Miguel D Bustamante | Searcharxiv

arXiv2016

Precession resonance in water waves

We describe the theory and present numerical evidence for a new type of nonlinear resonant interaction between gravity waves on the surface of deep water. The resonance constitutes a generalisation of the usual 'exact' resonance as we show that exchanges of energy between the waves can be enhanced when the interaction is three-wave rather than four and the linear frequency mismatch, or detuning, is non-zero i.e. $ω_1\pmω_2\pmω_3 \neq0.$ This is possible because the resonance condition is now a match between the so-called 'precession frequency' of a given $\textit{triad interaction}$ and an existent nonlinear frequency in the system. In the limit of weak nonlinearity this precession frequency is simply due to the linear 'drift' of the triad phase; therefore, it tends toward the detuning. This means precession resonance of this type can occur at finite amplitudes, with nonlinear corrections contributing to the resonance. We report energy transfer efficiencies of up to 40%, depending on the model options. To the authors' knowledge this represents the first new type of nonlinear resonance in surface gravity waves since the seminal work of Benjamin & Feir (1967).

physics.flu-dyn↗

arXiv2012

Quaternion Solution for the Rock'n'roller: Box Orbits, Loop Orbits and Recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock'n'roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

nlin.SI↗