Fig. 1 Acoustic space for the method of images: (a) 2D corner; (b) 3D corner; (c) infinite or 2D wedge; (d) semi-infinite or 3D wedge. Half-space is not illustrated. Real domains are smaller than image domains and are where spheres belong, that is, the concave part. Grey planes are unbounded except shared edges. Outstretched thicker axes are explained in the main texts From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 3 Schematic diagram showing the rotation of a coordinate system by an anticlockwise azimuthal angle of $\beta$ and the use of subscripts for position vectors. The $z$-axis is out of the paper. A perfectly-reflecting flat surface is the rotated $zx'$-plane. The subscripts $c$ and $\widetilde{c}$ represent the real and image spherical scatterers which are indicated by grey circles. $O_c$ and $O_{\widetilde{c}}$ are equidistant from the $zx'$-plane. The subscript for the global origin is $o$. $\mathbf{r}$ is a position vector whose azimuthal angle is denoted by $\phi$: the diagram is a projection to the $xy$-plane; hence, two spheres are subject to the same polar angle. $(x,y,z)$ indicates a general field point; $b$ is a field point on the $zx'$-plane From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 2 A coordinate orientation where a perfectly-reflecting flat surface is $z=0$: the $y$-axis is into the $zx$-plane. The subscripts $c$ and $\widetilde{c}$ represent the real and image spherical scatterers which are indicated by grey circles. $O_c$ and $O_{\widetilde{c}}$ are equidistant from the $xy$-plane. The subscript for the global origin is $o$. $\mathbf{r}$ is a position vector whose polar angle is denoted by $\theta$: the diagram is a projection to the $zx$-plane; hence, two spheres are subject to the same azimuthal angle. $(x,y,z)$ indicates a general field point; $b$ is a field point on the $xy$-plane. The continuous arrow lines show the use of double-subscript position vectors, while the use of single- and non-subscript position vectors is denoted by dashed and dash-dot arrow lines, respectively From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 4 Schematic diagram showing the location of image objects for a wedge angle of $\beta=\pi/5$. The $z$-axis is towards the reader; the diagram is a projection to the $xy$-plane. (a): wedges above the $xy$-plane; the real wedge is painted in grey. (b): wedges below the $xy$-plane. These three types of objects represent either scatterers or sources; each type is subject to the same polar angle within (a) or (b) From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 5 Schematic diagram showing the formation of image wedges with regard to boundary conditions. Each diagram is a projection to the $xy$-plane. Grey regions indicate real wedges. Continuous lines denote rigid planes; broken lines are of pressure-release. $+$sign indicates the corresponding image wedge has the same sign as the real wedge; $-$sign implies the opposite sign. (a) and (c): rigid real wedge cut half by pressure-release $xy$-plane; (b) pressure-release real wedge; (d) real wedge with both rigid and pressure-release sides. Wedge angles are $\beta=\pi/5$ for (a), (b), and (c); $\beta=\pi/6$ for (d). (a) can also represent a case of $-\infty<z<\infty$ From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 9 Comparison of PL due to 13 pool balls between measurements (in grey) and MSM (in black) for a semi-infinite wedge configuration in Fig. 6 with apex angle of $\beta=\pi/3$ rad: (a) frequency range in a 20-Hz increment from 0.1 to 5 kHz; (b) from 5 to 10 kHz From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 6 Near top view of a measurement setup for 13 pool balls in a 3D wedge with apex angle of $\beta=\pi/3$. The left vertical board is made of PVC; the other two by MDF. A source and microphone are also shown. Remnants of optical images of the balls can be glimpsed on the PVC board: see the electronic version for better resolution From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 7 Comparison of PL due to 13 pool balls between measurements (in grey) and MSM (in black): (a) half-space problem; (b) 2D corner From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 8 Comparison of PL due to 13 pool balls between measurements (in grey) and MSM (in black) on a rigid 3D corner: (a) frequency range in a 20-Hz resolution from 0.1 to 5 kHz; (b) from 5 to 10 kHz From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 11 Grey-scale representation at a fixed depth of the acoustic field in a water-filled pressure-release infinite wedge with the apex angle of $\beta=\pi/150$ excited by a 10-Hz point source. No scatterer is present. The abscissa denotes the range from the apex; the ordinate the cross range from the source. (a): field amplitude evaluated by the wedge Green’s function, referenced to the free-space amplitude 1-m away from the source. (b): difference of field amplitudes between (a) and the one calculated by a model from Doolittle et al. From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 10 Effect of underwater half-space boundaries on an air sphere (MSM in black, BEM in dashed grey) and a pressure-release sphere (MSM in continuous grey). Amplitudes are referenced to that 1-m away from a point source under free-space propagation. (a) free space; (b) pressure-release flat surface; (c) rigid flat surface. The black and dashed grey curves are difficult to separate in the graphical resolution provided From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 13 Comparison of the multiple scattering model (in dots) and BEM (in grey) for the acoustic response of a 3-m diameter rigid sphere in an infinite wedge with the apex angle of $\beta=\pi/150$, under a frequency sweep from 10 to 500 Hz in every 5 Hz by a spherical-wave source. (a) PL for a receiver at $(x,y,z)=(250,2,0)$ m; (b) at $(500,2,0)$ m From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com

Fig. 12 Comparison of the multiple scattering model (in dots) and BEM (in grey) for the acoustic response of a 3-m diameter rigid sphere in an infinite wedge with the apex angle of $\beta=\pi/150$, insonified by a 50-Hz spherical-wave source. (a) PL; (b) TL referenced to the free-space field 1 m away from the source From: Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces Q J Mechanics Appl Math. 2017;70(4):419-453. doi:10.1093/qjmam/hbx018 Q J Mechanics Appl Math | © The Author, 2017. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com