## 2 Materials and Methods
### Expressing the Virtual Forcing in Terms of Corner Modes
#### 2.3.1 Inverse Transformation of the Forced Plate  
We can now approximate the unknown functions from Section 2.1, \(s\), \(p\), \(q\), and \(r\), as a series of the similarity solutions associated with a corner of angle \(\xi=\pi/2\), found in Section 2.2, which we can extend mathematically beyond the physical corner. We use polar coordinates consistent with that defined in Figure 4, i.e., angles are measured from the respective corner bisectors as shown in Figure 5. Hence, the virtual upper boundary is defined by \(\theta=-3\pi/4\), whereas \(\theta=3\pi/4\) corresponds to the virtual lower boundary on the lines \(y=\pm 1/2\), \(x<0\), and \(\rho=-x\), so we can express the forcing terms in Cartesian form, which yield approximations, correct up to order \(o((-x)^{L})\), as  
\[\tau_{xy}\big{|}_{y=\frac{1}{2}\,\&\,x<0} =s(x)\approx\sum_{m=1}^{M(L)}\eta_{m}\mathfrak{f}_{m}\bigg{(}-x, -\frac{3\pi}{4};L\bigg{)}, \tag{49}\] \[\tau_{yy}\big{|}_{y=\frac{1}{2}\,\&\,x<0} =p(x)\approx\sum_{m=1}^{M(L)}\eta_{m}\mathfrak{g}_{m}\bigg{(}-x, -\frac{3\pi}{4};L\bigg{)},\] (50) \[\tau_{xy}\big{|}_{y=-\frac{1}{2}\,\&\,x<0} =q(x)\approx\sum_{m=1}^{M(L)}\eta_{m}\mathfrak{f}_{m}\bigg{(}-x, \quad\frac{3\pi}{4};L\bigg{)},\] (51) \[\tau_{yy}\big{|}_{y=-\frac{1}{2}\,\&\,x<0} =r(x)\approx\sum_{m=1}^{M(L)}\eta_{m}\mathfrak{g}_{m}\bigg{(}-x, \quad\frac{3\pi}{4};L\bigg{)}. \tag{52}\]  
The solutions in Section 2.1 were found in terms of the Fourier transform of these virtual conditions, which we write as  
\[S^{+}(\alpha) =\int_{-\infty}^{0}s(x)\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x \approx\sum_{m=1}^{M(L)}\int_{-\infty}^{0}\eta_{m}\mathfrak{f}_{m}\bigg{(}-x,-\frac{3\pi}{4};L\bigg{)}\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x=\sum_{m=1}^{M( L)}\eta_{m}\mathfrak{F}_{m}^{+}\bigg{(}\alpha,\frac{-3\pi}{4};L\bigg{)}, \tag{53}\] \[P^{+}(\alpha) =\int_{-\infty}^{0}p(x)\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x \approx\sum_{m=1}^{M(L)}\int_{-\infty}^{0}\eta_{m}\mathfrak{g}_{m}\bigg{(}-x,-\frac{3\pi}{4};L\bigg{)}\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x=\sum_{m=1}^{M( L)}\eta_{m}\mathfrak{G}_{m}^{+}\bigg{(}\alpha,\frac{-3\pi}{4};L\bigg{)}, \tag{54}\]  
\[Q^{+}(\alpha) =\int_{-\infty}^{0}q(x)\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x \approx\sum_{m=1}^{M(L)}\int_{-\infty}^{0}\eta_{m}\mathfrak{f}_{m}\bigg{(}-x,\quad\frac{3\pi}{4};L\bigg{)}\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x=\sum_{m=1} ^{M(L)}\eta_{m}\mathfrak{F}_{m}^{+}\bigg{(}\alpha,\frac{3\pi}{4};L\bigg{)}, \tag{55}\] \[R^{+}(\alpha) =\int_{-\infty}^{0}r(x)\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x \approx\sum_{m=1}^{M(L)}\int_{-\infty}^{0}\eta_{m}\mathfrak{g}_{m}\bigg{(}-x,\quad\frac{3\pi}{4};L\bigg{)}\mathrm{e}^{-\mathrm{i}ax}\mathrm{d}x=\sum_{m=1} ^{M(L)}\eta_{m}\mathfrak{G}_{m}^{+}\bigg{(}\alpha,\frac{3\pi}{4};L\bigg{)}, \tag{56}\]  
where the transforms can be determined analytically. Combining the transforms of the virtual boundary conditions (53)-(56) and expressions for the stresses (24) and (25), we now have a complete solution in Fourier space, given in terms of an infinite series of similarity solutions, which have, built-in, the irregular corner behaviour. To find the expressions for the potentials and stresses in physical space, we use the inverse Fourier transform defined by  
\[f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\alpha)\mathrm{e}^{\mathrm{i}ax} \mathrm{d}\alpha. \tag{57}\]  
We know that the integrands of the solution in this form will only contains pole-type singularities, not branch-cuts, and so we can evaluate these in \(x\geqslant 0\) using [PERSON]'s residue theorem, by deforming the contour up to a finite imaginary part of \(\alpha\), along the contour \(\mathcal{C}_{1}\), say, as shown in \[\tau_{xy}\big{|}_{y=\frac{1}{2}\,\&\,x<0} =s(x)\approx\sum_{m=1}^{M(L)}\eta_{m}\mathfrak{f}_{m}\bigg{(}-x, -\frac{3\pi}{4};L\bigg{)}, \tag{49}\]0 Note that it is straightforward to show that the contributions from the contours at \(\pm\infty\) are zero.