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Surface wave heights from pressure records
Coastal Engineering, 2 (1978) 55--67 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
55
SURFACE WAVE HEIGHTS FROM PRESSURE RECORDS
ROBERT A. GRACE
Department of Civil Engineering, University of Hawaii, Honolulu, Hawaii (U.S.A.) (Received July 5, 1977; accepted October 22, 1977)
ABSTRACT Grace, R.A., 1978. Surface wave heights from pressure records. Coastal Eng., 2: 55--67. Various studies in the past have considered the ability of the linear and other wave theories to predict surface wave conditions from near-bottom pressure records accurately. This paper is an extension of such work. In deterministic terms, considerations involve the ability of the linear and second-order cnoidal wave theories to predict accurately surface wave heights from individual, near-bottom, total pressure head ranges. In stochastic terms, the fidelity of the Gaussian random process in making predictions of a surface spectrum will be assessed. Both field and laboratory data are employed in this work. The field data were taken in the ocean off Honolulu, Hawaii, in 11.3 m of water, and involved swell with periods of from 12 to 17 sec and heights to 3.2 m. The laboratory data were taken for water depths of 2.9 and 3.5 m in the large Wave Research Facility of Oregon State University, Corvallis, Oregon, and involved periods of from 2 to 6 sec and heights to 1.5 m. It was found that the cnoidal wave theory is slightly superior to the linear theory in predicting surface heights for the ocean data. As found by other investigators, the linear theory was found to underestimate surface wave heights for low frequencies and overestimate them for high frequencies. It was found that the Gaussian random process model provided a theoretical surface spectrum that corresponded closely to that measured over a reasonable frequency band.
INTRODUCTION Waves can be m e a s u r e d in a variety o f ways (Grace, 1970), b u t t h e semis t a n d a r d m e t h o d o f carrying o u t such m e a s u r e m e n t s in r e c e n t years has involved use o f the floating Waverider b u o y . A p r o b l e m with slack-line-moored objects in the sea, however, is t h a t t h e y r u n the risk o f being s t r u c k b y ships or h o o k e d b y fishing trawls. In this regard, t h e Waverider o p e r a t e d b y t h e Maritime Services B o a r d o f N e w S o u t h Wales outside B o t a n y Bay, Australia, is t o r n loose f r o m its m o o r i n g s b y trawlers on t h e average o f f o u r times per year. T h e p a p e r o f M c L e o d ( 1 9 7 6 ) provides i n f o r m a t i o n o n such difficulties (as well as others) f o r Waveriders in Alaskan waters. A wave-measuring i n s t r u m e n t l o c a t e d o n t h e b o t t o m , such as a pressure sensor, is less susceptible t o being m a d e i n o p e r a t i v e b y ship a n d fishing ac-
56 tivities. However, there are still problems. One is the operational problem of protecting an electrical cable running to shore, if this approach is used for data transmission and power supply. Another problem, the topic of this paper, concerns the translation of information on wave pressure head variations at or near the b o t t o m into the concurrent surface wave heights. The conceptual problem with the pressure-measuring approach to waves is that the pressure head variations at the sea floor do not equal the surface level changes above - because of the vertical accelerations in the overlying water column. Questions about the correspondence between the two have led to doubts regarding the usefulness of pressure sensors in yielding valid results for the water surface. Concurrent surface and subsurface data on waves and wave pressures have been obtained by the writer both in the sea and in a large wave tank. These data have been processed for this paper to yield information on the ability of the linear wave t h e o r y (Ippen, 1966) and Laitone's (1960) second-order cnoidal wave theory to predict accurately surface wave heights from total pressure head variations at and near the bottom. Whereas this work emphasizes the prediction of surface wave characteristics from pressure records using deterministic approaches, a comparison between an observed surface spectrum and one predicted according to the zero-mean Gaussian wave model from the pressure spectrum is also included for the field data. THEORETICAL MODELS IN CONTEXT The linear (Airy) theory predicts that the surface wave height that gave rise to a total pressure head variation Hp a distance S above the (solid) b o t t o m is: cosh kd HAiry - cosh kS Hp
(1)
In eq. 1, d corresponds to the water depth and k is the wave number. This is defined by the equation:
k-
2~
(2)
LAiry where L Airy is the wave length predicted by the linear theory for a wave in a specified depth of water and having a particular period T or, equivalently, a given cyclical frequency f. Let the relationship between the linear-theorypredicted wave height and the measured (hopefully the actual) height H be defined by the relationship: H = n HAiry
(3)
Thus n = 1.0 corresponds to perfect surface-wave-height predictions from Hp and the Airy theory. The linear theory is supposed, on theoretical grounds, to be valid for:
57
H/d < < I
(4)
and only if: U < < Ulimi t where Ulimi t is some upper bound on the Ursell parameter defined by: (H/d) a
H/d u =
-
(d/LAiryY
(5)
(6)
(H/LAity) 2
Wiegel (1964), for example, has Ulimit ~ 105. Various other assumptions and constraints that are involved are discussed in Ippen (1966). For one thing, the water surface configuration is supposed to be sinusoidal - certainly not the case for shoaling waves. However, the practical utility of a theory is really demonstrated by its correspondence with experimental data on surface wave ordinate, velocities, accelerations, or pressures, independent o f theoretical niceties. The linear theory has been found to give excellent correspondence with measured laboratory and field wave data for near-bottom peak horizontal velocities (Goda, 1964; Grace, 1976). The waves for which such corre: spondence extends are those which, purely on theoretical grounds, would be a priori rejected from consideration according to the inequalities 4 and 5 and because of gross departures from a sinusoidal surface profile. For the most part, the laboratory and field waves used herein similarly do not conform with Airy theory requirements, but because of the encouraging results in terms of velocities, there is no reason to reject the linear theory from consideration. Interest herein will center on the readily-calculable predictions of the Airy theory. A simple relationship like eq. 1 does not exist for second-order cnoidal wave theory. However, this complicated theory has been programmed for computer solution by Isaacson (1976), and this approach can be employed to predict, for given T, d, Hp, and S, the corresponding theoretical wave height HcnoidaI. Let the relationship between this theoretical height and the actual one be defined by the relationship : H = m HcnoidaI
(7)
The customary probabilistic model for waves is the zero-mean Gaussian random process (G.r.p.). The surface is assumed to be composed of an infinity of infinitesimal-variance sinusoidal waves having a continuous spectrum of frequencies and with random phases uniformly distributed on the interval (0, 2~). Any individual c o m p o n e n t in this superposed array is an Airy wave. Thus the single-sided, mean-square spectral density of the sea surface can theoretically be obtained from that determined at the subsurface station, a distance S off the bottom, by the equation: ( cosh kd .f WG-r.P. (Jr) = cosh k S Wp (f) (8)
58
In eq. 8, Wp (f) and WG r p (f) are respectively the subsurface pressure head and the theoretical surface" ordinate spectra. The wave number k is a function of f. Let the actual surface spectrum be represented as:
