Novel closed non-harmonic solutions for 1D, 2D, and 3D-wave equations with time as a directional angle in generalized (n+1)-spherical coordinates

Wave equations are ubiquitous: classical physics, fluid dynamics, relativistic quantum theory. Standard solutions use n-spatial dimensions (n=1,2,3) plus one independent time t dimension. The method of descent connects solutions for n=3 down to n=2 and n=1, the latter solved by D’Alembert with canonical variables p=x+w and m=x-w, where x is position along x-axis, w=Ct and C a relevant speed, see Lessons 17-24 in [1]. Instead, we ascend from the one-dimensional (1D) wave equation in spherical coordinates, with radial distance 0<r instead of -∞<x<+∞. Time is treated here as a novel orientation angle whose tangent is q=w/r. The 1D-wave equation is thus solved as a superposition of a permanent background potential dependent on r only, plus a new entangled potential dependent on q only; meaning of the r-q plane is discussed. Solutions with same structure hold for n>1, with an azimuth angle added in the 2D-plane, plus an additional elevation angle for the 3D-case. First-time explicit closed solutions are reported here for wave equations with n=1,2,3. Our novel solutions are non-harmonic and depend upon two independent variables r and q, plus (n-1) spatial angles. Reference: [1] S. J. Fowler, Partial Differential Equations for Scientists and Engineers (Dover, New York 1992).