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uŽt A. D. D. Craik ‹³Žφ ( St. Andrews‘εŠw ”Šw‰Θ )
‘θ–Ϊ Aspects of subharmonic resonance with external forcing
“ϊŽž 2000”N1ŒŽ31“ϊ(ŒŽ) 13:00
κŠ ‰ž—p”—HŠw‰Θ‘ζ2ƒ[ƒ~ŽΊ (‰ž—p”—HŠw‰Θ“3ŠK)

Abstract

Two tpics will be considerd:
  1. subharmonic or second-harmonic resonance od two small-amplitude standing water waves with Faraday excitation; and
  2. the growth of localised disturbances in a Blasius boundary layer.
The first, which is joint work with H. Okamoto and H. Allen, concerns the equations \frac{d a}{d t} = - \sigma a + \mu a^* + \lambda a^* b, \frac{d b}{d t} = - \sigma_1 b + \nu a^2. These govern the complex amplitudes of a resonant pair of standing water waves with Faraday excitation. Identical equations govern small oscillations of forced, resonant, coupled pendula. Real positive \sigma and \sigma_1 correspond to linear damping with perfect tuning of both forcing and internal resonance. With detuning from resonance, \sigma and \sigma_1 are complex. A rich dynamical structure is revealed by computer solutions of the equations; and local stability analyses of fixed points and periodic orbits explain many observed features.

The second topic attempts to explain certain phenomena found in recent experiments of Medeiros & Gaster (JFM 1999) on the growth of localised disturbances in a Blasius boundary layer. Because their disturbance is localised in space, it has a spread of wavenumbers and frequencies: among these are components which can initiate resonant subharmonics with well-determied phase, which are then amplified by the familiar three-wave resonance mechanism.