Explicit expressions for pressure in acoustic waves multiply reflected from reflecting surfaces of canonical form

In framework of the geometrical diffraction theory, the explicit expressions for the pressure in waves arbitrarily re-reflected N times from a contour, boundary surfaces of the cylindrical and spherical reflectors are obtained. Pressure expressions in the reception point for reflectors of the canonical form are obtained on the basis of 2D and 3D solutions to the problems on the pressure determination in the acoustical wave re-reflected from a set of reflectors in the case of high-frequency oscillations. The problem in its general formulation is studied on the basis of a modified physical diffraction of Kirchhoff diffraction theory. Within the frames of the proposed modification, diffraction integrals which leading terms for asymptotic expansions are studied through the multidimensional stationary phase technique, are obtained. The developed analytical expressions for the pressure in the re-reflected wave conform to the GTD. For all three cases, these expressions are connected with calculating the N-th order determinant (in the 2D case), and of the 2N-th order determinants (for reflectors in 3D space). An analytical and numerical analysis of the obtained expressions versus distances between a source and a receiver from the reflecting surface is performed. The acoustic wave focusing points are marked. The problem of replacing non-plane reflectors by plane ones in the applied problems of acoustics is discussed.