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- Author or Editor: Kunio Rikiishi x

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## Abstract

Two conventional methods of computing the power spectrum, via the autocovariance function or via the fast Fourier transform (referred to as the lagged product method and the FFT method respectively for simplicity), have been examined analytically and numerically for equally spaced time series of finite length. It is found that the two methods are equivalent to each other, and that the only difference between them lies in regard to the spectral window. Spectral windows for the FFT method are superior to those for the lagged product method in that they do not show any negative values and that their influence is band-limited in frequency domain. There is little difference in spectral estimates between the two methods. In many cases the FFT method is economical in computation time, but for the case of large data points and small maximum lag the lagged product method is the more economical. It is proved that in the strict sense the power spectrum for higher frequencies than the Nyquist frequency is not folded linearly over lower frequencies both in the FFT method and the lagged product method. Finally it is discussed whether or not the use of original data repeatedly is consistent with the analysis of random phenomena.

## Abstract

Two conventional methods of computing the power spectrum, via the autocovariance function or via the fast Fourier transform (referred to as the lagged product method and the FFT method respectively for simplicity), have been examined analytically and numerically for equally spaced time series of finite length. It is found that the two methods are equivalent to each other, and that the only difference between them lies in regard to the spectral window. Spectral windows for the FFT method are superior to those for the lagged product method in that they do not show any negative values and that their influence is band-limited in frequency domain. There is little difference in spectral estimates between the two methods. In many cases the FFT method is economical in computation time, but for the case of large data points and small maximum lag the lagged product method is the more economical. It is proved that in the strict sense the power spectrum for higher frequencies than the Nyquist frequency is not folded linearly over lower frequencies both in the FFT method and the lagged product method. Finally it is discussed whether or not the use of original data repeatedly is consistent with the analysis of random phenomena.

## Abstract

A new method for measuring the directional spectrum, introduced in the preceding paper (Rikiishi, 1978), has been applied to actual wind waves in a large experimental tank 70 m Ã— 8 m with the water 3 m deep, and in a wind-wave tunnel 850 cm Ã— 60 cm with the water 35 cm deep. Measurements of the directional spectrum have shown that the mean wave direction of propagation agrees generally with the wind direction, that a bimodal distribution in the spectrum is not generally seen, and that the angular width of the directional spectrum is not correlated consistently with the wave frequency. These results conflict with the existence of Phillips resonant angle. Measurements of the phase velocity have shown that the phase velocity of the spectral-peak component wave is larger than that obtained from linear small-amplitude wave theory. that the measured phase velocity shows a roughly constant value among frequencies near the dominant frequency, and that the deviation of the constant value from the theoretical varies with fetch in proportion to the wind speed over the water surface. Based on these observational facts, it has been stressed that wind waves under the direct action of wind stress should not be regarded as the linear superposition of free Airy waves.

## Abstract

A new method for measuring the directional spectrum, introduced in the preceding paper (Rikiishi, 1978), has been applied to actual wind waves in a large experimental tank 70 m Ã— 8 m with the water 3 m deep, and in a wind-wave tunnel 850 cm Ã— 60 cm with the water 35 cm deep. Measurements of the directional spectrum have shown that the mean wave direction of propagation agrees generally with the wind direction, that a bimodal distribution in the spectrum is not generally seen, and that the angular width of the directional spectrum is not correlated consistently with the wave frequency. These results conflict with the existence of Phillips resonant angle. Measurements of the phase velocity have shown that the phase velocity of the spectral-peak component wave is larger than that obtained from linear small-amplitude wave theory. that the measured phase velocity shows a roughly constant value among frequencies near the dominant frequency, and that the deviation of the constant value from the theoretical varies with fetch in proportion to the wind speed over the water surface. Based on these observational facts, it has been stressed that wind waves under the direct action of wind stress should not be regarded as the linear superposition of free Airy waves.