W(f) = n 2 (f) WG"r.p. (f)
(9)
Just because real wave sequences are random does not mean ipso facto that a random wave model is any better than a deterministic one. The assumptions involved in the stochastic model may be grossly out of line with reality -more so in fact than applying a deterministic model to a random process. Consider the observation t h a t the distribution of surface wave ordinates, particularly in moderately shallow waters of engineering interest, is not even symmetric let alone Gaussian (e.g., Collins, 1967). However, in truth this may not be a serious problem; it was n o t for the Airy theory in deterministic form. The most important practical deficiency with the Gaussian random process model, in the writer's view, is the following. Assume a very narrow-band swell of mean radian frequency ¢o0 in moderately shallow water. Such swell is typified by broad, shallow troughs and very peaked crests. A spectral representation for such a configuration demonstrates appreciable values at frequencies substantially greater than ¢o0 -- in order to reproduce the peaked crest. But to associate that spectral value with the high frequency in terms of wave propagation is physically imperfect. The particular "wavelet" does n o t suffer dispersion -- but the crest remains an essentially stable configuration, over a very gradually shelving b o t t o m , and that spectral value for the high frequency is really more properly associated with w0. Higher frequency components are predicted by the theory to be rapidly attenuated with depth -- such that the spectral approach tends to underestimate, as an example, the a m o u n t of energy associated with near-bottom horizontal water motion (Grace, 1976). The standard spectral approach then suffers from its inherent assumption that the different frequency components are separate decoupled "wavelets" rather than simply a fitting of the profile of a non-linear surface wave of a more or less specific frequency. PAST RELATED EXPERIMENTAL WORK
Grace (1970) summarized the data available on deterministic and stochastic correlations between surface wave heights and pressure head variations through 1969. Some of the data sources are listed in the References. Although there was scatter in the data, the general trend was for the empirical correction factor n to increase through unity with decreasing wave frequency -- or equivalently with decreasing d/L. An important reference since that time is Esteva and Harris (1970). These authors compared theoretical predictions with measured results in spectral form.* Four gages were involved in this work: two pressure gages at different * T h e r e is s o m e c o n f u s i o n in t h e p a p e r o f Esteva a n d Harris ( 1 9 7 0 ) over w h e t h e r t h e y used eq. (9) for n2(f) or called this n(f).
59
depths of submergence; a step-resistance relay gage (Willimas, 1969); and a continuous-wire resistance gage. The mean water depth was 4.7 m, and one pressure sensor was immediately above the b o t t o m with the other 1.7 m above it. For the lower gage, n(f) dropped from about 1.05 at a frequency of approximately 0.07 Hz to n(f) ~ 0.95 for f ~ 0.32 Hz. The upper gage showed n(f) values consistently closer to unity. The most recent paper on this topic is apparently Tubman and Suhayda, 1976. EXPERIMENTAL ARRANGEMENTS
FOR PRESENT WORK
Ocean tests The field experimentation station for this study was located in 11.3 m of water, slightly over 400 m from the reclaimed coastline adjacent to Honolulu, Hawaii. The coral-rock b o t t o m in the vicinity of the pressure sensor, which was set on the sea floor, was comparatively smooth, and the mean b o t t o m slope was about 1 : 40. The surface sensor consisted of a commercial spirally wound resistance gage. This was m o u n t e d on a special tiltable wave mast described by Grace (1976). Field checks on calibration of the resistance wave gage were effected by comparing its traces to those read visually from the graduated portion of the wave mast. It has been established in tests by the (U.S.) National Oceanographic Instrumentation Center (1972) that the Bendix wave pressure system used somewhat underestimates the true pressure variations. A calibration was performed in the field, on a day of minor wave action, to discover what errors were involved. A second wave pressure system was used to approximately allow for whatever water level variations occurred during the tests. A diver continuously raised and lowered the Bendix pressure sensor a controlled distance (1.19 m) in three different series of tests of separate frequencies. The nominal raise--lower--raise periods involved were 12.6, 15.5, and 23.4 seconds, and the results are shown in Table I. Thereafter, supposed wave TABLE I Calibration of wave pressure system in the ocean T(sec)
Sample size
Actual distance (m)
Average indication on chart
Standard deviation (m)
Ratio
0.08 0.02 0.05
1.17 1.19 1.21
(m) 12.6 15.5 23.4
21 20 23
1.19 1.19 1.19
1.02 1.00 0.99
60 pressure heights on the chart were scaled up according t o the figures in the last c o l u m n o f Table I. Swell rather than wind waves comprised the experimental conditions. The periods o f this swell ranged f r om about 12 to 17 sec. Surface heights used for this paper ran f r o m 0.4 to 3.2 m. The ranges o f i m p o r t a n t wave parameters in the ocean experiments were as follows: 0.059 <~ d/LAity
<~ 0.106
0.032 ~< H/d
~< 0.284
3.4
< 54.2
~< U
Laboratory tests The lab o r ato r y work was done in the Wave Research Facility of Oregon State University at Corvallis, Oregon. This concret e wave flume is a unique facility because it can provide conditions approaching those f o u n d in the ocean. The overall length of the c onc r et e tank is 104.2 m, and the width and overall d ep th are 3.66 m and 4.57 m, respectively. Both periodic and random waves can be generated. F or the tests r e p o r t e d u p o n herein, the pressure sensor was m o u n t e d on a bracket th at was bolted t o the floor of the flume a b o u t 0.6 m from the side and 11.0 m f r o m the t oe of a beach used for wave dissipation. The b o t t o m 7.3 m o f this beach were set at a 1 on 6 slope with the remainder at a 1 on 12 slope. The pressure pickup was 0.18 m above the (horizontal) b o t t o m . Various calibrations o f the pressure sensor were effected by raising and lowering the device k n o w n distances in still water. T wo d if f er en t water depths were used for the tests: 2.90 and 3.51 m. Only m o n o c h r o m a t i c waves were used, and periods were selected so t h a t a comprehensive sequence of d/LAity values was obtained between the a p p r o x i m a t e end values of 0.1 and 0.5. Three separate wave trains were e m p l o y e d at each frequency. The first of these involved waves t hat satisfied the most limiting of t h e following three criteria: the largest wave t hat could be generated by the wave board; the largest wave t h a t would n o t spill over the t op of the flume; or the largest wave t ha t would n o t break. The second and third runs then involved waves nominally 0.15 and 0.30 m less in height than those of the first wave train. An e x c e p t i o n t o the above was made only for some waves of the highest frequencies where the waves in the three sequences were closer to each o t h e r in height. This was because the substantial pressure a t t e n u a t i o n of the higher freqencies made it desirable to obtain as high a pressure head variation as possible. Although the level of wave reflection f r om the beach was minor, waves used in the analysis were only those before such reflection would have exerted an influence -- and after the first atypical t w o waves for low frequencies and the first atypical f o u r f or high frequencies. An acoustic gage (in air) was
61 used to measure surface profiles, and heights were checked visually against staff.
