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INTERNAL DOCUMENT 1
WAVE MEASUREMENT FOR JAGUAR SEA TRIALS AT OREGRUND SWEDEN
17/18th June 1983
by
J.A.Swing & K.G.Birch
Internal Document No. 188
[ This document should not be cited in a published bibliography, and is supplied for the use of the recipient only].
INSTITUTE OF OCEAIMOQRAPHIC
SCIENCES
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03 VVi
INSTITUTE OF OCEANOGRAPHIC SCIENCES
Wormley, Godalming, Surrey GU8 5UB (042-879-4141)
(Director: Dr. A. S. Laughton)
Bidston Observatory,
Birkenhead,
Merseyside L43 7RA
(051-653-8633)
(Assistant Director: Dr. D. E. Cartwright)
Crossway,
Taunton,
Somerset TA1 2DW
(0823-86211)
(Assistant Director: M .J . Tucker)
WAVE MEASUREMENT FOR JAGUAR SEA TRIALS AT OREGRUND SWEDEN
17/18th June 1983
by
J.A.Ewing & K.G.Birch
Internal Document No. 188
Institute of Oceanographic Sciences, Brook Road Godalming, Surrey GU8 SUB, U.K.
Jaguar Sea Trials
Report on wave measurements made by the Institute of Oceanographic
Sciences, Wormley during the period 17-18 June 1983, at the request of
Burness, Corlett & Partners Ltd.
1. Introduction
The purpose of the work was to obtain estimates of the directional
wave spectrum in conjunction with sea trials of "Jaguar".
The system employed to obtain the above information was the
pitch-roll buoy of the I.O.S. The buoy, which measures 1.2 m diameter,
and which weighs approximately 70 kg, was deployed from the lifeboat
"K.A. Wallenberg". Measurements taken in the buoy were transmitted
along a buoyant cable to the lifeboat, thus enabling a continuous
monitoring of the data on the "K.A. Wallenberg".
Six measurements were made during the period 17-18 June 1983. The
results from these measurements, together with the method of analysis,
are discussed in the following sections.
2. Measurements
The pitch-roll buoy is a surface-following wave buoy which
measures the vertical acceleration of the surface together with the
pitch and roll angle of the surface using a gyroscope. Additionally, a
magnetic compass is used to transform the pitch and roll signals to
axes referred to true North-East axes. Thus, after transformation to
true North, three signals are obtained, namely the vertical
acceleration and two orthogonal components of surface slope. This
information is required in the analysis.
The sensors in the buoy were calibrated in the laboratory before
the trials and the calibrations are accurate to within ±2%. Linearity
of sensors is better than ±^%.
- 1
Data from the buoy were digitized at 0.5 sec intervals and each
recording lasted 35 mins to allow accurate estimates of the wave
parameters. The data were stored on magnetic tape ready for analysis
in the laboratory after the trials.
During the deployment of the buoy, care was taken to ensure that
the buoyant cable to the boat was at all times slack. Furthermore,
situations where the boat might shield the waves to the buoy or reflect
waves onto the buoy were avoided by careful positioning of
"K.A. Wallenberg".
All measurements were made at a fixed station 6(f 8'N, 19 4'E.
Magnetic variation was taken as f East. Water depth was 100 m.
Six records were taken on 17 and 18 June 1983. Table 1 gives
details of the times of these records (all times are Swedish Summer
Time) together with visual estimates of the wave characteristics and
wind speed and direction.
Table 1
Record Day Time at Wind speed Visual wave height number (June 1983) start of record and direction and direction (from)
1 17 1427 15 m/s. 35CP 1h m, 350° 2 17 1605 12 m/s. 33CP <13? m, 35CP
3 17 1806 11 m/s. 330P l-IJj m, 350°
4 18 0940 3-5 m/s. 330P h m, 350P 5 18 1504 1-2 m/s. 170° <1^m, 350°
6 18 1620 <1 m/s. 32CP 10 cm, 35CP
3. Theory and analysis of the pitch-roll buoy system
The method of analysis for obtaining wave directional spectra from
the pitch-roll buoy system has been described by Longuet-Higgins,
Cartwright and Smith (1963). Let the subscript 1 denote the series of
vertical acceleration (in units of 'g'). And let subscripts 2 and 3
- 2 -
denote the two components of surface slope in the North and East
directions respectively. (The pitch, roll and compass channels are
used to derive the components of surface slope). Then, it can be shown
that the six cross-spectra derived from this system are given by
C^^(f) = j^^(2%f)*F(f,8)d8,
Cggff) = /^^k^cos^G F(f,6)d0,
Cggff) = /^^k^sin^e F(f,0)d9,
~ /^^k(2%f)^cos8 F(f,9)d9,
= J^^k(2itf)^sin0 F(f,9)d9,
CggCf) = /^^k^cosG F(f,9)d9,
where Cj j and Qj j are co- and quad spectra of series i with j
respectively. F{f,9) is the directional wave spectrum with respect to
frequency f and direction of propagation 0. k is the wave number.
