Characteristics of Significant Wave Components in the Long Time Wave Evolution Process

Abstract

Spectral bandwidth is a relevant parameter of water wave evolution and is commonly used to represent the number of wave components involved in wave—wave interactions. However, whether these two parameters are equivalent is an open question. Following the high-order spectral method and taking the weakly modulated Stokes wave train as the initial condition, the relationship between the spectral bandwidth and the number of wave components is investigated in this work. The results showed that the number of wave components can vary with the same spectral bandwidth and that distinct wave profiles emerge from different numbers of wave components. With a new definition of significant wave components, the characteristics of this parameter in the long-time wave evolution are discussed, along with its relationship with common parameters, including the wave surface maximum and the wave height. The results reveal that the wave surface evolution trend of different numbers of significant wave components (N _s) is the same from a holistic perspective, while the difference between them also exists, mainly in locations where extreme waves occur. Furthermore, there is a negative correlation between r (a _j/a ₀) and wave surface maximum (η _max/a ₀) and wave height (H _max and H _s). The evolution trends of the relative errors (RE) of η _max/a ₀, H _max, and H _s of different N _s show the periodic recurrence of modulation and demodulation in the early stage when the Benjamin—Feir instability is dominated. The difference is that in the later stage, the RE of η _max/a ₀ and H _max is chaotic and irregular, while those of H _s gradually stabilize near an equilibrium value. Furthermore, we discuss the relationship between the mean relative error (MRE) and r. For η _max/a ₀, MRE and r show a logarithmic relationship, while for H _max and H _s, a quadratic relationship exists between them. Therefore, the choice of N _s is also important for extreme waves and is particularly meaningful for wave generation experiments in the wave flume.

Keywords

Spectral bandwidth / Significant wave components / Long-time wave evolution / Wave surface maximum / Maximum wave height