Spectral Tuning and Calibration of a Wave-Follower Buoy
Authors: Middleton F.H. , LeBlance L.R. , Czarnacki M.F. 
Published: December 2007

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A wave-follower buoy has many advantages over nearly all other types of wave-measuring devices, particularly on the open sea or far offshore. The buoy requires only the earth's gravitational field as a frame of reference instead of a pier, a piling, or perhaps a large spar buoy. Any of the ordinary-capacity or resistive-wave gauges must be attached to a solid place.
Any type of wave-follower (accelerometer) buoy will exhibit its own resonant dynamic behavior in the presence of a wave field. The engineer in need of wave information generally needs to know its power spectral density to be able to apply the data, whether the problem is wave forces on a structure or something else. It is not sufficient to simply double-integrate the vertical acceleration of the wave buoy and claim to have the desired wave-height information.
The objective of this paper is to compare an actual power spectral density in a wave field with that of the power spectral density in a wave field with that of the accelerometer-output power spectral density produced in the same wave field. We will show an empirical spectral transfer function that can be applied to the observed accelerometer spectrum to yield the desired power spectral density of the driving waves.
For this purpose, an ideal linear model for the buoy was proposed to keep the system design simple. Even with this incorrect model, the comparison of the precise spectrum with the compensated spectrum out of the buoy is quite good. The buoy used for this study was a Model WF-100 buoy manufactured by Coastal Data Service, Inc.
Let x(t) be the actual vertical displacement of the sea surface as a function of time. If y(t) is the output from the buoy accelerometer, then x(t) and Y\y(t) are related as in Eq. 1.
x(t) * h(t) = y(t),...................................(1)
where h(t) is the time domain representative of desired transfer function and * denotes convolution. In the frequency domain, we have Eq. 2.
X (f) x H(f) = Y(f)...................................(2)
Using the Fourier transform theory, one obtains Eq. 3.
Gy(f) H(f) 2 = ------------, Gx(f)
where
Gx(f) = X(f) 2 ..................................(3)
In this equation, means absolute value, means expected value, Gx(f) is the power spectral density of waves, and Gy(f) is the power spectral density of the accelerometer output. In this experiment, the two desired power spectral densities were generated by a minicomputer, power spectral densities were generated by a minicomputer, and the ratio of the two in Eq. 3 leads to the compensation function, H(f). The linear model is shown in Eq. 4. 1 1/2 sin (1.26f) H(f) = 32.2 [w2] -------- x ------------- 1 + 2f2 1.26f
I II III
...................................(4)
Factor I in Eq. 4 includes the accelerometer scale constant and w 2 corresponds to straight double integration of sinusoidal components.
P. 652