e
o (f)= principal propagation direction perfre-quency with respect to true North,
de-fined as the direction from which the
waves approach
direction of the line normal to the shore
with respect to true North (=275°)
- 28
-If ~t4
>
0 the wave gauge signal lags behind the velocity meter. The accuracy with which (~- Go(f» can be determined is limit-ed. Comparing the values of Go(f) given in table 2 maximal er-rors of : 5% seem realistic. This results in maximal errors of ~ t4: ± 45 ms, depending on the wave direction and the phase speed.In the phase spectra this causes an inaccuracy of ±16°/Hz. For the calculation of the phase speed the linear theory has been used as an approximation. According to Guza and Thornton (1980) the actual phase speed lies between 90% and 120% of the phase speed estimated by the linear theory, introducing apos-sible error of -60 ms to +80 ms or -2201Hz to +29°/Hz.
Tablè B summarizes the time delays and their inaccuracies.
Correction Maximal inaccuracucies
~t1 142 ms
-~t2 8 - 22 ms
-
10 ms - + 10 ms~t3 1
-
5 ms-~t4 depends on freq. -105 ms - +125 ms
Table B: lags of
ç
behind u,v,w.Corrected phase spectra
If the maximal inaccuracies of the phase corrections applied as mentioned above are considered as a 90% confidence interval the resulting 90% confidence interval, or based on these
inaccura-cies (ai) and the sampling variability (Os), can be calculated
as:
Fig.' 23a - 23d and 24a - 24d show the examples of the phase functions ~çV(f) and ~çW(f) corrected for the time delays with
"
this resulting 90% confidence interval. Since <p wvC f) has been calculated from the measurements of one velocity meter, no
cor-rections have to be applied to this phase function.
The estimated phase spectra ~çv(f) and ~çw(f) are in most cases
consistent with linear theory within the margin of the accuracy.
'"
<Pçv(f)of T5-1 st. 2 shows a significant discrepancy with linear
theory for the higher frequency part.
"
The results concerning <Pwv(f) are of particular interest since
these spectra are not influenced by a lag between two different
instruments and by a distance between two sensors: se'ethe fig.
25a - 25d. Shortcrestedness is also not a complicating factor
since the velocities have been measured in the same vertical.
"
For the estimated phase spectra <Pwv(f)it can be concluded that
the theoretically expected value of 90° is generally within the
90% confidence interval of the estimate. Of the examples given,
'"
<Pwv(f)ofT5-1 st. 2 is again the worst: a very broad confidence interval and nevertheless a frequency range where the
theoreti-cal value is outside the confidence interval. with the exception of this case the results of <Pwv(f) agree very well with expected
phase between wand v, even within the surf zone. This implies
that standing wave motions due to e.g. reflection off the coast
are virtually absent for the frequency range considered in this
report (0.1 Hz - 0.5 Hz).
8.3. Transfer functions at the peak frequency and its higher harmonies
8.3.1 Squared coherence
The peak frequencies of the surface elevation spectra have been
listed in table 2. The spectra of T1 show two peaks, viz. at 8.~f and
at 13.~f with ~f being the resolution ~ 1/60 Hz. For T1-1 the maximum
variance density occurs at 8.~f, while for T1-2 the peak at 13.~f is dominant. For practical reasons the peak at 13.~f is used as a first
30
-Since no coherence spectra can be calculated for z; and u, the squared coherence spectra of
z;
and vare used. This velocity component contains the major part of the energy in the horizontal plane.Table 8 and 9 give the squared coherence of z; and v for the first order peak frequencies and their higher harmonies. These tables show that the squared coherence for the measurements made at an eleva-tion of 0.20 m above the bottom differs not significantly from the squared coherence of measurements made at a higher elevation as far as the horizontal velocity components are concerned. Therefore, for these components no discrimination has been made between the results of the different elevations.
These tables give also the squared coherence of Z; and w for the same frequencies. For this component the results of the measurements made at the two different elevations differ significantly. Due to the nominally small vertical veloeities near the bottom the squared cohe-rences at an elevation of 0.20 m above the bottom are very smalle As a consequence the signal/noise ratio becomes too low. Therefore, the results of these measurements have not been used for the comparison of w with the linear theory.