a
RESULTS FOR DETERMINISTIC MODELS Pressure determinations for the second-order cnoidal wave theory were performed by Isaacson (personal communication, 1977) using his own computer program (Isaacson, 1976). Computations were run only for the ocean tests because Isaacson did not consider the laboratory waves consistent with theoretical constraints expressed for the second-order cnoidal theory in terms of T v f g / d and H / d , where g is the gravitational acceleration. The 141 ocean waves analyzed using both the linear and cnoidal wave theories gave the following comparative results in terms of mean values and standard deviations: n = 1.184 and s n = 0.127; m = 1.148 and s m = 0.125. Thus the second-order cnoidal theory is slightly superior to the linear theory in accurately predicting H from Hp. Whether this small advantage is worth the 1.60
!
!
Legend
x • o i
1.50 0
H ~ 0.61 m 0 . 6 2 - H ' - 1.22 m 1.23- H ~ 1.83 m 1 . 8 4 - H -~ 2 . 5 6 m 2.57- H ~ 3.55 m
5=I.18 1.40 x
•
•
n
a
•
•
•
1.30
~o -.~.
o
• • oO
•
1.2C
~e• • a o
•
_
• •
•
o
Io~
~
• x •x
•
I.IC •
t
x
•
•
I
x
x
!
e•
1.00
o
0.90
O.
80.• 5 0
I .060
I .0"70
I .080
I .090
I .100
I .I I0
.120
d/LAiry
Fig.1. Correction factors for Airy theory-predicted wave heights: ocean data.
62 considerable extra computational effort and expense is up to the individual and the situation. The variation in individual n values with the parameter d/LAity for the ocean tests is shown in Fig. 1. Delineating points in different wave height classes did n o t uncover any clear effect of wave height on the values obtained. The laboratory results are summarized in Fig.2 and in Table II. The 10series tests in Table II were for a water depth of 2.90 m, the 20-series for d = 3.51 m. The term U in Table II is the Ursell parameter defined in eq. 6 and e indicates the root-mean-square prediction error of the Airy theory. The clear trend for the data shown in Table II and Fig. 2 is for n to decrease with increasing d/LAity. The field data (Fig.l) were broken down into classes in an a t t e m p t to uncover any trends. The results are shown in Table III, and a tendency for n to decrease with increasing d/LAiry is apparent. Based on the trends shown in Fig.2 and Table III, it is suggested that the following equations be used for n: t 1.550-4.50
d/LAiry
, 0.06 ~
d/LAiry ~
0.10
(10a)
1.175-0.75
d/LAiry
, 0 . 1 0 % d/LAiry ~ 0.50
(10b)
n
The break in slope in eqs. 10 happens to correspond to the line of division between the laboratory and field data. This result is suspicious; however, the data of Shooter and Ellis (1967) show a similar trend. See Grace, 1970.
L.2
I
I
i
i
~
I
i
r
i
i
f
i
T I.I
r.O
i 0.9
T
0.8
0.7
0.1
'
'
0t 2
'
'
0i 5
. . . . 0.4 . .
0.5 d/LA,ry
Fig. 2. Correction f a c t o r s f o r A i r y theory-predicted wave heights: l a b o r a t o r y data. Bars i n d i c a t e 95% c o n f i d e n c e intervals.
(sec)
5.88 5.88 5.00 5.26 4.55 4.65 4.00 4.17 3.57 3.57 3.03 2.99 2.50 2.53 2.35 2.00 2.13
Te~t No.
11 21 12 22 13 23 14 24 15 25 16 26 17 27 28 18 29
T
10.8 9.8 9.2 8.8 8.4 7.8 7.3 7.0 6.5 6.0 5.6 5.0 4.6 4.2 3.9 3.7 3.6
T g~/-~
Remdts of laboratory tests
TABLE II
0.098 0.109 0.118 0.124 0.133 0.144 0.155 0.166 0.180 0.205 0.228 0.270 0.310 0.358 0.411 0.469 0.498
d/LAin, 0.277 0.276 0.335 0.329 0.352 0.355 0.383 0.416 0.353 0.407 0.429 0.348 0.336 0.290 0.255 0.257 0.230
Hmax/d 0.175 0.188 0.215 0.219 0.246 0.257 0.261 0.285 0.275 0.297 0.297 0.274 0.215 0.216 0.196 0.147 0.169
Hm~/d 28.8 23.2 24.1 21.4 19.9 17.1 15.9 15.1 10.9 9.7 8.3 4.8 3.5 2.3 1.5 1.2 0.9
Ureax
18.2 15.8 15.4 14.2 13.9 12.4 10.9 10.3 8.5 7.1 5.7 3.7 2.2 1.7 1.2 0.7 0.7
Um~,
6 6 9 9 9 9 7 8 9 9 8 9 9 9 9 9 9
Sample size 1.072 1.069 1.053 1.084 1.020 1.046 1.041 1.101 1.021 1.045 1.006 0.963 0.954 0.914 0.882 0.801 0.789
-~
0.049 0.026 0.038 0.020 0.024 0.026 0.050 0.047 0.043 0.034 0.026 0.019 0.030 0.022 0.026 0.061 0.054
s.
