## 3 Results
### Reconstructions  
The figures presented in this section show the tractions along the end face of the plate, as we wish to compare how well the different modal expansions recover the specified boundary conditions. The scattered stress field (65) and (66) on \(x=0\) should be equal to the negative of that for the incoming wave, as there is zero overall traction on the end face (see (67a) and (67b)). Note that we only need to consider the vertical end face, as the conditions at the top and bottom of the plate are automatically satisfied for each and every mode. In Figure 7, we choose the wavenumber \(k_{t}=4\) and the Poisson ratio \(\
u=0.3\), and vary only the number of modes. The plots show the scattered field stresses, i.e., Equations (65) and (66), where the coefficients were found by collocation forced by the incident propagating symmetric Lamb wave (this is the only propagating mode for this value of \(k_{t}\)). This is plotted against the negative of the incoming wave in Figure 7a,b; for an exact solution, these would be identical. Also plotted, in Figure 7c,d, are the total normal and shear stress fields (i.e., due to both the incoming and outgoing waves) in order to demonstrate how well the results approximate the zero traction conditions on the face. The results in Figure 7 compare the behaviour of four modal expansions; one is a standard Lamb wave expansion without corner modes that we deem to have converged and gives acceptable results in terms of conservation of energy and errors in the boundary conditions. The other results are with varying numbers of corner modes. It should be noted that, in the pure Lamb wave expansion, we need a considerable number of modes to ensure convergence; this means that the condition number of the collocation matrix is large in that case, demanding that high precision calculations be used to obtain good results. We note that the results show that we need significantly fewer total modes with our new expansion, i.e., even with a small number of corner modes, we can represent the near field behaviour with extremely small errors. We can therefore deduce that these modes are correctly capturing the key behaviour of the irregular fields near the corners. We also plot the error in traction reconstructions associated with a variety of wavenumbers, which is equivalent to changing the forcing frequency; higher \(k_{t}\) corresponds to higher frequencies where there are more propagating modes. In practice, it is found that higher frequency problems require greater numbers of Lamb modes to ensure convergence with the same accuracy. The plots in Figure 8 show, again, the tractions along the face \(x=0\), this time considering three different values of \(k_{t}\). As an additional illustration of the superiority of our method compared to the pure Lamb mode approach, we study the convergence of the reflection coefficients (and the energy) in the challenging case of \(k_{t}=8\), for which there are three propagating modes. The results are summarised in Figure 9. We show that, even with only \(M=4\) corner modes, our method displays results that are more than two orders of magnitude better than the pure Lamb mode approach with the maximum number of Lamb modes considered (\(N=80\)). Figure 8: A set of results showing the accuracy of the zero traction boundary condition for a variety of frequencies \(k_{t}\) for fixed Poisson ratio \(\
u=0.3\). The maximum values of the incoming field for both tractions are order one. There is one propagating mode apiece for \(k_{t}=2\) and \(4\), and three propagating modes when \(k_{t}=8\). The number of modes in each plot was chosen to maintain the same order of magnitude accuracy: for \(k_{t}=2\), \(N=2\) and \(M=2\); for \(k_{t}=4\), \(N=4\) and \(M=4\); and for \(k_{t}=8\), \(N=10\) and \(M=12\). Figure 7: The reconstructions of the stress fields using \(N\) symmetric Lamb modes and \(M\) corner modes.