Table C shows the means and the standard deviatons of the squared coherence of Z; and v and of Z; and w at the first order peak and its higher harmonies. Between the results of station 2 and 3 no discrimination has been made, since these results differ only slight-ly. "n" is the number of records used for the calculations of the means (Av) and standard deviations (St.dev.).
n Av St.dev. "2 fp} 26 0.93 0.05 Y Z;v( "2 26 0.82 0.09 Y Z;v(2fp} "2 19 0.59 0 13 Y z;v(3fp} "2 fp} 21 0.75 0.16 Y z;w( "2 21 0.76 0.14 Y z;w(2fp} y2z;w(3fp} 14 0.44 0.15
Table C: means and standard deviations of the squared coherence values at the peak frequency and its higher harmonies.
These values show high eoherenees between
ç
and v at the first order peak and a small deerease of eoherenee at the first harmonie. This implies that in this frequeney range an approximately linear re-lation exists betweenç
and v, as is shown in more detail below (par. 8.3.3). A eonsiderable deerease of eoherenee oeeurs at the seeond har-monie of the first order peak.For ç and w the squared eoherenee is substantially lower, whieh indieates that the vertieal velocity is relatively more eontaminated by turbulenee.
Sinee it is more likely that the influenee of turbulenee is eor-related with the absolute value of the frequeney the means and stan-dard deviations were also ealeulated for another grouping. The peak frequeneies of T1, T4 and T5 are all near 10.~f. The peak frequeneies of T3 are near f=20.~f, see table 7. Therefore, the peak frequeneies and their higher harmonies have been split in frequeneies near 10.~, 20.~f and 30.~f. The seeond harmonie of the first order peak of T3 has not been eonsidered sinee the energy level at this frequeney component is too low, resulting in an unaeeceptable low signal/noise ratio. Table D shows the results.
n Av St.dev "2 20 0.93 0.05 Y çv{ 10~f) "2 26 0.87 0.07 Y çv{20~f) "2 25 0.62 0.13 Y çv{30~f) "2 15 0.68 0.13 Y çw( 10~f)
'1
2çw(20~f) 19 0.75 0.17 "2 20 0.51 0.18 Y çw{30~f)Table D: Means and standard deviations of the squared coherenee values at different frequencies.
It appears that these results are not essentially different from
32
-8.3.2 Gain values
Table E shows the means and standard deviations of the
quanti-,... _ _ A
ties Rç~(f)
=
Hçu(f)/Hçu(f) and Rçw f)=
Hçw(f)/HCW(f) at the firstorder peak and its higher harmonies, eorreeted for the wave height
reduction eaused by the tube of the wave gauge. Between the results of station 2 and 3 again no diserimination has been made. The original measurement values, not eorrected for the wave height reduetion due to
the tube of the gauge, are listed in the tables 10 and 11.
n Av St.dev. Rçu{ fp> 26 0.83 0.09 Rçii{2fp> 26 0.83 0.11 Rçü{3fp> 19 1.18 0.30 Rçw{ fp> 21 0.59 0.20 Rçw{2fp> 19 0.53 0.09 RZ;;W{3fp> 14 0.49 0.13
Table E: Ratios measured to theoretical gain at the peak frequeney and its higher harmonies.
These values in table E show a systematic overprediction by the
linear theory of the horizontal velocity of 17% at the first order
peak and its first higher harmonie, changing in an underpredietion of 18% at the second higher harmonie.
The vertical velocity component is overpredicted by linear
thee-ry over the whole frequency range: 41% at the first order peak and 47%
and 51% at its higher harmonies.
Table F gives the corresponding results of these ratios for the
n Av St.dev. Rç\i(1011f) 20 0.84 0.09 RQi(2011f) 26 0.83 0.10 RQi(3011f) 25 1.18
o
32 Rçw(1011f) 15 0.55 0.25 Rçw(2011f) 19 0.53 0.07 R~3011f) 20 0.49 0.11Table F: Ratios measured to theoretical gain at different frequencies.
The standard deviations given in table 0 are comparable to those
in table F.
