Abstract
An asymptotic theory for cross-waves generated by an oscillating wavemaker in a semi-infinite rectangular wave tank is derived for the limit of large mode number. The possibility of multiple mode excitation is included by introducing a spanwise modulation. The partial differential equations governing the evolution of inviscid cross-waves are shown to be two coupled nonlinear Schrödinger equations. Energy dissipation in the system is taken into account including a linear damping term. A center manifold analysis is used to reduce the PDEs to a system of coupled Landau equations in the neighborhood of a codimension-two point where two adjacent spanwise modes are marginally stable. Four possible steady states of the system are found, one of which is a mixed-mode superposition of two spanwise modes. A Hopf bifurcation from the mixed mode is predicted for some parameters; extending the system to higher order allows the stability of this bifurcation to be determined in terms of perturbations to a Hamiltonian system. Both subcritical and supercritical bifurcation are possible. An experimental study in the neighborhood of a codimension-two point shows good agreement with the theoretical predictions including the discovery of a mixed-mode state.
Original language | English (US) |
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Pages (from-to) | 87-104 |
Number of pages | 18 |
Journal | Physica D: Nonlinear Phenomena |
Volume | 43 |
Issue number | 1 |
DOIs | |
State | Published - May 1990 |
Funding
The experimentald ata presentedh ere were taken and compiled by Bill Underhill This re-searchw ass upportedb y NSF grantM SM-8611379 and by ONR grants NO001486-K-0617 and NOOO14-89-J-174O8.n e of us (A.J.B.) would like to thank Dave Paisley for his helpful physical intuition.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics