An Efficient Semi-Analytical Scheme for Determining the Reflection of Lamb Waves in a Semi-Infinite Elastic Waveguide

Abstract

The classical problem of reflection of Lamb waves from a free edge perpendicular to the centre line of an elastodynamic plate is studied. It is known that Lamb wave expansions for the displacement and stress fields poorly represent the irregular behaviour near corners, leading to the slow convergence of a series of such waves. The form of the irregularity for an elastodynamic corner is derived asymptotically, and a new solution method, which incorporates this corner behaviour analytically, is then implemented. Results are presented showing that this new approach represents the near-field and far-field behaviour very accurately, requiring very modest numbers of Lamb wave and corner modes. Further, it is revealed that the method can recover the trapped-mode phenomenon encountered in this configuration at the Lamé frequency and a specific Poisson’s ratio that we find to be approximately 0.224798.

Edge resonance: exact results and fresh insights

“…The accuracy of the root-free approach enables an improved estimate of the resonant value of PR to be obtained by employing an iterative process with a 80 × 80 truncation of the matrix. It is found that ν r ≈ 0.224798037, which is identical to six decimal places to the value recently presented in the discussion of [12]. On using this value of ν r with the modal method of §4 with 30 terms |P| = 2.30214 × 10 −7 , whereas the root-free approach with the same number of terms gives |P| = 2.20861 × 10 −8 .…”

“…Thus, provided the function f o (y) is odd, differentiable and has the property f o (±1) = 0, the two series of (A 2) converge point-wise to −f o (y). The requirement that f o (±1) = 0 is strictly necessary for the case in which the expansion is in terms of σ 12 n (y). However, for the expansion in terms of V n (y) numerical experimentation suggests that point-wise convergence holds not only when f o (±1) = 0 but also when f o (±1) = 0 (whether or not f o (±1) = 0).…”

“…Perhaps the best-known property of the eigensystem is the bi-orthogonality relation ∫−11falsefalse{σn11false(yfalse)Umfalse(yfalse)−σm12false(yfalse)Vnfalse(yfalse)falsefalse} dy=Enδmn,where δmn is the Kronecker delta and En=iUnfalse(1false)normal∂σ12normal∂sfalse(sn,1false).Although expression (3.1) is well known, the relationship between En and the derivative of the characteristic equation (see (2.9)) given by (3.2) is less so. This result, which is crucial to the following analysis, can be established using a fairly standard approach (although the algebra is somewhat tedious) and for the sake of brevity the reader is referred to the appendices in [12,17] for details. In this section, following [12–14], a number of identities are proved.…”

“…In 2011, Cès et al [11] executed an experimental investigation into the local resonances of a free isotropic elastic plate. More recently, Davey et al [12] used a method involving corner modes to show that, at the second real edge resonance, ν ≈ 0.224798. In this paper, edge resonance is studied using methods developed by the author for addressing waveguide problems in structural acoustics.…”

“…In this paper, edge resonance is studied using methods developed by the author for addressing waveguide problems in structural acoustics. In that context, the author has established that the eigenmodes for acoustic waveguides with flexible walls are linearly dependent, and that this property is manifest through certain special sums of the eigenmodes that are identically zero [12–14]. The key step in deriving such identities is to recognize the link between the (generalized) orthogonality relation and the derivative of the characteristic equation that defines permissible wavenumbers.…”