Resonant oscillation of surface waves in an annular tank

We consider the resonant oscillation of surface waves in a shallow annular tank. This axisymmetric configuration provides a simple yet nontrivial setting in which to investigate the long-time interplay between geometric and nonlinear effects. Since, unlike the rectangular case, standard analytic approaches based on underlying characteristic structures are no longer applicable, we employ an alternative and more general analytic framework based upon the underlying spectral decomposition of spatial eigenfunctions. In the limit that geometric effects vanish, we show that this reduces to the known forced Korteweg–de Vries (fKdV) equation for the rectangular case. We then apply this approach to the annular configuration both in general and in the limit that the curvature scales with the forcing amplitude. For the latter, we derive an associated hierarchical system of algebraic solvability conditions for the modal amplitudes as a correction to that for the underlying rectangular case. The subsequent solution of these systems in turn provides direct characterization of the geometric effects to this order. In parallel, and in the absence of a simplified long-time balance akin to the fKdV, a general numerical implementation involving the splitting of the elliptic and evolutionary aspects of the full equations for surface waves is developed and implemented. This is used to validate and extend these analytic results and provide further insights into the fundamental ways in which geometric variation impacts resonant behavior.