## Abstract

A new method for measuring the directional wave spectrum is introduced, and the usefulness of the method is examined through numerical analysis of synthetic wave fields. The principle of the method consists of the direct application of the fast Fourier transform technique to the two-dimensional spectrum estimation. Unlike other known methods, the procedure is not based explicitly on the evaluation of the cross-covariance function or the cross spectrum.

Two examples of wave gage arrangement for the method have been presented. The first army (Array A) consists of a set of twelve wave detectors equally spaced around a circle, and the second (Army B) consists of eight detectors on an outer circle and four detectors on an inner circle with half the radius of the outer. Denoting the nondimensional scale of the wave detector array by = 2*r*/*L* [where *r* is the radius of the (outer) wave detector circle and *L* the wavelength], the ranges in which the method is useful turn out to be 1.0 â‰¤ â‰¤ 1.18 and 1.25 â‰¤ â‰¤ 1.60 for Array A and 0.6 â‰¤ â‰¤ 1.7 for Array B.

In this method, the use of an improper dispersion relation for actual wind waves is a possible factor in methodological error. Many of the spurious estimates obtained can be ascribed to the use of improper dispersion relations. Using this fact, it is possible to determine a reasonable dispersion relation for actual wind waves.

## Abstract

A new method for measuring the directional wave spectrum is introduced, and the usefulness of the method is examined through numerical analysis of synthetic wave fields. The principle of the method consists of the direct application of the fast Fourier transform technique to the two-dimensional spectrum estimation. Unlike other known methods, the procedure is not based explicitly on the evaluation of the cross-covariance function or the cross spectrum.

Two examples of wave gage arrangement for the method have been presented. The first army (Array A) consists of a set of twelve wave detectors equally spaced around a circle, and the second (Army B) consists of eight detectors on an outer circle and four detectors on an inner circle with half the radius of the outer. Denoting the nondimensional scale of the wave detector array by = 2*r*/*L* [where *r* is the radius of the (outer) wave detector circle and *L* the wavelength], the ranges in which the method is useful turn out to be 1.0 â‰¤ â‰¤ 1.18 and 1.25 â‰¤ â‰¤ 1.60 for Array A and 0.6 â‰¤ â‰¤ 1.7 for Array B.

In this method, the use of an improper dispersion relation for actual wind waves is a possible factor in methodological error. Many of the spurious estimates obtained can be ascribed to the use of improper dispersion relations. Using this fact, it is possible to determine a reasonable dispersion relation for actual wind waves.

## Abstract

The power spectra of typical sets of ocean wave data obtained in the open ocean using a cloverleaf buoy are analyzed to determine an idealized form for the spectrum of ocean surface waves. It is shown that most of the single-peaked spectra observed in a generation area can be described well by the spectral form of the JONSWAP type. Two parameters Î± and Î³ characterizing the spectral form are calculated for each spectrum measured. Their relations to the dimensionless peak frequency *fÌ„ _{m}
* (=

*f*U/

_{m}*g*) are then determined. These relations are further converted into fetch relations for Î± and Î³ through a relation between fÌ„ and a dimensionless fetch

*FÌ„*(=

*gF*/

*U*

^{2}).

Another spectral form proposed by Toba (1978) is examined and shown to fit as well to the observed spectra at high frequencies This fact shows quasi-equivalence of the JONSWAP spectrum and Toba's spectrum in the high-frequency range. On the basis of the agreements of both spectral forms at high frequencies, properties of the dimensionless constant Î±&prime in Toba's spectrum are examined. It is shown that Î±&prime depends very weakly on the dimensionless fetch *FÌ„*.

