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Improved approximations for the rayleigh wave velocity in [ 1, 0.5]

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In the present paper we derive improved approximations for the Rayleigh wave velocity in the interval ν ∈ [−1, 0.5] using the method of least squares. In particular: (i) We create approximate polynomials of order 4, 5, 6 whose maximum percentage errors are 0.035 %, 0.015 %, 0.0083 %, respectively. (2i) Improved approximations in the form of the inverse of polynomials of order 3, 5 are also established | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 347 – 358 Special Issue of the 30th Anniversary IMPROVED APPROXIMATIONS FOR THE RAYLEIGH WAVE VELOCITY IN [-1, 0.5] Pham Chi Vinh Hanoi University of Science, Vietnam Peter G. Malischewsky Institute for Geosciences, Friedrich-Schiller University Jena Burgweg 11, 07749 Jena, Germany Abstract. In the present paper we derive improved approximations for the Rayleigh wave velocity in the interval ν ∈ [−1, 0.5] using the method of least squares. In particular: (i) We create approximate polynomials of order 4, 5, 6 whose maximum percentage errors are 0.035 %, 0.015 %, 0.0083 %, respectively. (2i) Improved approximations in the form of the inverse of polynomials of order 3, 5 are also established. They are approximations with very high accuracy. (3i) By using the best approximate second-order polynomial of the cubic power in the space C[0.474572, 0.912622], we derive an approximation that is the best, so far, of the approximations obtained by approximating the secular equation. 1. INTRODUCTION Elastic surface waves in isotropic elastic solids, discovered by Lord Rayleigh [1] more than 120 years ago, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example. It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as noted by Samuel [2]. For the Rayleigh wave, its velocity is a fundamental quantity which is significance in practical applications, so researchers have attempted to find its analytical approximate expressions which are of simple forms and accurate enough for practical purposes. Let c be the Rayleigh wave velocity in isotropic elastic solids and x(ν) = c/β, .