8.3.3 Correlation with turbulent energy
A closer analysis of the comparison between measured and theore-tical gain spectra may be based on investigating the correlation of
the discrepancies with breaking induced turbulence. To this end a
dimensionless turbulent energy rate is introduced defined as:
A
q2 = phq2/Ekin inc
,
in which p= the water density {kg/m3}, q2 = the turbulent kinetic energy {m/s}and Ekin inc
,
=
the incident kinetic energy = 1/16pgH2r.m.s. {J/m2}. The turbulent kinetic energy is estimated as (Battjes, 1975):q2 ~ (D/p)2/3
- 34
-,,'
Here D is approximated by (Battjes and Janssen, 1978 'and Stive and Battjes, 1984):
where Qb is the fraction of breaking waves and Hm the local maximum wave height.
"2
Fig. 26 shows the results of
Y
r"v at fp and its higher harmon-ies. y2r"v(fp) shows no trend with q2, the results at f~2fp a slight trend and at f~3fp a distinct trend with relative turbulent energylevel: y2r"v decreases with increasing relative turbulent energy lev-el. It is noted that this more or less confirms Thornton IS (1979)
suggestion to distinguish between wave induced and turbulence induced kinetic 'energy i.e.:
1 -
~
?
r"v=s
v'v'(f)S (f)
vv
where Vi is the turbulence induced component of the offshore horizon-tal velocity.
Fig. 27 gives the results of y2çw at fp' 2fp and 3fp• No trend with relative turbulent energy is found for these coherences. Fig. 28
shows the results of Rr,,~and fig. 29 those of Rr"w. For Rr,,~the same holds as for y2r"v in the understanding that Rr,,~increases with increa-sing relative turbulence energy level. For R r"w no distinct relation
with turbulence has been found.
In general all results indicate a consistent overestimation of the linear theory prediction of both horizontal and vertical veloci-ties. There is one exception, namely at three times the peak frequency for the horizontal velocity gain ratio. Since for the horizontal gain
function we have to rely on the gain spectrum derived from the auto-spectra (see par. 5.5) contamination of the result by noise (Iturbu-lence') is inevitable, it may logically be concluded that the findings for the horizontal velocity gain are influenced by this contamination.
Based on the above findings we may now give an explanation of the results found for the ratio of measured to theoretical gain values at the energy density peaks: the trends mentioned above can be
as-cribed to the presence of turbulent kinetic energy in the higher frequency ranges.
8.4. The r.m.s. fluctuations
A scatter diagram of the r.m.s. fluctuations of
u
obtained fram the measurements and those calculated from the measured surface eleva-tion spectrum using the linear theory is given in fig. 30. From this diagram it appears that the deviations are generally smaller than 10% with some exceptions, which indicates deviations up to 20%. This conclusion holds too if the correction for the wave height reduction of 5% has been applied. The regression line is Ur.m.s.,meas{m/s} 0.82 ür.m.s.,theor{m/s} + 0.05{mis}
and the correlation coefficient 0.98.Fig. 31 shows the ratio of the measured to theoretical r.m.s. values of
Ü
as a function of the relative turbulent energy level, q2. There is a clear trend of this ratio with q2, which may be explained from contamination by turbulence as described above.Fig. 32 shows an identical diagram for the vertical velocity. In this diagram the results of T5-3, T5-4 and T5-5 of station 2 and those of T5-4 and T5-5 of station 3 have not been taken into consideration,
since the vertical velocity component is nearly zero at the elevation
used in these measurements (0.20 m above the bottom) • The ratio of wr. m ,s.,meas to wr.m .s ,,theor as a function of q2 is shown in fig. 33. The trend with increasing relative turbulent energy level is obvious, which mayalso be explained as contamination by turbulence.
- 36
-9. Wave direction results
As shown in par. 8.3 the vertical velocity component is small campared with the horizontal velocity components and substantially determined by tur-buIence. Since the formulae given in par. 5 were derived for spectra linearly correlated with waves, formulae containing the auto spectrum of the vertical velocity component cannot be used. This means that the spreading parameter of the cos 2s (8/2) can only be calculated using the second two Fourier coeffi-cients, s02. For 80 the average value of 801 and 802 has been used with 802 corrected for an ambiguity of 180°.
