On the Fractal Nature of Sea Surface Reverberation

Here, Ns is the so-called strength of sound scattering by the sea surface in decibels, f is the sound frequency, h is the amplitude of sea waves, and Θ is the glancing angle. Up to now there has been no explanation of the physical mechanisms of the “origin” or nature of this dependence. Meanwhile, the fact that dependence (1) is described by a power law with a fractional (nonintegral) index attracts one’s attention. This kind of dependence is typical of the wave scattering by fractal structures and surfaces (see, e.g., [2, 3]). It is known that a power dependence of the intensity of wave scattering on the frequency (the wavelength) and the scattering angle with a fractional power index is characteristic of fractals. It is natural to associate dependence (1) with the fractal characteristics of the sea surface. Rather convincing evidence of the fact that the sea surface is characterized by fractal properties exists now. For example, Barenblatt and Leœkin [4] have demonstrated the self-similarity of the high-frequency spectrum of the wind waves on the sea surface and presented a formula describing the frequency spectrum of wind waves. The wind wave spectrum [4] is characterized by a power law with the index ν which can take on fractional or integral values, and, in particular, at ν = 5, it describes the Phillips spectrum [5] and, at ν = 4, the Zakharov–Filonenko spectrum [6]. Barenblatt and Leœkin [4] focused their attention on the many existing experimental observations which indicate that the index ν for the frequency spectrum of the waves on the sea surface assumes a nonintegral (i.e., fractional) value. In other words, the spectrum of the sea-surface waves is described by a fractal law. Expressions characterizing the elevations and the frequency spectrum of a rough sea surface in the space E = 3 were obtained by West [7] on the basis of the modified Weierstrass–Mandelbrot function, which is frequently used for describing fractal surfaces. It was Ns 10 fh Θ sin ( ) 0'99 log 45.3. – =