Kelvin ship-wave pattern

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21: 10.73 Physical Applications

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§10.73(iii) Kelvin Functions

►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …The McLachlan reference also includes other applications of Kelvin functions. …

22: 10.75 Tables

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§10.75(xi) Kelvin Functions and their Derivatives

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Young and Kirk (1964) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

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Zhang and Jin (1996, p. 322) tabulates $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, $x=0(1)20$, 7S.

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§10.75(xii) Zeros of Kelvin Functions and their Derivatives

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, 8D.

23: 36.14 Other Physical Applications

… ►Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. … ►, swimming-pool patterns), and water-droplet “lenses” (e. …

24: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. …

25: 27.17 Other Applications

… ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

26: 10.74 Methods of Computation

… ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►Similar considerations apply to the spherical Bessel functions and Kelvin functions. … ► ►

§10.74(vi) Zeros and Associated Values

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27: 7.3 Graphics

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. Fresnel (1818) introduced the integral $ℱ\left(x\right)$ in his study of the interference pattern at the edge of a shadow. … Magnify

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28: 10.77 Software

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§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions

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29: Bibliography Y

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

Encodings:

BibTeX

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30: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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trig$	any word that matches the pattern “trig$”, that is, that starts with “trig”, such as “trigonometric” and “trigonometry”, including the various trigonometric functions such as $\sin$ and $\cos$.
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… ►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function): …