The results of these calculations have been plotted in the figures 34a - 34d. The spreading is expressed in degrees using:
360
27T to = half the total spread)
The example of T1-1 shows that two different wave fields occurred, one non-Iocally generated with a frequency of 0.10 Hz to 0.30 Hz and another caused by local winds at higher frequencies. The direction of this wave field is the same as the direction of the wind. The wave directions at lower fre-quencies show clearly refraction effects.
The directional width of the waves shows a minimum at the peak frequen-cy, which is in accordance with results found at deeper water, see Mitsuyasu et al., 1975. Another phenomenon which is clearly shown by this example is the decreasing spreading as the waves travel towards the shore, which can be ascribed to refraction.
At the other examples, T3-1, T4-1 and T5-1, the same features can be seen, however sometimes not so distinctly since the direction of the waves coming from deeper water are at the same time affected by local wind. The direction of the waves is not always equal to the wind direction due to re-fraction effects , inhomogeneity of the wind field and the spreading of the waves (Kuik and Holthuysen, 1981).
From these results it appears that the directional width in the surf zone is rather broad. The minimum spreading is about 15°, so that the total spreading at fp is about 30°.
10. Radiation stress results
In a statistically homogeneous and stationary wave field the principal radiation stress component in the main wave direction is gi ven, correct to second order in wave elevation (Battjes, 1972) by:
S11, meas.
=
1 0 __
pgÇ2 +
pI
lv*22 -d
w2) dz (1)
where v*2 is the variance of the orbital velocity in the main wave direction,
which term may be derived from the measured variances u2, v2 and u.v in the
horizontal plane (see Battjes and Van Heteren, 1980). The variances and
co-variances are listed in table 12.
Assuming a linear and unidirectional wave field the main principal'co~ ponent may be written in the form:
lNyq
S11, theor. =
I
sçç (f). (2kh/sinh(2kh) + ~,)df (2) owhere k is the linear wave number. Battjes (1972) shows how the reducing
effects of shortcrestedness on S11,theor may be incorporated. For the
simpli-fied case of shallow water the reduction factor is given by (1-2/3E), where
E = (2so+1>.(so+1>:1 (so+2) in which So is the spreading parameter of the cos2s(G/2)-model.
The radiation stresses have been estimated from the measurements
accor-ding to both expressions (1) and (2). The estimations on the basis of (1) may
be considered as more general, i.e. (1) implies only weakly restrictive
as-sumptions so that the results are labelled "measured".
The estimation according to (2) implies at least two restrictive
as-sumptions, i.e. linearity and unidirectionality, so that the result is
Ia-belled "theoreticai" • The unidirectionality assumption can be relaxed by
introduction of the reduction factor E due to shortcrestedness, described
above, leading to an improvement of the theoretical result. The results are
given in fig. 35. Inspection of this figure indicates that the effect of
shortcrestedness is 10% typically.
The correlation between the ratio of measured to theoretical estimates
of the principal radiation stress component and the relative turbulent energy
- 38
-This result clearly shows a trend with the rate of turbulent energy, confirming that in this case also contamination due to turbulence plays a role.
The unidirectional linear theory overestimates the radiation stress by 40% at negligible turbulent energy rates and by 25% at higher rates. Corrected for shortcrestedness these percentages decrease to 35% and 15% respectively. These percentages have not been corrected for the reduction of the surface elevation caused by the tube of the wave gauge, which was estimated to be maximally 10%, see par. 4.3. If a correction is applied for the mean value (5%), accepting a random error of 5%, the percentages mentioned above become 45% and 30% for unidirectional waves and 40% and 20% for shortcrested waves.
11. Summary and conclusions
11.1 Introduction
Measurements have been performed of surface elevations and internal velocities generated by wind waves close to and in the surf zone of'a typical beach on the Dutch coast. Velocities were measured at three locations (20 m apart) along a survey line in offshore direction mainly at an elevation of 0.8 - 1.0 m above the sea bed. The data cover a range of mild to moderate wind conditions with offshore significant wave heights up to 1.85 m. The most
energetic frequency band typically extends from 0.15 to 0.25 Hz.
