Surface Shear Waves in a Half-Plane with Depth-Variant Structure

Andrey Sarychev (), Alexander Shuvalov () and Marco Spadini ()
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Andrey Sarychev: Università di Firenze
Alexander Shuvalov: Université de Bordeaux
Marco Spadini: Università di Firenze

Journal of Optimization Theory and Applications, 2020, vol. 184, issue 1, No 3, 42 pages

Abstract: Abstract We consider the propagation of surface shear waves in a half-plane, whose shear modulus and density depend continuously on the depth coordinate. The problem amounts to studying the parametric Sturm–Liouville equation on a half-line with frequency and wave number as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum in the wave number/frequency plane for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of a finite number of surface wave solutions; the number grows and tends to infinity with the wave number. The most intriguing result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behaviour of the shear modulus and density close to infinite depth.

Keywords: Functionally graded medium; Surface shear waves; Parametric Sturm–Liouville problem; 74J15; 34B08; 34B24 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10957-019-01501-2

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