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ut A. D. D. Craik ³φ ( St. Andrewsεw wΘ )
θΪ Aspects of subharmonic resonance with external forcing
ϊ 2000N131ϊ() 13:00
κ pHwΘζ2[~Ί (pHwΘ3K)
Abstract
Two tpics will be considerd:
- subharmonic or second-harmonic resonance od two small-amplitude
standing water waves with Faraday excitation; and
- 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.