On the modulational stability of traveling and standing water waves

Abstract

Asymptotically exact evolution equations are derived for trains of small amplitude counterpropagating water waves over finite depth. Surface tension is included. The resulting equations are nonlocal and generalize the equations derived by Davey and Stewartson for unidirectional wave trains. The stability properties of stationary standing and quasiperiodic waves are determined as a function of surface tension and fluid depth for both long wavelength longitudinal and transverse perturbations.

Nearly Inviscid Faraday Waves

“…In particular, the cubic coeñicients coincide with those obtained in strictly inviscid formulations (Pierce and Knobloch [1994]; see also Miles [1993], Hanscn and Alstrom [1997] and references therein). The coefficient a$ diverges at (cxcluded) resonant wavenumbers satisfying Lü(2k) = 2w(fe).…”

“…The mean flow variables in the bulk depetid weakly on time but strongly on both x and y, and evolve accordíng to the equations Thus the mean ftow is forced by the surface waves in two ways. The right sides of the boundary conditions (2.19a) and (2.20) provide a normal forcíng rnechanism; this mechanism is thc only one present in strictly inviscid thcory ; Pierce and Knobloch [1994]) and does not appear unless the aspect ratio is large. The right sides of the boundary conditions (2.19b) and (2.21c) describe two shear forcíng mechanisms, a tangential stress at the free surface and a tangential velocity at the bottom wall.…”

“…Substitution of these expressions into (3.12) followed by integration of the resulting equations yields The particular solution of (3.19)-(3.20) yields the usual inviscid mean flow included in nearly inviscid theories (see Pierce and Knobloch [1994] and references therein); the averaged terms are a consequence of the conditions (3.15), i.e., of volume conservation (cf. Pierce and Knobloch [1994]) and the requircment that the nearly inviscid mean flow has a zero mean on the timescale T; the latter condition is never imposed in strictly inviscid theories but is essential in the limit we are considering, as explained above.…”

“…Thus the relevan t nearly inviscid modes are either of long wavelength (k -> 0) or are concentrated aronnd the two counterpropagating modes. The long wave modes constitute the nearly inviscid mean flow; in the strictly inviscid case, this flow is the mean flow considered in inviscid theories ; Pierce and Knobloch [1994]). However, because of its long wavelength this mean flow does not appear if the aspect ratio is of order unity (Nicolás and Vega [1996]; Higuera, Nicolás. and ), where for each k > 0, q n > 0 is the n-th root of q tanh k -k tan q, and henee decay orí an 0{C a ) timescale, i.e., more slowly than the surface modes when C g is sufficiently small.…”

“…In systems of small to modérate aspect ratio such mean flows are entirely of viscous origin (Nicolás and Vega [1996]; Higuera, Nicolás, and Vega [2000]), but in the larger aspect ratio systems of interest below the situation is rather more subtle because of the presence of an additional inviscid mean flow. For inviscid free waves this mean flow is associated with spatial modulation of a single mode, as described by the celebrated Davey-Stewartson equations ; Pierce and Knobloch [1994]). If viscosity is retained and the system forced, as in a shear flow, a similar set of equations but with complex coefficients can be derived (Davey, Hocking, and Stewartson [1974]).…”