## 2 Materials and Methods
### Expressing the Virtual Forcing in Terms of Corner Modes
#### 2.3.2 Satisfaction of the Free-End Boundary Conditions by Collocation  
We now wish to determine the unknown coefficients in the modal expansion, i.e., the Lamb mode constants \(\zeta_{n}\) and the corner mode constants \(\eta_{m}\). To do so, we need to satisfy the boundary conditions on \(x=0\), where the tractions of the total field must be zero; this implies that our scattered field must satisfy  
\[\tau^{\text{sca}}_{xx}(0,y) =\sum_{n=1}^{N}\zeta_{n}t_{x}^{l}(\alpha_{n},0,y)+\sum_{m=1}^{M} \eta_{m}t_{x}^{c}(\mathcal{C}_{1}(N),\gamma_{m},0,y;L)= -\tau^{\text{inc}}_{xx}(0,y), \tag{67a}\] \[\tau^{\text{sca}}_{xy}(0,y) =\sum_{n=1}^{N}\zeta_{n}t_{y}^{l}(\alpha_{n},0,y)+\sum_{m=1}^{M} \eta_{m}t_{y}^{c}(\mathcal{C}_{1}(N),\gamma_{m},0,y;L)= -\tau^{\text{inc}}_{xy}(0,y), \tag{67b}\]  
where a superscript \({}^{\text{inc}}\) denotes the incoming wave. We need \((N+M)/2\) collocation points, on which we will enforce the two boundary conditions (67a) and (67b) exactly. This yields a linear system of \((N+M)\) equations that can be solved to find the \((N+M)\) unknown coefficients \(\zeta_{n}\) and \(\eta_{m}\).  
The locations of the collocation points were found by trial and error, determining the best by inspection of the reconstructions. The optimally-positioned collocation points were found to vary, depending on which modal expansion we take.  
For expansions involving just Lamb modes \((M=0)\), the ideal points lie away from the corner, as these modes poorly represent the field near these points of irregularity. Therefore, we chose to use linear spacing for the \(N/2\) collocation points in this case. If, instead, we chose a Chebyshev distribution of collocation points for the pure Lamb expansion, the convergence of the method would become very poor indeed.  
When using an expansion including corner modes, the best points to take are those nearer to the corner; hence the optimal results were found by employing Chebyshev points. With corner modes included, we also have a choice of the number of Lamb modes \(N\) and the number of corner modes \(M\), to use; we generally found that the best results are obtained when \(M\) is large and \(N\) is small, although \(N\) must always be large enough to include all the propagating modes.  
It should be noted that the corner expansion method, unlike the pure Lamb mode expansion, seems to be very stable with respect to the choice of collocation points. Indeed, if, instead of a Chebyshev distribution, we used a linear distribution as in the pure Lamb case, then the results are only marginally worse. This may be expected as, effectively, once we have removed the corner singularities thanks to our corner modes, the remaining part of the field to be approximated is very regular indeed. We should also note at this stage that this novel expansion also worked well when choosing more collocation points than actually needed and solving the overdetermined system via a least squares approach.  
A discussion on how the two different approaches (pure Lamb and with the corner expansion) behave and compare as \(N\) and \(M\) increase will be given in the following section;we will show that, as expected, the corner mode expansion method has much better convergence properties than the pure Lamb approach.