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A WKB approximation of elastic waves travelling on a shell of revolution

Accepted version
Peer-reviewed

Repository DOI


Type

Article

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Authors

Morsbøl, JO 
Sorokin, SV 

Abstract

This paper is concerned with the elastic waveguide properties of an infinite pipe with circular cross section whose radius varies slowly along its length. The equations governing the elastodynamics of such shells are derived analytically, approximated asymptotically in the limit of slow axial variation, and solved by means of the WKB-method (Wentzel–Kramers–Brillouin). From the derived solution the dispersion relation, modal coefficients, and wave amplification at each location along the structure are extracted, allowing identification of which types of waves are able to propagate along the structure at a given frequency. A key feature in the formulation of the model and the solution is that the radius and its variation are not specified in advance. Two characteristic examples of shells of revolution are presented to illustrate some general features of the waveguide properties, demonstrating how the evolution of the waves depends on the axial variation of the shell radius. It is explained how local resonances can be excited by the travelling waves and how strong amplifications of displacement can be produced. Specifically, for the axial/breathing wave it is shown that a local resonance is excited at the location where the frequency of the travelling wave and the radius of the shell exactly match the ring-frequency.

Description

Keywords

WKB, Dispersion diagram, Waveguide, Cylindrical shell, Shell of revolution, Elastodynamics, Ring-frequency

Journal Title

Journal of Sound and Vibration

Conference Name

Journal ISSN

0022-460X
1095-8568

Volume Title

375

Publisher

Elsevier
Sponsorship
This research has been founded by The Danish Council for Technology and Innovation, under Grant no. 10-083896, which is gratefully acknowledged.