oscillations of chains

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1: 10.73 Physical Applications

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§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 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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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

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Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

Encodings:

BibTeX

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An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 2002 (37), pp. 1945–1977.

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External Links:

ISSN 1073-7928, Document, MathReview (S. Ole Warnaar), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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7: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ 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 $\omega =2\pi \nu =\sqrt{k/m}$ is the circular frequency of oscillation (with $\nu$ the ordinary frequency), independent of the amplitude of the oscillations. … ►argument b) The Morse Oscillator …The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, 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 $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …

10: 22.19 Physical Applications

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§22.19(ii) Classical Dynamics: The Quartic Oscillator

… ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. Such oscillations, of period $2K\left(k\right)/\sqrt{\eta }$, with modulus $k=1/\sqrt{2-{\eta }^{-1}}$ are given by: …