## 2 Theoretical Analysis  
### Extended BTM Technique  
The extended BTM technique is fully described by [PERSON] al. in [17]. For the diffraction problem of an infinite wedge with geometry as described in Figure 1a, where a point source S with a source signal \(q(t)=\rho_{0}A(t)/4\pi\) is located at \((r_{S},\theta_{S},0)\) and a receiver \(R\) is at \((r_{R},\theta_{R},z_{R})\) in the cylindrical coordinate system, an explicit solution for the diffracted sound pressure received at position \(R\) can be written as [17]:  
\[p_{diff}(t)=\int_{-\infty}^{\infty}q\left[t-\frac{m(z)+l(z)}{c}\right].\frac{D (m(z),\ l(z),\ \theta_{S},\ \theta_{R})}{m(z)l(z)}dz \tag{1}\]  
where \(D(m(z),\ l(z),\ \theta_{S},\ \theta_{R})\) is the directivity function with  
\[D(m(z),\ l(z),\ \theta_{S},\ \theta_{R})=-\ \frac{v\beta(m(z),\ l(z),\ \theta_{S},\ \theta_{R})}{4\pi} \tag{2}\]  
\[\beta(\tau)=\beta_{++}(\tau)+\beta_{+-}(\tau)+\beta_{-+}(\tau)+\beta_{--}(\tau) \tag{3}\]  
\[\beta_{\pm\pm}(\tau)= \frac{\sin[v(\pi\pm\theta_{S}\pm\theta_{R})]}{\cosh[v\eta(\tau)] -\cos[v(\pi\pm\theta_{S}\pm\theta_{R})]} \tag{4}\]  
\[\eta(\tau)=\cosh^{-1}\frac{c^{2}\tau^{2}-(r_{S}^{2}+r_{R}^{2}+z_{R}^{2})}{2r_ {S}r_{R}} \tag{5}\]  
where \(v=\pi/\theta_{w}\) is the wedge index; \(\tau=[m(z)+l(z)]/c.\ c\) is the speed of sound. For finite-wedge cases, the limits of the integral in Equation (1) are replaced by the two endpoints of the wedge.  
The integral in Equation (1) can be considered as a convolution integral of the source signal \(q(t)\) with an impulse response describing the diffraction effect of the wedge. It is convenient to divide the edges into equally sized elements of size \(\Delta z\) for numerical im  
Figure 1: Geometries for describing the BTM technique: **(a)** infinite wedge with a source and receiver for the first-order diffraction, and **(b)** truncated wedge with a source and two receivers for the second- and third-order diffractions.  
plementation, and the contribution of each element to the total impulse response can be expressed as [17]:  
\[\Delta h_{i}\approx-\frac{v}{4\pi}\frac{\beta(m(z),\,l(z),\,\theta_{S},\,\theta_{ R})}{m(z)l(z)}\Delta z_{i} \tag{6}\]  
The solution described by Equations (1)-(6) can be extended to the application of higher-order diffractions. Considering a truncated wedge with two parallel edges, as depicted in Figure 1b, \(R_{2}\) and \(R_{3}\) are the receivers of the second- and third-order diffractions, respectively. Treating each element on edge \(z_{1}\) as a secondary source for edge \(z_{2}\), the second-order diffraction contributions of the propagation path between \(S\) and \(R_{2}\) can be derived as [16]:  
\[\Delta h_{\bar{\mathrm{ij}}}\approx\frac{v_{1}v_{2}}{\left(4\pi\right)^{2}} \frac{\beta(m_{1}(z_{i}),\,m_{2}(z_{i},\,z_{j}),\,\theta_{S1},\,\theta_{R1}) \beta(m_{2}(z_{i},\,z_{j}),\,l_{2}(z_{j}),\,\theta_{S2},\,\theta_{R2})}{2m_{1 }(z_{i}),\,m_{2}(z_{i},\,z_{j})l_{2}(z_{j})}\Delta z_{i}\Delta z_{j} \tag{7}\]  
where the factor of 2 in the denominator compensates for the doubling in pressure generated by an acoustic source when mounted on a baffle [17]. The third-order diffraction contributions of the propagation path between \(S\) and \(R_{3}\) can be derived similarly as follows [16]:  
\[\Delta h_{ijk}\approx-\frac{v_{1}v_{2}v_{3}}{\left(4\pi\right)^{3}}\frac{\beta (m_{1}(z_{i}),\,m_{2}(z_{i},\,z_{j}),\,\theta_{S1},\,\theta_{R1})\beta(m_{2}( z_{i},\,z_{j}),m_{3}(z_{j},\,z_{k}),\,\theta_{S2},\,\theta_{R2})\beta(m_{3}(z_{j},\,z_{k}), \,l_{3}(z_{k}),\,\theta_{S3},\,\theta_{R3})}{2m_{1}(z_{i}),\,m_{2}(z_{i},\,z_{ j})2m_{3}(z_{j},\,z_{k})l_{3}(z_{k})}\Delta z_{i}\Delta z_{j}\Delta z_{k} \tag{8}\]