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oscillations of chains


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1: 10.73 Physical Applications
§10.73(i) Bessel and Modified Bessel Functions
Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation
10.73.3 4 W + λ 2 2 W t 2 = 0 .
2: Sidebar 9.SB2: Interference Patterns in Caustics
The oscillating intensity of the interference fringes across the caustic is described by the Airy function.
3: 17.17 Physical Applications
See Kassel (1995). … It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
4: 6.17 Physical Applications
Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
5: 17.12 Bailey Pairs
When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … The Bailey pair and Bailey chain concepts have been extended considerably. …
6: Bibliography S
  • B. Simon (2005c) Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.
  • V. P. Spiridonov (2002) An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 2002 (37), pp. 1945–1977.
  • 7: 8.24 Physical Applications
    The function γ ( a , x ) appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …
    8: 18.39 Applications in the Physical Sciences
    argument a) The Harmonic Oscillator …Then ω = 2 π ν = k / m is the circular frequency of oscillation (with ν the ordinary frequency), independent of the amplitude of the oscillations. … argument b) The Morse Oscillator …The finite system of functions ψ n is orthonormal in L 2 ( , d x ) , see (18.34.7_3). …The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). …
    9: 7.21 Physical Applications
    Fried and Conte (1961) mentions the role of w ( z ) in the theory of linearized waves or oscillations in a hot plasma; w ( z ) is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …
    10: 22.19 Physical Applications
    §22.19(ii) Classical Dynamics: The Quartic Oscillator
    For an initial displacement with 1 / β | a | < 2 / β , bounded oscillations take place near one of the two points of stable equilibrium x = ± 1 / β . Such oscillations, of period 2 K ( k ) / η , with modulus k = 1 / 2 η 1 are given by: …