## Abstract

The power spectra of typical sets of ocean wave data obtained in the open ocean using a cloverleaf buoy are analyzed to determine an idealized form for the spectrum of ocean surface waves. It is shown that most of the single-peaked spectra observed in a generation area can be described well by the spectral form of the JONSWAP type. Two parameters Î± and Î³ characterizing the spectral form are calculated for each spectrum measured. Their relations to the dimensionless peak frequency *fÌ„ _{m}
* (=

*f*U/

_{m}*g*) are then determined. These relations are further converted into fetch relations for Î± and Î³ through a relation between fÌ„ and a dimensionless fetch

*FÌ„*(=

*gF*/

*U*

^{2}).

Another spectral form proposed by Toba (1978) is examined and shown to fit as well to the observed spectra at high frequencies This fact shows quasi-equivalence of the JONSWAP spectrum and Toba's spectrum in the high-frequency range. On the basis of the agreements of both spectral forms at high frequencies, properties of the dimensionless constant Î±&prime in Toba's spectrum are examined. It is shown that Î±&prime depends very weakly on the dimensionless fetch *FÌ„*.

## Abstract

Analysis of the directional spectra of typical sets of surface wave data obtained in the open sea as well asa bay using a cloverleaf buoy system are reported.

It is shown that the directional wave spectrum can be approximated by the product of the frequencyspectrum and a unimodal angular distribution with mean direction approximately equal to that of thewind, and that various forms of frequency spectra exist, even in relatively simple wave systems, dependingon their generating conditions. Ocean waves at fairly short dimensionless fetches show spectral forms withvery narrow spectral width, which are similar to those of laboratory wind waves. On the other hand, thespectral forms for ocean waves at very long dimensionless fetches are quite similar to the Pierson-Moskowitzspectra, which are considered, within our present data, to be the wave spectra with the largest spectral width.Finally, there exist many ocean waves at moderate dimensionless fetches, which show spectral forms with interminate spectral widths lying between the above two extremes. However, a definite relationship betweenthe spectral width and the dimensionless fetch has not been obtained in the present study.

Concerning the angular distribution, it is shown that the shape of the angular distribution is dependenton the frequency of the spectral component even in a simple wave system in a generating area, althoughthe mean directions ot the spectral components are independent of the frequency and approximately equalto the wind direction. The angular distribution is very narrow for frequencies near the dominant peak of thefrequency spectrum, whereas it widens rapidly toward high and low frequencies. Thus, the major energy-containing frequency components propagate in almost the same direction as the wind with the least angularspreading.

Finally, it is shown that a similarity law is satisfied for the angular distributions, and an idealized formof the angular distribution function is derived for practical purposes.

## Abstract

Analysis of the directional spectra of typical sets of surface wave data obtained in the open sea as well asa bay using a cloverleaf buoy system are reported.

It is shown that the directional wave spectrum can be approximated by the product of the frequencyspectrum and a unimodal angular distribution with mean direction approximately equal to that of thewind, and that various forms of frequency spectra exist, even in relatively simple wave systems, dependingon their generating conditions. Ocean waves at fairly short dimensionless fetches show spectral forms withvery narrow spectral width, which are similar to those of laboratory wind waves. On the other hand, thespectral forms for ocean waves at very long dimensionless fetches are quite similar to the Pierson-Moskowitzspectra, which are considered, within our present data, to be the wave spectra with the largest spectral width.Finally, there exist many ocean waves at moderate dimensionless fetches, which show spectral forms with interminate spectral widths lying between the above two extremes. However, a definite relationship betweenthe spectral width and the dimensionless fetch has not been obtained in the present study.

Concerning the angular distribution, it is shown that the shape of the angular distribution is dependenton the frequency of the spectral component even in a simple wave system in a generating area, althoughthe mean directions ot the spectral components are independent of the frequency and approximately equalto the wind direction. The angular distribution is very narrow for frequencies near the dominant peak of thefrequency spectrum, whereas it widens rapidly toward high and low frequencies. Thus, the major energy-containing frequency components propagate in almost the same direction as the wind with the least angularspreading.

Finally, it is shown that a similarity law is satisfied for the angular distributions, and an idealized formof the angular distribution function is derived for practical purposes.