0.055 0.060 0.052 0.078 0.028 0.053 0.060 0.126 0.039 0.064 0.027 0.045 0.047 0.088 0.110 0.154 0.185
(m)
e
64 TABLE III Class breakdown of the Airy theory correction factors for ocean data
d/LAir] class mark 0.060 0.070 0.080 0.090 0.100
Sample size 3 30 60 44 4
~
sn
95% confidence interval
1.273 1.241 1.173 1.166 1.032
0.025 0.090 0.112 0.150 0.150
1.211--1.335 1.207--1.275 1.144--1.202 1.120--1.212 0.792--1.272
141 (total)
RESULTS FOR GAUSSIAN RANDOM PROCESS MODEL A pair o f c o n c u r r e n t surface and pressure records was digitized at 1/~-second intervals. The distribution of surface wave ordinates was n o t Gaussian but skewed to the right with a coefficient of skewness (Benjamin and Cornell, 1970) o f 0.26. However, researchers c o n t i n u e t o e m p l o y the Gaussian r a n d o m process mo d el despite this lack of c o n f o r m a n c e with a basic underlying p r o p e r t y o f the model. The coefficient of skewness for the pressure history was - 0 . 0 1 and the f r e q u e n c y distribution for these data was adequately fitted by t h e normal distribution. Spectra were d eter m i ned for the surface and pressure records using the Fast Fourier Tr an s f or m approach (Bendat and Piersol, 1971; Kinariwala et al., 1973) for sequences of 1,024 points. T he result of a point-by-point comparison of the two spectra is shown in Fig.3, and there is a notable correspondence between the measured and predicted surface spectra over an appreciable band o f d/LAir$. This precision would n o t have been anticipated from the deterministm results. The trend shown in Fig.3 is virtually identical to t h a t for one case o f Takahashi et al., 1967 (see Grace, 1970) for 0.3 <~ d/LAity ~ 0.5. Otherwise, the curve in Fig.3 is well below ot her spectral c o r r e c t i o n factor trends given in Grace (1970) and the need f or more, pert i nent i n f o r m a t i o n is indicated.
CONCLUSIONS Airy t h e o r y can be used t o predict individual surface wave heights from c o n c u r r e n t total pressure head variations at or near the sea fl oor as long as an empirical co r r ectio n f a c t or is included. This multiplicative f a c t o r n has been f o u n d to be greater than u n i t y for small ratios of water d e p t h t o wave length
65
2.0
,
,
1.8 1.6
1.4 n
1.2
•
I.C
. .
•
° o~
•
o
°*
0.~
0.6
O.4
0.2
O.C
.0
o'.i
012
013
o.~4
o.5
d/LAiry
Fig. 3. Correction factors for surface wave spectrum: ocean data.
and smaller than unity for larger values of this parameter. This trend has been quantified in eq. 10. The second-order cnoidal wave theory provides slightly better accuracy than the Airy theory but only over a certain band of wave parameters. Considering the limited extent of this applicability and the substantial complexity in evaluating pressures according to the cnoidal theory, this approach is not considered as useful practically as the linear technique. The approach of accurately predicting surface wave spectra from pressure spectra using the Gaussian random process approach has been found wanting by some investigators and promising by others. More data and analysis in this area would be useful. ACKNOWLEDGEMENTS
The National Sea Grant Program, of the National Oceanic and Atmospheric Administration, and the Marine Affairs Coordinator's Office, State of Hawaii,
66
provided funds for the field research reported upon herein and for the leave of absence wherein the experiments at Oregon State University were carried out. The Marine Affairs Coordinator was John Craven. The Directors of the Sea Grant Programs offices at the University of Hawaii and Oregon State University were respectively Ronald Linsky and William Wick. The associated grant numbers were 04-3-158-29 and 04-5-158-17. Assistance with the operations involved in the ocean experiments and with the taking, reducing, and presentation of the data was provided by Kent Reinhard, Joseph Castiel, Steven Nicinski, Arthur Shak, Elizabeth Leis, Frederick Casciano, Gabriel Zee, Charles Schuster, James Sands, Michael Rayfuse, Edward Noda, Edgar Bilderback, Henry Ho, and Sylvia Khong. John Nath and Terence Dibble provided extensive assistance with the Oregon State University tests. Michael Isaacson carried out the second-order cnoidal theory calculations using his own computer program.
REFERENCES Bendat, J.S. and Piersol, A.G., 1971. Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York, N.Y. Benjamin, J.tL and Cornell, C.A., 1970. Probability, Statistics and Decision for Civil Engineers. McGraw-Hill, New York, N.Y. Collins, J.I., 1967. Wave statistics from hurricane Dora. Am. Soc. Cir. Eng., J. Waterways Harbors Div., 93(WW2): 5236. Esteva, D. and Harris, D.L., 1970. Comparison of pressure and staff wave gage records. Proc. Coastal Eng. Conf., 12th, Washington, D.C., pp.101--116. Goda, Y., 1964. Wave Forces on a Vertical Circular Cylinder: Experiments and a Proposed Method of Wave Force Computation. Port and Harbour Technical Research Institute, Yokosuka, Japan, Report No. 8. Grace, R.A., 1970. How to measure waves. Ocean Ind., 9(2): 65--69. February. Grace, R.A., 1976. Near-bottom water motion under ocean waves. Proc. Conf. Coastal Eng., 15th, Honolulu, Hawaii, 3, pp. 2371--2386. Ippen, A.T. (Editor), 1966. Estuary and Coastline Hydrodynamics. McGraw-Hill, New York, N.Y. Isaacson, M., 1976. The second approximation to mass transport in cnoidal waves. J. Fluid Mech., 78: 445--457. Kinariwala, B.K., Kuo, F.F. and Tsao, N-K, 1973. Linear Circuits and Computation. John Wiley, New York, N.Y. Laitone, E.V., 1960. The Second Approximation to Cnoidal and Solitary Waves. J. Fluid Mech., 9: 430--444. McLeod, W.R., 1976. Operations experience with a wave and wind measurement program in the Gulf of Alaska. Proc. Eighth Ann. Offshore Technol. Conf., 8th, Houston, Texas, pp. 719--733. National Oceanographic Instrumentatior~,Center, 1972. Instrument Fact Sheet: M o d e l A-2/Q-6 Wave Recording System, The Bendix Corp., Environmental Science Division, IFS-73002, Washington, D.C. August. Shooter, J.A. and Ellis, G.E., 1967. Surface Waves and Dynamic Bottom Pressure at Buzzard's Bay, Massachusetts. Acoustical Report No. 292, University of Texas at Austin, Defense Research Laboratory.
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Takahashi, T., Suzuki, Y. and Nakai, T., 1967. Method of Analysis of the Data Observed with Wave Meters. Technical Note No. 39, Port and Harbour Research Institute of Japan, 11 (in Japanese; partial translation by Y. Goda and Y. Suzuki). Tubman, M.W. and Suhayda, J.N., 1976. Wave action and bottom movements in fine sediments. Proc. Coastal Eng. Conf., 15th, Honolulu, Hawaii, Vol. 2, pp. 1168--1183. Wiegel, ILL., 1964. Oceanographical Engineering. Prentice-Hall, Englewood Cliffs, N.J. Williams, L.C., 1969. CERC Wave Gages. Technical Memorandum No. 30, U.S. Army, Corps of Engineers, Coastal Engineering Research Center.
