## 1 Introduction
For sufficiently low frequencies, the wavelengths of the propagating modes are much longer than the corner lengthscales, therefore the local behaviour around the corner is unimportant away from it; however, even in such low-frequency regimes, the behaviour near sharp corners strongly affects the accuracy of methods for determining the global solution [6]; we shall see that this is true even for irregular, but non-singular, corner behaviour. In this article, as a first step towards the tamper model, and as a useful way to demonstrate the novel solution method, we examine a simpler problem that incorporates corner behaviour. We will consider the reflection of an incoming time-harmonic, symmetric wave in a semi-infinite elastic waveguide, as shown in Figure 2. In this problem we have symmetry about the mid-plane and hence no asymmetric mode conversion; however, it will allow us to study how sharp corners affect the reflection problem. The corners in this problem each have an internal angle less than \(\pi\), which means that the stress field around them is bounded. This problem may be solved by traditional methods, such as spectral collocation [7], allowing us to generate solutions to compare against; hence, this is a good test case to implement and validate our new method to deal with irregular corner behaviour. In addition to this goal, this canonical problem is also associated with the important question of the existence of a trapped mode, which in this case occurs at the Lame frequency and at a particular value of Poisson's ratio, which we will discuss in Section 3.2[7; 8; 9]. Figure 1: An exaggerated diagram showing a ‘hidden tamper’. The tamper is not symmetrically placed about the mid-plane and will therefore induce mode conversion between the two types of guided elastic waves. We wish to find the solution to this problem in terms of the in-plane natural modes of elastic waveguides, as these are the waves that are of experimental interest. These were first studied by [PERSON] over one hundred years ago [10] and are still an active area of research due to their very wide range of applications in areas such as nondestructive testing, seismology, and vibration control. The two sets of modes (in this article, we may ignore the third set of modes, the out-of-plane polarized shear waves, which are decoupled from the other two in the problem of interest) are the faster symmetric and the slower anti-symmetric modes, the wavenumbers of which satify different dispersion relations. Finding the roots of these relations is non-trivial, but methods relying on finite-product formulae [11], asymptotic analysis, or projection onto Chebyshev polynomials [7] can be used efficiently to this end. The solutions to each dispersion relation consist of a single propagating mode that always exists (the \(A_{0}\) or \(S_{0}\) mode), a finite number of higher propagating modes that cut on depending on frequency, and an infinite number of evanescent modes. It has been known for some time [6; 12] that the solution to the present canonical problem in terms of a modal expansion of Lamb waves is slowly convergent near the free end, and poorly represents the behaviour of the field near the corners. This is essentially due to the Gibbs phenomenon. There are a variety of numerical treatments that can be applied in this and other waveguide problems to aid convergence, including the Lanczos \(\sigma\)-approximation (see, e.g., [13; 14]). Note that difficulties arising from geometric singularities in waveguides when using a naive modal expansion do not only occur in the elastodynamic case considered presently. In fact, the corner mode expansion method presented in this paper was first conceived in the context of easier, scalar, acoustic problems [15]. Moreover, other authors have considered such problems in, for instance, acoustic waves [16; 17], water waves [18; 19], horizontal shear waves [20], and electromagnetic waves [21; 22].