DEPENDENCIES OF BREAKING TYPE, BREAKING CRITERIA AND ENERGY DISSIPATION ON AMPLITUDE-PHASE FREQUENCY STRUCTURE OF WAVES

Type of wave breaking - plunging or spilling - depends on symmetry of waves. The spilling waves are asymmetric against horizontal axis and are practically symmetric against vertical axis so the phase shift between first and second nonlinear harmonics (or biphase) is close to zero. The plunging breaking waves have larger asymmetry against vertical axis, (biphase is close to -pi/2), and near symmetric on horizontal axis (close to saw-toothed form). Non-linear wave transformation influences on depth-induced wave breaking. Breaking index depends on relation of wave energy in frequency range of second nonlinear harmonics to wave energy in frequency range of main harmonic and on biphase. The dissipation rate of spilling breaking waves energy quadratically depends on frequency, while in plunging breaking, this dependency is practically linear for all frequencies.

A novel singular spectrum analysis–based baseline-free approach for fatigue-breathing crack identification

Breathing cracks are the most common type of damages that occur in structures subjected to fatigue loading. These breathing cracks induce nonlinearity in the dynamic signatures of the cracked structure which carry useful information about the damage. In this article, we present a new baseline-free algorithm using singular spectrum analysis for breathing crack detection, localization, and characterization. The major advantage of using singular spectrum analysis is that it has the ability to reliably extract the nonlinear harmonics and intermodulation components buried in the noisy components of the response of the structure with breathing crack. A new damage index based on singular spectrum analysis exploiting the nonlinear harmonics and intermodulations in the response is proposed. Numerical simulation studies are carried out to evaluate the proposed damage diagnostic algorithm and later complemented with experimental studies to demonstrate its practical application.

PHYSICAL INTERPRETATION OF WAVE BREAKING CRITERIA

On the base of experimental data it was revealed that type of wave breaking depends on wave asymmetry against the vertical axis at wave breaking point. The asymmetry of waves is defined by spectral structure of waves: by the ratio between amplitudes of first and second nonlinear harmonics and by phase shift between them. The relative position of nonlinear harmonics is defined by a stage of nonlinear wave transformation and the direction of energy transfer between the first and second harmonics. The value of amplitude of the second nonlinear harmonic in comparing with first harmonic is significantly more in waves, breaking by spilling type, than in waves breaking by plunging type. The waves, breaking by plunging type, have the crest of second harmonic shifted forward to one of the first harmonic, so the waves have "saw-tooth" shape asymmetrical to vertical axis. In the waves, breaking by spilling type, the crests of harmonic coincides and these waves are symmetric against the vertical axis. It was found that limit height of breaking waves in empirical criteria depends on type of wave breaking, spectral peak period and a relation between wave energy of main and second nonlinear wave harmonics. It also depends on surf similarity parameter defining conditions of nonlinear wave transformations above inclined bottom.

PHYSICAL INTERPRETATION OF WAVE BREAKING CRITERIA

On the base of experimental data it was revealed that type of wave breaking depends on wave asymmetry against the vertical axis at wave breaking point. The asymmetry of waves is defined by spectral structure of waves: by the ratio between amplitudes of first and second nonlinear harmonics and by phase shift between them. The relative position of nonlinear harmonics is defined by a stage of nonlinear wave transformation and the direction of energy transfer between the first and second harmonics. The value of amplitude of the second nonlinear harmonic in comparing with first harmonic is significantly more in waves, breaking by spilling type, than in waves breaking by plunging type. The waves, breaking by plunging type, have the crest of second harmonic shifted forward to one of the first harmonic, so the waves have "saw-tooth" shape asymmetrical to vertical axis. In the waves, breaking by spilling type, the crests of harmonic coincides and these waves are symmetric against the vertical axis. It was found that limit height of breaking waves in empirical criteria depends on type of wave breaking, spectral peak period and a relation between wave energy of main and second nonlinear wave harmonics. It also depends on surf similarity parameter defining conditions of nonlinear wave transformations above inclined bottom.

PARAMETERIZATION OF EVOLUTION OF BIPHASE DURING NONLINEAR TRANSFORMATION OF WAVES IN COASTAL ZONE

Based on experimental data, the problem of parametrization of spatial variation of the phase shift (biphase) between the first and second nonlinear harmonics of wave motion during wave transformation over sloping bottom in the coastal zone is discussed. It is revealed that the biphase values vary in the range [–π/2, π/2]. Biphase variations rigorously follow fluctuations in amplitudes of the first and second harmonics and the periodicity of energy exchange between them. The empirical relation applied in modern practice to calculate the biphase, which depends on the Ursell number, is incorrect for calculating the biphase for wave evolution in the coastal zone, because it does not take into account periodic energy exchange between the nonlinear harmonics. The new approximations of the biphase values for typical scenarios of wave transformations are suggested. It was demonstrated that the biphase of breaking waves defines breaking index and breaking type.