55
SURFACE WAVE HEIGHTS FROM PRESSURE RECORDS
ROBERT A. GRACE
Department of Civil Engineering, University of Hawaii, Honolulu, Hawaii (U.S.A.) (Received July 5, 1977; accepted October 22, 1977)
ABSTRACT Grace, R.A., 1978. Surface wave heights from pressure records. Coastal Eng., 2: 55--67. Various studies in the past have considered the ability of the linear and other wave theories to predict surface wave conditions from near-bottom pressure records accurately. This paper is an extension of such work. In deterministic terms, considerations involve the ability of the linear and second-order cnoidal wave theories to predict accurately surface wave heights from individual, near-bottom, total pressure head ranges. In stochastic terms, the fidelity of the Gaussian random process in making predictions of a surface spectrum will be assessed. Both field and laboratory data are employed in this work. The field data were taken in the ocean off Honolulu, Hawaii, in 11.3 m of water, and involved swell with periods of from 12 to 17 sec and heights to 3.2 m. The laboratory data were taken for water depths of 2.9 and 3.5 m in the large Wave Research Facility of Oregon State University, Corvallis, Oregon, and involved periods of from 2 to 6 sec and heights to 1.5 m. It was found that the cnoidal wave theory is slightly superior to the linear theory in predicting surface heights for the ocean data. As found by other investigators, the linear theory was found to underestimate surface wave heights for low frequencies and overestimate them for high frequencies. It was found that the Gaussian random process model provided a theoretical surface spectrum that corresponded closely to that measured over a reasonable frequency band.
INTRODUCTION Waves can be m e a s u r e d in a variety o f ways (Grace, 1970), b u t t h e semis t a n d a r d m e t h o d o f carrying o u t such m e a s u r e m e n t s in r e c e n t years has involved use o f the floating Waverider b u o y . A p r o b l e m with slack-line-moored objects in the sea, however, is t h a t t h e y r u n the risk o f being s t r u c k b y ships or h o o k e d b y fishing trawls. In this regard, t h e Waverider o p e r a t e d b y t h e Maritime Services B o a r d o f N e w S o u t h Wales outside B o t a n y Bay, Australia, is t o r n loose f r o m its m o o r i n g s b y trawlers on t h e average o f f o u r times per year. T h e p a p e r o f M c L e o d ( 1 9 7 6 ) provides i n f o r m a t i o n o n such difficulties (as well as others) f o r Waveriders in Alaskan waters. A wave-measuring i n s t r u m e n t l o c a t e d o n t h e b o t t o m , such as a pressure sensor, is less susceptible t o being m a d e i n o p e r a t i v e b y ship a n d fishing ac-
56 tivities. However, there are still problems. One is the operational problem of protecting an electrical cable running to shore, if this approach is used for data transmission and power supply. Another problem, the topic of this paper, concerns the translation of information on wave pressure head variations at or near the b o t t o m into the concurrent surface wave heights. The conceptual problem with the pressure-measuring approach to waves is that the pressure head variations at the sea floor do not equal the surface level changes above - because of the vertical accelerations in the overlying water column. Questions about the correspondence between the two have led to doubts regarding the usefulness of pressure sensors in yielding valid results for the water surface. Concurrent surface and subsurface data on waves and wave pressures have been obtained by the writer both in the sea and in a large wave tank. These data have been processed for this paper to yield information on the ability of the linear wave t h e o r y (Ippen, 1966) and Laitone's (1960) second-order cnoidal wave theory to predict accurately surface wave heights from total pressure head variations at and near the bottom. Whereas this work emphasizes the prediction of surface wave characteristics from pressure records using deterministic approaches, a comparison between an observed surface spectrum and one predicted according to the zero-mean Gaussian wave model from the pressure spectrum is also included for the field data. THEORETICAL MODELS IN CONTEXT The linear (Airy) theory predicts that the surface wave height that gave rise to a total pressure head variation Hp a distance S above the (solid) b o t t o m is: cosh kd HAiry - cosh kS Hp
(1)
In eq. 1, d corresponds to the water depth and k is the wave number. This is defined by the equation:
k-
2~
(2)
LAiry where L Airy is the wave length predicted by the linear theory for a wave in a specified depth of water and having a particular period T or, equivalently, a given cyclical frequency f. Let the relationship between the linear-theorypredicted wave height and the measured (hopefully the actual) height H be defined by the relationship: H = n HAiry
(3)
Thus n = 1.0 corresponds to perfect surface-wave-height predictions from Hp and the Airy theory. The linear theory is supposed, on theoretical grounds, to be valid for:
57
H/d < < I
(4)
and only if: U < < Ulimi t where Ulimi t is some upper bound on the Ursell parameter defined by: (H/d) a
H/d u =
-
(d/LAiryY
(5)
(6)
(H/LAity) 2
Wiegel (1964), for example, has Ulimit ~ 105. Various other assumptions and constraints that are involved are discussed in Ippen (1966). For one thing, the water surface configuration is supposed to be sinusoidal - certainly not the case for shoaling waves. However, the practical utility of a theory is really demonstrated by its correspondence with experimental data on surface wave ordinate, velocities, accelerations, or pressures, independent o f theoretical niceties. The linear theory has been found to give excellent correspondence with measured laboratory and field wave data for near-bottom peak horizontal velocities (Goda, 1964; Grace, 1976). The waves for which such corre: spondence extends are those which, purely on theoretical grounds, would be a priori rejected from consideration according to the inequalities 4 and 5 and because of gross departures from a sinusoidal surface profile. For the most part, the laboratory and field waves used herein similarly do not conform with Airy theory requirements, but because of the encouraging results in terms of velocities, there is no reason to reject the linear theory from consideration. Interest herein will center on the readily-calculable predictions of the Airy theory. A simple relationship like eq. 1 does not exist for second-order cnoidal wave theory. However, this complicated theory has been programmed for computer solution by Isaacson (1976), and this approach can be employed to predict, for given T, d, Hp, and S, the corresponding theoretical wave height HcnoidaI. Let the relationship between this theoretical height and the actual one be defined by the relationship : H = m HcnoidaI
(7)
The customary probabilistic model for waves is the zero-mean Gaussian random process (G.r.p.). The surface is assumed to be composed of an infinity of infinitesimal-variance sinusoidal waves having a continuous spectrum of frequencies and with random phases uniformly distributed on the interval (0, 2~). Any individual c o m p o n e n t in this superposed array is an Airy wave. Thus the single-sided, mean-square spectral density of the sea surface can theoretically be obtained from that determined at the subsurface station, a distance S off the bottom, by the equation: ( cosh kd .f WG-r.P. (Jr) = cosh k S Wp (f) (8)
58
In eq. 8, Wp (f) and WG r p (f) are respectively the subsurface pressure head and the theoretical surface" ordinate spectra. The wave number k is a function of f. Let the actual surface spectrum be represented as:
W(f) = n 2 (f) WG"r.p. (f)
(9)
