Hysteresis phenomena in gravity--capillary waves on deep water generated by a moving two-dimensional/three-dimensional air-blowing/air-suction forcing*

Hysteresis phenomena in forced gravity--capillary waves on deep water where the minimum phase speed $c_{\mathrm{min}}=$23cm/s are experimentally investigated. Four kinds of forcings are considered; 2-D/3-D air-blowing/air-suction forcings. For a still water initial condition, as the forcing speed increases from zero towards a certain target speed ($U)$, there exists a certain critical speed ($U_{\mathrm{crit}})$ at which the transition from linear to nonlinear states occurs. When $U$\textless $U_{\mathrm{crit}}$, steady linear localized waves are observed (state I). When $U_{\mathrm{crit}}$\textless $U$\textless $c_{\mathrm{min}}$, steady nonlinear localized waves including steep gravity--capillary solitary waves are observed (state II). When $U\approx c_{\mathrm{min}}$, periodic shedding phenomena of nonlinear localized depressions are observed (state III). When $U$\textgreater $c_{\mathrm{min}}$, steady linear non-local waves are observed (state IV). Next, with these state-II, III and IV waves as new initial conditions, as the forcing speed is decreased towards a certain target speed ($U_{\mathrm{final}})$, a certain critical speed ($U_{\mathrm{crit,2}})$ is identified at which the transition from nonlinear to linear states occurs. When $U_{\mathrm{crit,2}}$\textless $U_{\mathrm{final}}$\textless $U_{\mathrm{crit}}$, steeper gravity--capillary solitary waves are observed. When $U_{\mathrm{final}}$\textit{\textless U}$_{\mathrm{crit,2}}$, linear state-I waves are observed. These are hysteresis phenomena which show the dependence of a state on its history starting from different initial conditions.