Connection between the Rayleigh and the Schr{umlt o}dinger equations
- University of California, Lawrence Livermore National Laboratory, Livermore, California 94550 (United States)
We make a connection between the Schr{umlt o}dinger equation {ital D}{sup 2}{Psi}+({ital E}{minus}{ital V}){Psi}=0 and the Rayleigh equation {ital D}({rho}{ital DW})+({ital k}{sup 2}/{Gamma}{sup 2}){ital WD}{rho}{minus}{ital k}{sup 2}{rho}{ital W}=0 which is used to study the Rayleigh-Taylor instability of fluids in a gravitational field. Here {ital D} is the differential operator {ital d}/{ital dy}, {rho}({ital y}) is the density profile of the fluid, {ital W}({ital y}) is the perturbed fluid velocity, {ital k} is the wave number of the perturbation, and {Gamma}{sup 2}={gamma}{sup 2}/{ital g}, where {gamma} is the growth rate of the instability and {ital g} is the strength of the gravitational field. The connection between the Rayleigh and the Schr{umlt o}dinger equations is made by defining a potential {ital V}({ital y}) associated with {rho}({ital y}), a wave function {Psi}({ital y}) associated with {ital W}({ital y}), and an energy {ital E} associated with {ital k}. We consider several examples of the Rayleigh equation and show that they correspond to well-known problems in quantum mechanics such as a particle in a box, the harmonic oscillator, the Coulomb potential, etc. We illustrate the inversion symmetry of the Rayleigh equation under {rho}({ital y}){r_arrow}1/{rho}({minus}{ital y}), and in an appendix we give and illustrate the more general potential {ital V}({ital y}), which includes surface tension and shear flow, the latter associated with the Kelvin-Helmholtz instability. {copyright} {ital 1996 The American Physical Society.}
- Research Organization:
- Lawrence Livermore National Laboratory
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 283811
- Journal Information:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 4 Vol. 53; ISSN PLEEE8; ISSN 1063-651X
- Country of Publication:
- United States
- Language:
- English
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