Just because real wave sequences are random does not mean ipso facto that a random wave model is any better than a deterministic one. The assumptions involved in the stochastic model may be grossly out of line with reality -more so in fact than applying a deterministic model to a random process. Consider the observation t h a t the distribution of surface wave ordinates, particularly in moderately shallow waters of engineering interest, is not even symmetric let alone Gaussian (e.g., Collins, 1967). However, in truth this may not be a serious problem; it was n o t for the Airy theory in deterministic form. The most important practical deficiency with the Gaussian random process model, in the writer's view, is the following. Assume a very narrow-band swell of mean radian frequency ¢o0 in moderately shallow water. Such swell is typified by broad, shallow troughs and very peaked crests. A spectral representation for such a configuration demonstrates appreciable values at frequencies substantially greater than ¢o0 -- in order to reproduce the peaked crest. But to associate that spectral value with the high frequency in terms of wave propagation is physically imperfect. The particular "wavelet" does n o t suffer dispersion -- but the crest remains an essentially stable configuration, over a very gradually shelving b o t t o m , and that spectral value for the high frequency is really more properly associated with w0. Higher frequency components are predicted by the theory to be rapidly attenuated with depth -- such that the spectral approach tends to underestimate, as an example, the a m o u n t of energy associated with near-bottom horizontal water motion (Grace, 1976). The standard spectral approach then suffers from its inherent assumption that the different frequency components are separate decoupled "wavelets" rather than simply a fitting of the profile of a non-linear surface wave of a more or less specific frequency. PAST RELATED EXPERIMENTAL WORK
Grace (1970) summarized the data available on deterministic and stochastic correlations between surface wave heights and pressure head variations through 1969. Some of the data sources are listed in the References. Although there was scatter in the data, the general trend was for the empirical correction factor n to increase through unity with decreasing wave frequency -- or equivalently with decreasing d/L. An important reference since that time is Esteva and Harris (1970). These authors compared theoretical predictions with measured results in spectral form.* Four gages were involved in this work: two pressure gages at different * T h e r e is s o m e c o n f u s i o n in t h e p a p e r o f Esteva a n d Harris ( 1 9 7 0 ) over w h e t h e r t h e y used eq. (9) for n2(f) or called this n(f).
59
depths of submergence; a step-resistance relay gage (Willimas, 1969); and a continuous-wire resistance gage. The mean water depth was 4.7 m, and one pressure sensor was immediately above the b o t t o m with the other 1.7 m above it. For the lower gage, n(f) dropped from about 1.05 at a frequency of approximately 0.07 Hz to n(f) ~ 0.95 for f ~ 0.32 Hz. The upper gage showed n(f) values consistently closer to unity. The most recent paper on this topic is apparently Tubman and Suhayda, 1976. EXPERIMENTAL ARRANGEMENTS
FOR PRESENT WORK
Ocean tests The field experimentation station for this study was located in 11.3 m of water, slightly over 400 m from the reclaimed coastline adjacent to Honolulu, Hawaii. The coral-rock b o t t o m in the vicinity of the pressure sensor, which was set on the sea floor, was comparatively smooth, and the mean b o t t o m slope was about 1 : 40. The surface sensor consisted of a commercial spirally wound resistance gage. This was m o u n t e d on a special tiltable wave mast described by Grace (1976). Field checks on calibration of the resistance wave gage were effected by comparing its traces to those read visually from the graduated portion of the wave mast. It has been established in tests by the (U.S.) National Oceanographic Instrumentation Center (1972) that the Bendix wave pressure system used somewhat underestimates the true pressure variations. A calibration was performed in the field, on a day of minor wave action, to discover what errors were involved. A second wave pressure system was used to approximately allow for whatever water level variations occurred during the tests. A diver continuously raised and lowered the Bendix pressure sensor a controlled distance (1.19 m) in three different series of tests of separate frequencies. The nominal raise--lower--raise periods involved were 12.6, 15.5, and 23.4 seconds, and the results are shown in Table I. Thereafter, supposed wave TABLE I Calibration of wave pressure system in the ocean T(sec)
Sample size
Actual distance (m)
Average indication on chart
Standard deviation (m)
Ratio
0.08 0.02 0.05
1.17 1.19 1.21
(m) 12.6 15.5 23.4
21 20 23
1.19 1.19 1.19
1.02 1.00 0.99
60 pressure heights on the chart were scaled up according t o the figures in the last c o l u m n o f Table I. Swell rather than wind waves comprised the experimental conditions. The periods o f this swell ranged f r om about 12 to 17 sec. Surface heights used for this paper ran f r o m 0.4 to 3.2 m. The ranges o f i m p o r t a n t wave parameters in the ocean experiments were as follows: 0.059 <~ d/LAity
<~ 0.106
0.032 ~< H/d
~< 0.284
3.4
< 54.2
~< U
Laboratory tests The lab o r ato r y work was done in the Wave Research Facility of Oregon State University at Corvallis, Oregon. This concret e wave flume is a unique facility because it can provide conditions approaching those f o u n d in the ocean. The overall length of the c onc r et e tank is 104.2 m, and the width and overall d ep th are 3.66 m and 4.57 m, respectively. Both periodic and random waves can be generated. F or the tests r e p o r t e d u p o n herein, the pressure sensor was m o u n t e d on a bracket th at was bolted t o the floor of the flume a b o u t 0.6 m from the side and 11.0 m f r o m the t oe of a beach used for wave dissipation. The b o t t o m 7.3 m o f this beach were set at a 1 on 6 slope with the remainder at a 1 on 12 slope. The pressure pickup was 0.18 m above the (horizontal) b o t t o m . Various calibrations o f the pressure sensor were effected by raising and lowering the device k n o w n distances in still water. T wo d if f er en t water depths were used for the tests: 2.90 and 3.51 m. Only m o n o c h r o m a t i c waves were used, and periods were selected so t h a t a comprehensive sequence of d/LAity values was obtained between the a p p r o x i m a t e end values of 0.1 and 0.5. Three separate wave trains were e m p l o y e d at each frequency. The first of these involved waves t hat satisfied the most limiting of t h e following three criteria: the largest wave t hat could be generated by the wave board; the largest wave t h a t would n o t spill over the t op of the flume; or the largest wave t ha t would n o t break. The second and third runs then involved waves nominally 0.15 and 0.30 m less in height than those of the first wave train. An e x c e p t i o n t o the above was made only for some waves of the highest frequencies where the waves in the three sequences were closer to each o t h e r in height. This was because the substantial pressure a t t e n u a t i o n of the higher freqencies made it desirable to obtain as high a pressure head variation as possible. Although the level of wave reflection f r om the beach was minor, waves used in the analysis were only those before such reflection would have exerted an influence -- and after the first atypical t w o waves for low frequencies and the first atypical f o u r f or high frequencies. An acoustic gage (in air) was
61 used to measure surface profiles, and heights were checked visually against staff.