All given values in this paragraph have been corrected for the ré duc-tion of the surface elevaduc-tion caused by the protective tube mounted around the wave gauges which was estimated to be maximally 10%. For this correction half the maximum value was used (5%) accepting a random error of ± 5%. Fur-thermore, the inaccuracy of the velocity meter must be taken into account, which was found to be also ± 5%.
These surf zone data have been analyzed with respect to:
(A) spectral transfer functions, in an effort to investigate the per-formance of linear theorYi
(B) wave directions, in an effort to derive directional parametersi (c) radiation stresses, in an effort to investigate the performance of
linear theory.
The results are summarized in the following.
11.2 Spectral transfer functions
In the summary of these results a distinction is made between relations of the horizontal veloeity to the surface elevation, those of the vertical velocity to the surface elevation and those between the horizontal and verti-cal velocity. These cases are affected differently by directional spreading, noise and calibration errors.
- 40
·-I Relation between horizontal velocity and surface elevation
Because of the directional width of the waves, the gain function is estimated from autospectra, rather than cross spectra, so that the results can be biassed by noise. Noise may be expected to be of minor importance when the estimated squared coherence between horizontal velocity and sur-face elevation is high, say exceeding 0.9. It appears th at for the present surf zone measurements this is generally only the case in a narrow fre-quency interval around the energy density peak. At higher frequencies the estimated squared coherence decreases significantly.
Results
(A 1) The estimated squared coherence between horizontal velocity surface elevation decreases with increasing frequencies.
(A 2) The set of estimated values of squared coherence at the peak frequency, fp for all analyzed records has an average of 0.93 and a standard deviation of 0.05.
(A 3) The set of the above values, evaluated at fp is not correlat-ed with the rate of turbulent energy.
(A 4) The set of estimated values of squared coherence at approxi-mately 2fp and 3fp' for all analyzed records, have averages of 0.82 and 0.59 and standard deviations of 0.09 and 0.13 respectively.
(A 5) Each set of the above values, estimated at 2fp and 3fp is correlated with the rate of turbulent energy.
(A 6) The theoretical phase difference between offshore velocity component and surface elevation is by and large within the margin of the sampling variability and the accuracy of the phase corrections applied.
(A 7) The average value of the estimated phase di'fference is for the frequency band 0.10 Hz - 0.20 Hz about 10° (the theoreti-cal value: 0°), for higher frequencies the ave rage values of the discrepancies decrease and become smaller than the inac-curacies of the phase corrections.
(A 8) The ratio of estimated ga in function to the theoretical one shows a trend from theoretical overprediction to underpredic-tion of the velocities with increasing frequency.
(A 9) The set of values of said ratio at the peak frequency, fp for all analyzed records, has an ave rage of 0.83 and a standard deviation of 0.09.
(A 10) The set .of said ratios, evaluated at the peak frequency, is not correlated with the rate of turbulent energy.
(A11) The set of values of said ratio at approximately 2fp and 3fp
for all analyzed'records, have averages of 0.83 and 1.18 and standard deviations of 0.11 and 0.30 respectively.
(A12) Each set of said ratios, evaluated at 2fp and 3fp , is
corre-lated with the rate of turbulent energy.
(A13) The ratio of measured r.m.s. fluctuation of the horizontal
velocity to the theoretical one shows a trend correlated with
the turbulent energy rate from 25% theoretical overprediction
at negligible turbulent energy rates to 5% underprediction at
high turbulent energy rates.
(A14) For the relation between surface elevation and horizontal
ve-locity no significant difference has been found between the
results of measurements made at different elevations, as
42
-These results indicate that linear theory systematically overpre-dicts the horizontal velocities in the frequency range of high squared co-herence where the comparison is most realistic • The trends mentioned in items (A 1), (A 5)I (A 12) and (A 13) can be ascribed to the presence of
turbulent kinetic energy in the higher frequency ranges, as confirmed by the correlation of the trend rates with the turbulent energy rates.
The discrepancy between theoretical expected phase difference
be-tween the offshore velocity component and surface elevation and the phase difference estimated from the measurements mày be caused by the tube and electronics of the wave gauge, since the phase difference between the ver-tical velocity component and the offshore velocity component (measured by the same velocity meters) is within the 90% confidence band of the estima-te (based on sampling variability), see iestima-tem (A25), (A26) and (A27).