Only five of the cross-spectra are independent. This allows estimation
of the five Fourier coefficients in the expansion of F(f,6), namely,
a + ib = — f^^e^'^^F(f,9)d9 ; n = 0,1,2 n n ic o
In the calculations, it is convenient to compute the normalized angular
harmonics A-j = a-j/aQ etc. a^ is the one-dimensional spectrum
obtained by integrating F(f,9) over all,directions. Thus,
^12 ^13 1 " /LC,,(C22 + C„)j 1 - /LC,,(C22 + C,,)]-
=22 - =33 _ 2=23 A = B = ^ =22 =33 ^ =22 * =33
- 3 -
In the above, use is made of the relation
11 _ {2%f) ^ tanh kh. (1) =22 =33 9k
Equation (1) is, in effect, the dispersion relation for waves of small
amplitude in water of depth h. It provides a check on the correct
functioning of the system and on the analysis. This normalization of
the angular harmonics removes the dependence on the wave number k which
occurs in the derivation of the cross-spectra.
The cross-spectral analysis of the vertical acceleration and two
components of surface slope were made using the Fast Fourier Transform
(FFT). Each signal of length 2000 s was digitized at 0.5 s interval
and transformed using the FFT. The resulting cross-spectra were then
averaged over 20 successive harmonics to produce smoothed cross-spectra
defined at 0.01 Hz interval with 40 degrees of freedom. The wave
height spectrum was obtained by dividing the acceleration spectrum by
( f r e q u e n c y ) ^ before smoothing over successive harmonics. The above
procedure was adopted for records 3-6.
For records 1 and 2 it was found that the accelerometer signals
occasionally went off scale. This was subsequently found to be due to
condensation in the instrument compartment. Erroneous parts of records
1 and 2 were readily identified and by-passed using a computer program
which was based on an ensemble-averaging scheme using sections of
length 128 sec. The results for records 1 and 2 therefore have a
resolution of 1/128 Hz with degrees of freedom of 26 and 22
respectively.
If we assume a unimodal directional distribution of the form
cos^s^O - 6) (2)
then a number of statistical parameters may be derived (Cartwright,
1963).
- 4
These parameters, which are displayed in the computer print-out,
are
(a) the mean wave direction G-] (= TH1) from the first-order
angular harmonics (direction towards which the waves are travelling),
(b) the mean wave direction 8^2 TH12) from the second-order
angular harmonics (0-j2 is ambiguous to 18(P). Direction towards
which the waves are travelling.
(c) spread parameters S-] (= SP1) and S2 (=SP2) from the first
and second order angular harmonics.
(d) spread parameter 82 (=TH2) which for a narrow directional
distribution, is equal to the root-mean-square angular spread.
- tanh kh fc,, + Cg,]'
From (1), the expected value of RATIO is thus unity.
4. Results
The main results from the analysis are contained in the computer
listings at the end of this report. The print-out gives the
one-dimensional wave spectrum, E(f), the angular harmonics
and statistical parameters which may be used for interpreting the
results. We suggest an appropriate directional distribution function
of the form (2) may be used with s = % ( s + S2) and 8 = 8-|.
Table 2 sets out some of the reduced statistics obtained from the
analysis of the six wave buoy records. The significant wave height was
defined as 4 x(variance)^ over the range of frequencies 0.09 Hz to 0.50
Hz. Mean zero-crossing period was obtained from evaluating the second
moment of the spectrum over the same range of frequencies.
- 5 -
Table 2
Sig. wave Mean zero- Spectral Direction of Record height crossing period peak period propagation at number (m) (sec) (sec) spectral peak period
1 0.97 5.1 8.0 18^ 2 0.98 5.5 7.5 196P 3 0.76 5.1 8.0 19"P 4 0.45 4.5 5.4 200P 5 0.21 3. 7 4 . 7 195P 6 0.16 3.7 4.4 191°
Statistical errors associated with the analysis of these data are
summarized in Table 3; the standard deviation of the parameters were
obtained using the work of Long (1980).
Table 3
Standard deviation of
Record Sig. wave 9 -] at spectral S-],S2 number ht. (m) peak (deg)
1 ) 3 35% of s 2 ) 3 38% of s 3 ) 3% of sig. 3
4 ) wave ht. 3 5 ) 3 30% of s 6 ) 3
References
Institute of Oceanographic Sciences, Wormley, Surrey, England, 9th August 1983.
Cartwright, D.E. "The use of directional spectra in studying the output
of a wave recorder on a moving ship". In Ocean Wave Spectra,
Prentice-Hall, 203-218, 1963.
Long, R.B. "The statistical evaluation of directional spectrum
estimates derived from pitch/roll buoy data". J. Phys. Oceanogr.,
9, 373-381,1979.
Longuet-Higgins, M.S., D.E. Cartwright and N.D. Smith. "Observations
of the directional spectrum of sea waves using the motions of a
floating buoy". In Ocean Wave Spectra, Prentice-Hall, 111-136,
1963.
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