Linear stability of finite-amplitude capillary waves on water of infinite depth

• Published in: Journal of fluid mechanics Vol. 696; pp. 402 - 422
• Main Authors: Tiron, Roxana, Choi, Wooyoung
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.04.2012
• Subjects: Amplitudes; Capillary waves; Deep water; Evolution; Exact sciences and technology; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; Mathematical analysis; Mathematical models; Perturbation methods; Physics; Stability; Stability analysis; Superharmonics; Water; Water depth; capillary waves; Theoretical study; Free surface flow; Linear stability; Capillary waves; Deep water; Disturbances
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2012.56

We study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).