a
RESULTS FOR DETERMINISTIC MODELS Pressure determinations for the second-order cnoidal wave theory were performed by Isaacson (personal communication, 1977) using his own computer program (Isaacson, 1976). Computations were run only for the ocean tests because Isaacson did not consider the laboratory waves consistent with theoretical constraints expressed for the second-order cnoidal theory in terms of T v f g / d and H / d , where g is the gravitational acceleration. The 141 ocean waves analyzed using both the linear and cnoidal wave theories gave the following comparative results in terms of mean values and standard deviations: n = 1.184 and s n = 0.127; m = 1.148 and s m = 0.125. Thus the second-order cnoidal theory is slightly superior to the linear theory in accurately predicting H from Hp. Whether this small advantage is worth the 1.60
!
!
Legend
x • o i
1.50 0
H ~ 0.61 m 0 . 6 2 - H ' - 1.22 m 1.23- H ~ 1.83 m 1 . 8 4 - H -~ 2 . 5 6 m 2.57- H ~ 3.55 m
5=I.18 1.40 x
•
•
n
a
•
•
•
1.30
~o -.~.
o
• • oO
•
1.2C
~e• • a o
•
_
• •
•
o
Io~
~
• x •x
•
I.IC •
t
x
•
•
I
x
x
!
e•
1.00
o
0.90
O.
80.• 5 0
I .060
I .0"70
I .080
I .090
I .100
I .I I0
.120
d/LAiry
Fig.1. Correction factors for Airy theory-predicted wave heights: ocean data.
62 considerable extra computational effort and expense is up to the individual and the situation. The variation in individual n values with the parameter d/LAity for the ocean tests is shown in Fig. 1. Delineating points in different wave height classes did n o t uncover any clear effect of wave height on the values obtained. The laboratory results are summarized in Fig.2 and in Table II. The 10series tests in Table II were for a water depth of 2.90 m, the 20-series for d = 3.51 m. The term U in Table II is the Ursell parameter defined in eq. 6 and e indicates the root-mean-square prediction error of the Airy theory. The clear trend for the data shown in Table II and Fig. 2 is for n to decrease with increasing d/LAity. The field data (Fig.l) were broken down into classes in an a t t e m p t to uncover any trends. The results are shown in Table III, and a tendency for n to decrease with increasing d/LAiry is apparent. Based on the trends shown in Fig.2 and Table III, it is suggested that the following equations be used for n: t 1.550-4.50
d/LAiry
, 0.06 ~
d/LAiry ~
0.10
(10a)
1.175-0.75
d/LAiry
, 0 . 1 0 % d/LAiry ~ 0.50
(10b)
n
The break in slope in eqs. 10 happens to correspond to the line of division between the laboratory and field data. This result is suspicious; however, the data of Shooter and Ellis (1967) show a similar trend. See Grace, 1970.
L.2
I
I
i
i
~
I
i
r
i
i
f
i
T I.I
r.O
i 0.9
T
0.8
0.7
0.1
'
'
0t 2
'
'
0i 5
. . . . 0.4 . .
0.5 d/LA,ry
Fig. 2. Correction f a c t o r s f o r A i r y theory-predicted wave heights: l a b o r a t o r y data. Bars i n d i c a t e 95% c o n f i d e n c e intervals.
(sec)
5.88 5.88 5.00 5.26 4.55 4.65 4.00 4.17 3.57 3.57 3.03 2.99 2.50 2.53 2.35 2.00 2.13
Te~t No.
11 21 12 22 13 23 14 24 15 25 16 26 17 27 28 18 29
T
10.8 9.8 9.2 8.8 8.4 7.8 7.3 7.0 6.5 6.0 5.6 5.0 4.6 4.2 3.9 3.7 3.6
T g~/-~
Remdts of laboratory tests
TABLE II
0.098 0.109 0.118 0.124 0.133 0.144 0.155 0.166 0.180 0.205 0.228 0.270 0.310 0.358 0.411 0.469 0.498
d/LAin, 0.277 0.276 0.335 0.329 0.352 0.355 0.383 0.416 0.353 0.407 0.429 0.348 0.336 0.290 0.255 0.257 0.230
Hmax/d 0.175 0.188 0.215 0.219 0.246 0.257 0.261 0.285 0.275 0.297 0.297 0.274 0.215 0.216 0.196 0.147 0.169
Hm~/d 28.8 23.2 24.1 21.4 19.9 17.1 15.9 15.1 10.9 9.7 8.3 4.8 3.5 2.3 1.5 1.2 0.9
Ureax
18.2 15.8 15.4 14.2 13.9 12.4 10.9 10.3 8.5 7.1 5.7 3.7 2.2 1.7 1.2 0.7 0.7
Um~,
6 6 9 9 9 9 7 8 9 9 8 9 9 9 9 9 9
Sample size 1.072 1.069 1.053 1.084 1.020 1.046 1.041 1.101 1.021 1.045 1.006 0.963 0.954 0.914 0.882 0.801 0.789
-~
0.049 0.026 0.038 0.020 0.024 0.026 0.050 0.047 0.043 0.034 0.026 0.019 0.030 0.022 0.026 0.061 0.054
s.
