## 4 Discussion  
In this article, we have asymptotically determined the form of the similarity solutions that are associated with the behaviour close to a corner in a semi-infinite elastodynamic thick plate. We recognized that the irregular corner behaviour causes difficulties for Lamb mode expansion techniques and so, to overcome this, we have introduced a new method for constructing corner modes that not only accurately capture the behaviour near the corner, but that also satisfy the plate boundary and radiation conditions. Additionally, because these modes are found using the same Fourier transform approach as for the Lamb waves, adding them does not result in an over-complete representation of the field.  
As found in the results Section 3, our numerical work shows that including the corner modes into the expansion allows us to accurately capture the near-field behaviour, and as a consequence we need include far fewer modes to achieve comparable or superior results. Therefore, we believe that these corner mode solutions can be utilised in a wide variety of Lamb wave or other, scattering problems for which standard plate or duct wave modal expansion techniques converge slowly [15; 23]. We have also demonstrated that, by considering these extra modes, we can more accurately capture the trapped-mode behaviour associated with this configuration; this allows us to quantify the Poisson's ratio with more precision than other extant methods, yielding the new result \(v\approx 0.224798\) at which this trapped-mode occurs. This result has recently been confirmed in a private communication with [PERSON] [26], who has approached the localised mode problem in a quite distinct way to the present work or to that described in [7; 8; 9].  
Conceptualization, R.C.A. and I.D.A.; methodology, R.C.D., R.C.A., and I.D.A.; software, R.C.D. and R.C.A.; validation, R.C.D., R.C.A., and I.D.A.; formal analysis, R.C.D., R.C.A., and I.D.A.; investigation, R.C.D., R.C.A., and I.D.A.; resources, R.C.A. and I.D.A.; writing--original draft preparation, R.C.D.; writing--review and editing, R.C.A. and I.D.A.; supervision, R.C.A. and I.D.A.; project administration, R.C.A. and I.D.A.; and funding acquisition, R.C.A. and I.D.A. All authors have read and agreed to the published version of the manuscript.  
[PERSON] was in receipt of funding from AWE and EPSRC to support his postgraduate studentship, [PERSON] undertook part of the work under EPSRC grants EP/N013719/1 and EP/W018381/1, and [PERSON] undertook part of the work under EPSRC grants EP/K032208/1 and EP/R014604/1.  
The authors are grateful to [PERSON] for his comments on early drafts of this manuscript and for facilitating the relationship with the industrial partner through his William Penney Fellowship.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; or in the writing of the manuscript. The funders approved the manuscript for publication.  
Figure 10: The absolute value of the determinant of the collocation matrix, excluding the propagating mode, for different numbers of corner modes. The respective lines correspond to four Lamb modes and \(M=4\) (), \(M=6\) (), \(M=8\) (), and \(M=10\) () corner modes. The dotted lines are the projected values of the determinant, found by extending the curves on either side of zero; this aids the determination of location of zero. The value of Poisson’s ratio at which the determinant vanishes can be seen to have converged, once \(M=6\) or larger, to \(v\approx 0.224798\).