Three-wave interactions among surface gravity waves in a cylindrical container

The motion of a container filled with fluid perturbs the free surface and may result in spilling. In practice, most of the energy is localized in the modes of lowest frequencies (the gravest modes), and sloshing can be predicted once the dynamics of these modes is known. In this Rapid Communication, we investigate the nonlinear interactions between such grave modes in a cylindrical container. We first show that energy can be transferred from modes to modes with three-wave interactions: we derive the resonance conditions and characterize the early stage of this interaction. This result strongly contrasts with resonant interactions between surface gravity waves in extended domains such as the ocean, which involve at least four waves and are thus less efficient. An experiment is then performed to provide evidence of these nonlinear interactions.

Figure 1

Experimental setup (sketch and photography). The notations $(r,\theta ,z)$ in the text correspond to the usual cylindrical coordinate with $z=0$ at the interface. In this experiment, $h\simeq 3\mathrm{cm}$ and $R\simeq 9\mathrm{cm}$, while a typical value of $\mathrm{\Omega }$ is $2\pi \times 250\mathrm{mHz}$.

Figure 2

Left: Surface elevation power spectrum. The black circles are the theoretical resonance frequencies (see right graph). Right: Experimental natural frequencies of the gravest modes, compared to the theoretical prediction of Eq. (6) derived in the limit of small Froude numbers

Figure 3

Left: surface elevation spectra and phase of the cross-spectral density for $\mathrm{\Omega }=2\pi \times 0.240\mathrm{mHz}$. Right: Evolution of a measure of the energy of the daughter wave with the ones of the mother waves.