On the properties of the Rational Function Spectrum describing waves near islands and reefs in South China Sea

On-site measurements show that water waves near islands and reefs in South China Sea exhibit different properties of wave energy distributions with regard to wave frequencies,among which the most prominent factor is the interplay of swells arising from the West Pacific Ocean and the local wind waves.Observations also show that the breaking waves continuously appear,containing more energy in high frequency components,and the nonlinear characteristics of the waves are important in adjusting the energy distribution.These properties may explain the large discrepancies between the well-accepted wave spectra(for example the P-M spectrum,Neumann spectrum,ITTC spectrum etc.)and the measured wave spectra near islands and reefs in South China Sea.Therefore,a new Rational Function Spectrum is proposed in this paper to describe waves near islands and reefs which turns out to show satisfactory accuracy.It well captures wave power distributions in the form of single and double peaks,at low-and high-frequency regions,as well as nonlinear scale power law.Based on the investigation of the measured data near an island in South China Sea,the relation between the parameters used in the Rational Function Spectrum and the statistical parameters of water waves(significant wave height and wave period)is established.It is noted that the wave properties at low-and high-frequencies are controlled by the local wind velocities at the wave growth stage,but remain constant at the wave decay stage.The parameter peak frequency is only dependent on the wave period corresponding to the maximum wave height.The parameter spectral peak is determined from the wave height and the wave period.These relations help to clarify the physical meanings of the parameters used in the Rational Function Spectrum,and thus provide an alternative spectral form to describe random waves near islands and reefs.

The Spectrum of a Rational Function

Let k be an infinite field,A be a finite set of k,and Q∈k[x](with x=(x_(1),...,X_(n))and n≥2)be a noncons taut polynomial.The main goal of this paper is to construct a polynomial P(x)∈k[x]with suitably large partial degrees in x_(1),...,x_(n-1)such that P and Q axe coprime,and P-aQ is reducible for all a in A.