, Volume 3, Issue 5, pp 343-360

On the computation of special functions with applications to non-linear and inhomogeneous waves

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Abstract

Large amplitude and inhomogeneous oscillations in compressible fluids, are described by wave equations which are non-linear or have variable coefficients, and whose solutions can be obtained exactly, in specific cases, in terms of special functions, usually particular or generalized forms of the hypergeometric type. We present a simple method for the calculation of generalized hypergeometric functions, including a statement of the conditions of convergence for all values of the variable and parameters, an estimate of the rate of convergence and upper bound for the error of truncation, and a discussion of rounding-off errors and ways to reduce them. The method is illustrated by three applications, namely, a non-linear pulse in a viscous fluid, resonant modes of an acoustic horn, and oscillations in a column of gas stratified in density and temperature. These three cases illustrate the acoustics of linear and non-linear waves, propagating or standing, in inhomogeneous, viscous or heated fluid, in free space or confined in ducts. We conclude with an indication of the efficiency of the algorithm in obtaining the preceding results, i. e., the accuracy after a given number of iterations, for some of the preceding problems.

Work supported by C.A.U.T.L./I.N.I.C.
Communicated by S. N. Atluri, May 17, 1987