1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

Overall properties of a cracked solid

When the surface of a body is subjected to transient heating (e.g., by electron bombardment or rf absorption) elastic waves are produced as a result of surface motion due to thermal expansion. This process is analyzed, with particular emphasis on the case of an input heat flux varying harmonically with time, to relate the elastic wave amplitude to the characteristics of the input flux and the thermal and elastic properties of the body. Experiments performed with both electron impact and rf absorption verify the proportionality of the stress wave amplitude and the absorbed power density, and correlate well with the thermal and elastic properties of the heated medium. Comparison of the elastic wave stress amplitude with radiation pressure shows that the former may be much greater than the latter, as experiments have demonstrated. When a barium titanate crystal was used to detect the elastic waves produced, heating by a single 2‐μsec pulse of electrons or microwave radiation produced easily detectible signal...

Generation of Elastic Waves by Transient Surface Heating

Fluxmetric (ballistic) and magnetometric demagnetizing factors N/sub f/ and N/sub m/ for cylinders as functions of susceptibility chi and the ratio gamma of length to diameter have been evaluated. Using a one-dimensional model when gamma >or=10, N/sub f/ was calculated for -1 >

Demagnetizing factors for cylinders

SUMMARY A new approach for viscoacoustic wave propagation is developed. The Boltzmann’s superposition principle based on the general standard linear solid rheology is implemented in the equation of motion by the introduction of memory variables. This approach replaces the conventional convolutional rheological relation, and thus the complete time history of the material is no longer required, and the equations of motion become a coupled first-order linear system in time. The propagation in time is done by a direct expansion of the evolution operator by a Chebycheff polynomial series. The resulting method is highly accurate and effects such as the numerical dispersion often encountered in time-stepping methods are avoided. The numerical algorithm is tested in the problem of wave propagation in a homogeneous viscoacoustic medium. For this purpose the l-D and 2-D viscoacoustic analytical solutions were derived using the correspondence principle.

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Wave propagation simulation in a linear viscoacoustic medium

This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;90), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0

On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions

This paper is concerned with the determination of the distribution of stress in an infinite elastic solid when time-dependent body forces act upon certain regions of the solid. It is assumed throughout that the strains are small. In §2 a general solution of the equations of motion for any distribution of body forces is derived by the use of four-dimensional Fourier transforms, and from that is derived the general solution for an isotropic solid (§ 3). From the latter solution are deduced the general solution of the statical problem (§4) and the two-dimensional problem (§5). The solution of the equations of motion in the case in which the distribution of body forces is symmetrical about an axis is derived in §6. The remainder of the paper consists in deducing the solution of special problems from these general solutions. In §§7 to 13 some typical two-dimensional problems are considered and exact analytical expressions found for the components of the stress tensor. In §§14 to 16 examples are given of the use of the general non-symmetrical three-dimensional solution derived in § 3, and in §§17 to 19 examples are given to illustrate the use of the general solution of the axially symmetrical problem. A certain amount of numerical work (presented in graphical form) is quoted to give some idea of the physical nature of the solutions.

G. Eason

Papers

On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions

The Generation of Waves in an Infinite Elastic Solid by Variable Body Forces