The measurement of power spectra is a problem of steadily increasing importance which appears to some to be primarily a problem in statistical estimation. Others may see it as a problem of instrumentation, recording and analysis which vitally involves the ideas of transmission theory. Actually, ideas and techniques from both fields are needed. When they are combined, they provide a basis for developing the insight necessary (i) to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates and (ii) to interpret these estimates correctly and usefully. This account attempts to provide and relate the necessary ideas and techniques in reasonable detail — Part I of this article appeared in the January, 1958 issue of THE BELL SYSTEM TECHNICAL JOURNAL.

The Measurement Of Power Spectra: From The Point Of View Of Communications Engineering

1. Introduction 2. Observation techniques 3. Description of ocean waves 4. Statistics 5. Linear wave theory (oceanic waters) 6. Waves in oceanic waters 7. Linear wave theory (coastal waters) 8. Waves in coastal waters 9. The SWAN wave model Appendices References Index.

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Waves in Oceanic and Coastal Waters

Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences

The cold upper halcoline of the Arctic Ocean is maintained by large-scale lateral advection from the adjoining continental shelves, where dense and saline shelf water is produced during freezing; the salinization of the water column is especially pronounced in certain areas of persistent ice divergence. Estimates show the annual rate at which the dense shelf water feeds into the Polar Basin is probably in the neighborhood of 2.5 x 106 m3 s−1; this is of the same order as the inflow of warm and saline water from the Atlantic. A consequence of this process is that the halocline must be a heat sink for the underlying Atlantic water, thereby shielding the ice cover from an upward heat flux. The Atlantic water is thus linked rather directly to the enormous shelf seas that border the Polar Basin. Proposed massive river diversions in the Arctic could, by increasing the shelf salinities and driving a deeper flow into the interior, cause a thinning of the halocline and place the Atlantic water in more direct contact with the surface mixed layer.

On the halocline of the Arctic Ocean

The statistical density function is derived for a variable (such as the surface elevation in a random sea) that is ‘weakly non-linear’. In the first approximation the distribution is Gaussian, as is well known. In higher approximations it is shown that the distribution is given by successive sums of a Gram–Charlier series; not quite in the form that has sometimes been used as an empirical fit for observed distributions, but in a modified form due to Edgeworth. It is shown that the cumulants of the distribution are much simpler to calculate than the corresponding moments; and the approximate distributions are in fact derived by inversion of the cumulant-generating function. The theory is applied to random surface waves on water. The third cumulant and hence the skewness of the distribution of surface elevation is evaluated explicity in terms of the directional energy spectrum. It is shown that the skewness λ 3 is generally positive, and positive upper and lower bounds for λ 3 are derived. The theoretical results are compared with some measurements made by Kinsman (1960). It is found that for free, undamped surface waves the skewness of the distribution of surface slopes is of a higher order than the skewness of the surface elevation. Hence the observed skewness of the slopes may be a sensitive indicator of energy transfer and dissipation within the water.

The effect of non-linearities on statistical distributions in the theory of sea waves

Principles of Physical Oceanography

: Contents: The properties of ocean waves; The generation and forecasting of sea waves; Wave propagation and forecasting swell waves--simple models; Wave observation techniques; Synoptic forecasting methods and weather map analysis; Wave refraction; Operational applications of forecasting methods; Verification by forecasts and observations.

Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics

Publisher Summary The purpose of this chapter is to describe the new theoretical developments, which give a more complete and more accurate description of the analysis, generation, propagation, and refraction of wind generated gravity waves. A special path of development through hydrodynamic theory permits a link with time series theory that yields a realistic and practical description of actual ocean waves. This description explains the irregularity of these waves, it describes their short crestedness, and it illustrates the change from sea waves to swell waves. Because of the essential irregularity of actual wind generated gravity waves, the finer developments and the refinements of nonlinear theory have to be sacrificed. Linear theories are used in order to employ the principle of superposition to the best advantage. The irregularity and the marked variation in height from wave to wave in an actual wave record are explained before minor effects of nonlinearity can be taken into consideration. The plan of this chapter is to outline the many different kinds of inquiry pursued in classical hydrodynamics with reference to waves; to show that simple harmonic progressive waves must be selected as the link with time series theory; to show how Fourier integral theory leads more generally to time series theory; and finally, to show how a model of wind generated gravity waves can be constructed, which will explain many of the features of actual waves.

Wind Generated Gravity Waves

Techniques of power spectrum analysis developed by Tukey and Hamming for the analysis of noise in electronic circuits are applied to ocean-wave records The power spectrum of a sea-surface record is computed directly from the power spectrum of a pressure record. Statistical properties of the wave records are deduced and verified within the limitations of a linearized theory. The implications of the results with respect to the problem of wave forecasting and wave refraction are discussed. The relationship between the non-normalized autocorrelation function and the power spectrum is given, and the results of Deacon and Seiwell are interpreted in connection with the results of the analysis.

The power spectrum analysis of ocean-wave records

A study of the forces on a pile caused by irregular waves was conducted in terms of time series analysis, for the pile section that is continually under water. A probability density function of force at a particular depth on the pile has been derived and is shown to be a function of two parameters only. Records of forces on a pile, of up to 20 minutes duration, were analyzed, and a good fit was found between the measured density distributions of force and those predicted from the theory. It was found that the distribution is not normal and that the probability of an extreme value of force is on the order of ten times greater than it would be if the data were fitted to a normal distribution.

Willard J. Pierson

Papers

Principles of Physical Oceanography

Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics

Wind Generated Gravity Waves

The power spectrum analysis of ocean-wave records

Irregular Wave Forces On A Pile