## 1 Introduction
Most of these methods take an ingenious numerical approach relying on the use of an expansion in terms of a carefully chosen family of orthogonal polynomials (Chebyshev, Gegenbauer, Jacobi, etc. ), which can effectively be used to tackle certain corner singularities and 'fix' the naive modal expansions. In the present work, rather than trying (in vain) to find known special functions with the required singular behaviour, we let them emerge naturally from the physical problem, giving rise to what we call the corner modes. This method has the advantage of not requiring an a priori knowledge of the singularities at hand and can accommodate rather complicated singular behaviours. As hinted above, the aim of this paper is to apply a new approach tailored to the specific physical nature of the irregularity. We will introduce new modes that accurately represent the irregular behaviour near the corners and add them to the Lamb wave terms, which well represent the smooth part of the field elsewhere. This paper offers an application of the new method in the context of elastodynamics, whilst contemporaneous applications are presented in [15], for acoustic problems, and [23], for a model problem in elastostatics. However, the present work also requires an interesting additional step. That is, it is convenient for mathematical purposes to imagine that the plate shown in Figure 2 is infinite in extent, and has specified traction boundary forcing on the top and bottom faces in \(x<0\) to ensure that the prescribed boundary conditions on \(x=0,\ |y|\leq 1/2\) are satisfied (i.e., that the total normal and shear stresses on \(x=0\) are zero.) This geometry is illustrated  
Figure 2: A simple canonical problem, containing two corners with locally irregular behaviour, at the points of intersection of the traction-free surfaces. A propagating symmetric Lamb wave is incoming from the right, and we wish to determine the resulting scattered field. The method we introduce will also work for an incoming asymmetric Lamb wave. in Figure 3, noting that the extension of the plate is a purely virtual construct; however, from [PERSON]'s theorem, it can easily be shown that there will always exist forcing functions, \(p(x),q(x),r(x),s(x)\) that ensure the correct boundary conditions on the actual plate edge, \(x=0\). The remainder of this paper is organised as follows. In Section 2.1, we recast the problem as an infinite plate, with (as yet unknown) boundary tractions on the faces of the semi-infinite virtual extension of the plate. We then write the solution to this new problem in Fourier space. In Section 2.2, we find an asymptotic representation of the new unknown boundary conditions in terms of the behaviour associated with the corners. Next, we summarise the known similarity solutions of the stress field associated with a static elastic corner before going on to implement an asymptotic technique to recover the similarity solutions for the stresses in an elastodynamic corner. In Section 2.3, we discuss how the corner behaviour, as determined in Section 2.2, can be expressed as a sum of corner modes. These new corner modes will enable us to write the field as an expansion of the form  
\[\text{Field}=\sum\text{Lamb Modes}+\sum\text{Corner Modes}.\]  
In this representation, the Lamb modes capture well the far field behaviour, which is of interest experimentally, while the corner modes accurately describe the near field behaviour, which is important when applying the edge boundary conditions to determine the reflection coefficients. We then highlight the excellent convergence properties of such an expansion in Section 3. We present some results obtained by including both corner and Lamb modes, and illustrate the superiority of our method (compared to the pure Lamb mode expansion) when it comes to the reconstruction of the stresses on the free end of the plate and the prediction of the propagating waves reflection coefficients.