Solitary waves in boundary layer induced by a travelling wave with increasing amplitude
Abstract The stability of solitary waves in an incompressible boundary layer, induced by a driving wave on the wall, is studied in order to investigate the generation and development of solitary waves. The nonlinear waves in the boundary layers, excited by a travelling wave with an increasing amplitude, rather than a standing wave, are described by forced Benjamin–Ono equation. In the study, pseudo-spectral method is used to approach to the solution of the governing equation. The results show that there are several wave energy wave branches and energy jumps to higher branch via violent oscillation. The wave pattern remains similar at every branch but the number of spike increases by one with every wave oscillation. In order to study the stability of the nonlinear waves, the perturbed wave equation is derived and a complex oscillator system is obtained from the linear perturbed wave equation. The stability analysis shows that the original wave loses its stability and wave energy begins to oscillate, and a new spike-like wave is born, when imaginary part of eigenvalue reaches to zero. Indeed, spile-like solitary waves can be induced by a travelling wave during the wave energy oscillations in boundary layers.