Guided Elastic Waves along Periodically Corrugated Structures

Doctoral thesis, 1994

The present thesis consider time-harmonic surface and guided elastic waves propagating along periodically corrugated structures. The problems discussed deal with various configurations where the surfaces are sinusoidally - singly or doubly - periodic, and the interest is focused on investigating the roots of the dispersion relations and in particular the stopband and cross-over resonances. In all cases the results are presented for varying frequencies, corrugation heights, and, when relevant, angles of propagation. The analyses and results may be applicable in a variety of fields: ultrasonic nondestructive testing and evaluation of materials, ultrasonic devices, mode converters and energy transformation, filters, etc. Firstly, we consider the two-dimensional problem of the periodically corrugated parallel plate with traction-free surfaces. Using the technique based on the Rayleigh hypothesis (also called a modal approach) the dispersion relation for the symmetric Rayleigh-Lamb modes is obtained. The symmetric sinusoidally corrugated plate is studied numerically and the behaviour of the three lowest modes in both stopbands and passbands is closely investigated. The existence and behaviour of surface waves along an infinite, periodically corrugated, cylindrical cavity with traction-free boundary in an elastic medium is next investigated. The dispersion relation is derived by use of both the modal and the formally exact null- field approaches to demonstrate the equivalence of the two techniques. The sinusoidally corrugated cavity is studied numerically, and the dispersion relation is solved for roots on the physical Riemann sheet for the first three modes and on some of the nonphysical sheets for the axially symmetric one. The three-dimensional geometry of a, on both surfaces, double corrugated layer on a half-space is then considered. The dispersion relation for the full problem is derived using the null-field approach. The stopband and passband structure of the physical modes are investigated numerically for two different doubly sinusoidally corrugated geometries. First the Rayleigh-like surface waves that exist when there is no layer are investigated for varying frequency, corrugations heights, and angle of propagation. Afterwards, with the layer present, some of the lowest Love and Rayleigh-Lamb-like waves are studied. In the last part of the thesis the excitation of generalized Rayleigh waves by a time- harmonic source located below the infinite, doubly periodically corrugated free boundary of an elastic half-space is studied. The full elastodynamic equations are solved using the null-field approach, and the contribution of the surface waves to the total displacement field is obtained for the sinusoidally corrugated geometry with equal periods of corrugation. For different corrugation heights and frequencies and both a vertical and an arbitrarily directed horizontal point force, numerical results are presented for the angular dependence of the surface field far away from the source where the generalised Rayleigh mode contribution dominates.