Efficient measurement of the vibrational rogue waves by compressive sampling based wavelet analysis

Nonlinear rogue waves appear as a result of spectral stochastic supercontinuum generation in nonlinear dynamical models, such as the nonlinear Schrödinger equation (NLSE). They are observed in the fields including but are not limited to fluid mechanics and optics. Rogue wave phenomena can also be observed in solid mechanics, where the envelope of a surface vibrational wave packet of an Euler–Bernoulli beam or Kirchhoff–Love plate can be modeled in the frame of the NLSE. Their efficient measurement and early detection is important to analyze critical displacements, stresses, and resonance. One of the possible techniques for their measurement and detection is to measure the triangular wavenumber (Fourier) spectra that become evident at the early stages of their development. One can treat such a spectra as a sparse signal due to energy located at the few central modes. Therefore compressive sensing can be used as an efficient tool for the measurement and prediction of vibrational rogue waves. However, Fourier analysis can only detect whether a vibrational rogue wave will develop or not in the vibrating medium. In order to locate its occurrence location, wavelet analysis deems necessary if a spectral analysis approach is utilized. In this paper, we discuss the possible usage of the compressive sampling based wavelet analysis for the efficient measurement and for the early detection of one-dimensional (1D) vibrational rogue waves. We study the construction of the triangular (V-shaped) wavelet spectra using compressive samples of rogue waves that can be modeled as Peregrine and Akhmediev–Peregrine solitons. We show that triangular wavelet spectra can be acquired by compressive sampling at the early stages of the emerging vibrational rogue waves. Our results may lead to development of the efficient vibrational rogue wave measurement and early sensing systems with reduced memory requirements which use the compressive sampling algorithms. In typical solid mechanics applications, compressed measurements can be acquired by randomly positioning single sensor and multi-sensors.