Gravity–capillary jet-like surface waves generated by an underwater bubble

Jet-like surface waves generated by an electric-spark-generated underwater bubble are experimentally studied. Three different motions of jet-like surface waves are observed depending on the inception position of the bubble ($d$: 0.28–7 mm) below the free surface and the maximum radius of the bubble ($R_{m}$: 1.5–3.6 mm). When $d/R_{m}>1.3$, the surface wave shows a simple smooth hump (case 1). When $0.82<d/R_{m}<1.3$, a single droplet or multiple droplets are pinched off sequentially or simultaneously at the tip or from some points of the jet-like surface wave (case 2). Finally, when $d/R_{m}<0.82$, a series of squirting and jetting phenomena are observed at the top of the jet-like surface wave (case 3). For case 1, a proportional relationship is found between $\unicode[STIX]{x1D70C}gh/\unicode[STIX]{x0394}p$ and $(d/R_{m})^{-4.4}$, where $\unicode[STIX]{x1D70C}$ is the density of the fluid, $g$ is the gravitational acceleration and $\unicode[STIX]{x0394}p$ is the difference between the reference atmospheric pressure and the vapour pressure inside a bubble. This proportional relationship is explained semi-analytically using a scaling argument and conservation of momentum and energy, with the help of the Kelvin impulse theory. In addition, we solve the relevant axisymmetric Cauchy–Poisson problem where the initial condition is a jet-like surface wave near its maximum height. By comparing the analytical wave solution with the observed surface wave pattern, it is found that the resultant surface waves are indeed gravity–capillary waves where both the gravity and the surface tension are equally important.