11 Relation between vertical velocity and surface elevation
In this case there is theoretically no bias due to directional ener-gy spreading and noise, so that the gain function may be estimated from the cross spectra. The estimated squared coherence between vertical velo-city and surface elevation appears to be low. This results in a high sam-pling variability in the estimates of phase and gain spectra. The total widths of the 90% confidence interval for these quantities in the energet-ic frequency band are 150 for the phase and 30% for the gaine
Results
(A1S) The estimated squared coherence between vertical velocity and surface elevation shows a trend with increasing frequencies.
(A16) The set of estimated values of squared coherence at fp' 2fp' .and 3fp' for all analyzed records, have averages of 0.75, 0.76 and 0.44 and standard deviations of 0.16,0.14 and 0.1S respectively.
(A17) Each set of the above values evaluated at fp' 2fp and 3fp is correlated with the rate of turbulent energy.
(A18) The theoretical phase difference of 90 ° is generally within the 90% confidence band of the estimate and the accuracy of the corrections applied.
(A19) The ave rage value of the estimated phase difference is for f
<
0.20 Hz about 10° higher than the theoretical value of 90°, for higher frequencies this discrepancy is smaller and lies within the margin of the accuracy of the phase correc-tions applied.(A20) The ratio of estimated gain function to the theoretical one shows a 40% theoretical overestimation in the energetic fre-quency range 0.10 - 0.40 Hz.
(A21) The set of said ratios, evaluated at fp' 2fp and 3fp' for all analyzed records, have averages of 0.59, 0.53 and 0.49 and standard deviations of 0.20, 0.09 and 0.13 respectively.
(A22) Each set of said ratios, evaluated at fp' 2fp and 3fp' is not correlated with the rate of turbulent enerqy.
(A23) The ratio of measured r.m.s. fluctuation of the vertical ve-locity to the theoretical one shows a trend correlated with the turbulent enerqy rate from 45% theoretical overprediction at negligible turbulent enerqy rates to 75% underprediction at high turbulent energy rates.
(A24) For the relation between vertical velocity component and the surface elevation significant differences have been found between measurernents at different elevations.
The low coherence (item A16) implies an only weakly linear relation between vertical velocity and surface elevation. This may be ascribed to the relatively large contribution of turbulent kinetic energy to the ver-tical velocity due to wave breaking, as confirmed by the correlation of the trend rates with the rate of turbulent energy. Since the phase and gain relations are based on the cross spectra they are theoretically not affected by the low coherence, which is confirmed by the results A18, A19 en A22.
- 44
-The estimated phases are as close to the linear theory prediction as can be expected in view of sampling variability and the phase correction applied. The discrepancy of these phases and the theoretical phases (A19) may be caused by the tube of the wave gauges used, since the phase differ-ences between vertical velocity component and offshore velocity component
(measured by the same velocity meter) is consistent with linear theory
within the sampling variability and since the same discrepancies were
found between the offshore velocity component and surface elevation (item A7, A2S, A2G and A27).
As far as the gain function is concerned item (A20) confirrns the
overprediction by linear theory as found in case of the horizontal veloci-ty. It is remarked that the overprediction is significant despite the re-latively large 90% confidence interval of 30% for the gaine
As far as the r.m.s. fluctuation of the vertical velocity is
con-cerned the overprediction by linear theory is only noticeable at low rates of turbulent energy. Due to higher turbulent rates the overprediction of the r.m.s. fluctuation may change into underprediction as high as 100%.
111 Relation between horizontal and vertical velocities
Results with respect to the gain function for this category were not derived, since the derivation has to be based on the auto spectra of two signals relatively strongly contarninatedby physical noise (turbulence). Instead the phase relationship was investigated, since the estimations in this case are not biassed by directional energy spreading nor by a possi-bLe systematic error in the estimated phase response of the wave gauge. However, the presence of significant turbulent kinetic energy - as indica-ted by the relatively low coherence - can introduce a bias in the estimat-ed phase relation.
Results
(A2S) The theoretical phase difference of 90° is generally within the 90% confidence band of the estimate.