0.055 0.060 0.052 0.078 0.028 0.053 0.060 0.126 0.039 0.064 0.027 0.045 0.047 0.088 0.110 0.154 0.185
(m)
e
64 TABLE III Class breakdown of the Airy theory correction factors for ocean data
d/LAir] class mark 0.060 0.070 0.080 0.090 0.100
Sample size 3 30 60 44 4
~
sn
95% confidence interval
1.273 1.241 1.173 1.166 1.032
0.025 0.090 0.112 0.150 0.150
1.211--1.335 1.207--1.275 1.144--1.202 1.120--1.212 0.792--1.272
141 (total)
RESULTS FOR GAUSSIAN RANDOM PROCESS MODEL A pair o f c o n c u r r e n t surface and pressure records was digitized at 1/~-second intervals. The distribution of surface wave ordinates was n o t Gaussian but skewed to the right with a coefficient of skewness (Benjamin and Cornell, 1970) o f 0.26. However, researchers c o n t i n u e t o e m p l o y the Gaussian r a n d o m process mo d el despite this lack of c o n f o r m a n c e with a basic underlying p r o p e r t y o f the model. The coefficient of skewness for the pressure history was - 0 . 0 1 and the f r e q u e n c y distribution for these data was adequately fitted by t h e normal distribution. Spectra were d eter m i ned for the surface and pressure records using the Fast Fourier Tr an s f or m approach (Bendat and Piersol, 1971; Kinariwala et al., 1973) for sequences of 1,024 points. T he result of a point-by-point comparison of the two spectra is shown in Fig.3, and there is a notable correspondence between the measured and predicted surface spectra over an appreciable band o f d/LAir$. This precision would n o t have been anticipated from the deterministm results. The trend shown in Fig.3 is virtually identical to t h a t for one case o f Takahashi et al., 1967 (see Grace, 1970) for 0.3 <~ d/LAity ~ 0.5. Otherwise, the curve in Fig.3 is well below ot her spectral c o r r e c t i o n factor trends given in Grace (1970) and the need f or more, pert i nent i n f o r m a t i o n is indicated.
CONCLUSIONS Airy t h e o r y can be used t o predict individual surface wave heights from c o n c u r r e n t total pressure head variations at or near the sea fl oor as long as an empirical co r r ectio n f a c t or is included. This multiplicative f a c t o r n has been f o u n d to be greater than u n i t y for small ratios of water d e p t h t o wave length
65
2.0
,
,
1.8 1.6
1.4 n
1.2
•
I.C
. .
•
° o~
•
o
°*
0.~
0.6
O.4
0.2
O.C
.0
o'.i
012
013
o.~4
o.5
d/LAiry
Fig. 3. Correction factors for surface wave spectrum: ocean data.
and smaller than unity for larger values of this parameter. This trend has been quantified in eq. 10. The second-order cnoidal wave theory provides slightly better accuracy than the Airy theory but only over a certain band of wave parameters. Considering the limited extent of this applicability and the substantial complexity in evaluating pressures according to the cnoidal theory, this approach is not considered as useful practically as the linear technique. The approach of accurately predicting surface wave spectra from pressure spectra using the Gaussian random process approach has been found wanting by some investigators and promising by others. More data and analysis in this area would be useful. ACKNOWLEDGEMENTS
The National Sea Grant Program, of the National Oceanic and Atmospheric Administration, and the Marine Affairs Coordinator's Office, State of Hawaii,
66
provided funds for the field research reported upon herein and for the leave of absence wherein the experiments at Oregon State University were carried out. The Marine Affairs Coordinator was John Craven. The Directors of the Sea Grant Programs offices at the University of Hawaii and Oregon State University were respectively Ronald Linsky and William Wick. The associated grant numbers were 04-3-158-29 and 04-5-158-17. Assistance with the operations involved in the ocean experiments and with the taking, reducing, and presentation of the data was provided by Kent Reinhard, Joseph Castiel, Steven Nicinski, Arthur Shak, Elizabeth Leis, Frederick Casciano, Gabriel Zee, Charles Schuster, James Sands, Michael Rayfuse, Edward Noda, Edgar Bilderback, Henry Ho, and Sylvia Khong. John Nath and Terence Dibble provided extensive assistance with the Oregon State University tests. Michael Isaacson carried out the second-order cnoidal theory calculations using his own computer program.
REFERENCES Bendat, J.S. and Piersol, A.G., 1971. Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York, N.Y. Benjamin, J.tL and Cornell, C.A., 1970. Probability, Statistics and Decision for Civil Engineers. McGraw-Hill, New York, N.Y. Collins, J.I., 1967. Wave statistics from hurricane Dora. Am. Soc. Cir. Eng., J. Waterways Harbors Div., 93(WW2): 5236. Esteva, D. and Harris, D.L., 1970. Comparison of pressure and staff wave gage records. Proc. Coastal Eng. Conf., 12th, Washington, D.C., pp.101--116. Goda, Y., 1964. Wave Forces on a Vertical Circular Cylinder: Experiments and a Proposed Method of Wave Force Computation. Port and Harbour Technical Research Institute, Yokosuka, Japan, Report No. 8. Grace, R.A., 1970. How to measure waves. Ocean Ind., 9(2): 65--69. February. Grace, R.A., 1976. Near-bottom water motion under ocean waves. Proc. Conf. Coastal Eng., 15th, Honolulu, Hawaii, 3, pp. 2371--2386. Ippen, A.T. (Editor), 1966. Estuary and Coastline Hydrodynamics. McGraw-Hill, New York, N.Y. Isaacson, M., 1976. The second approximation to mass transport in cnoidal waves. J. Fluid Mech., 78: 445--457. Kinariwala, B.K., Kuo, F.F. and Tsao, N-K, 1973. Linear Circuits and Computation. John Wiley, New York, N.Y. Laitone, E.V., 1960. The Second Approximation to Cnoidal and Solitary Waves. J. Fluid Mech., 9: 430--444. McLeod, W.R., 1976. Operations experience with a wave and wind measurement program in the Gulf of Alaska. Proc. Eighth Ann. Offshore Technol. Conf., 8th, Houston, Texas, pp. 719--733. National Oceanographic Instrumentatior~,Center, 1972. Instrument Fact Sheet: M o d e l A-2/Q-6 Wave Recording System, The Bendix Corp., Environmental Science Division, IFS-73002, Washington, D.C. August. Shooter, J.A. and Ellis, G.E., 1967. Surface Waves and Dynamic Bottom Pressure at Buzzard's Bay, Massachusetts. Acoustical Report No. 292, University of Texas at Austin, Defense Research Laboratory.
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Takahashi, T., Suzuki, Y. and Nakai, T., 1967. Method of Analysis of the Data Observed with Wave Meters. Technical Note No. 39, Port and Harbour Research Institute of Japan, 11 (in Japanese; partial translation by Y. Goda and Y. Suzuki). Tubman, M.W. and Suhayda, J.N., 1976. Wave action and bottom movements in fine sediments. Proc. Coastal Eng. Conf., 15th, Honolulu, Hawaii, Vol. 2, pp. 1168--1183. Wiegel, ILL., 1964. Oceanographical Engineering. Prentice-Hall, Englewood Cliffs, N.J. Williams, L.C., 1969. CERC Wave Gages. Technical Memorandum No. 30, U.S. Army, Corps of Engineers, Coastal Engineering Research Center.