Modulations of Deep Water Waves and Spectral Filtering

• Published in: Studies in applied mathematics (Cambridge) Vol. 111; no. 3; pp. 301 - 313
• Main Author: Kliakhandler, I. L.
• Format: Journal Article
• Language: English
• Published: 350 Main Street , Malden , MA 02148 , USA , and 9600 Garsington Road , Oxford OX4 2DQ , UK Blackwell Publishing, Inc 01.10.2003; Blackwell
• Subjects: Bandwidth; Computational methods in fluid dynamics; Deep water; Exact sciences and technology; Filtering; Filtration; Fluid dynamics; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; Mathematical analysis; Mathematics; Modulation; Partial differential equations; Physics; Schroedinger equation; Sciences and techniques of general use; Spectra; Nonlinear equations; Spectral filtering; Partial differential equations; Four wave resonance; Bandwith selection; Wave modulation; Stokes wave; Deep water; Stability criteria; Applied mathematics; Schroedinger equation; Zakharov equation; Sideband; Single mode equation
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/1467-9590.t01-1-00235

Modulations of deep water waves are studied by a new formalism of spectral filtering. For single‐mode dynamics, spectral filtering results in computable equations, which are counterpart to the nonlinear Schrödinger (NLS) equations. An essential feature of new equations is that bandwidth limitation is decoupled from small‐amplitude assumption. The filtered equations have a substantially broader range of validity than the NLS equations, and may be viewed as intermediate between the NLS and Zakharov equations. The new single‐mode equations reproduce exactly the conditions for nonlinear four‐wave resonance (“figure 8” of Phillips [1]) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean [2].