Maxwell's square roots from the metric tensors of wave surfaces and branches of solutions of the photon and phonon wave equations

For the wave equations of optics and acoustics of isotropic media the infinite families of three-dimensional plane wave Cauchy operators are found by a direct tensor method. These families form involutive Lie groups. Their generators N can be found by taking square roots from the unit tensors of the wavefront subspaces with outer normal n. In optics the basic structural elements of N are complex involutive operators (reflection isometries) described by pairs of complex vectors S and C, which satisfy the metric condition S.C = 1 (S.n = C.n = 0), and also by a pair of projective operators +/-(1 - n x n) of the two-dimensional space of a plane orthogonal to n. In the acoustics of isotropic media, in view of the inequality of the longitudinal and transverse wave velocities, the generators N are represented as a linear combination of an involutive operator and a diad n x n. It is shown that the projection of the average energy flux (P)n of the wave is conserved in the general case N+ not equal N. The families of vectors S = 1/root 2(e(1) - i alpha e(2)), C = 1/root 2(e(1) + (i/alpha)e(2)); e(1).e(2) = 0, e(1) = e(1)*, e(2) = e(2)*, e(1)(2) = e(2)(2) = 1, alpha = alpha* being a part of N, are indicated. For these families the global operators exp[ikN(z - z(0))] acting on initial-field vectors give states described by the right-hand and left-hand elliptical helices. The wave normal n characterizes the direction of the angular momentum of the field and for the case alpha = 1 turns out to be equivalent to the Darboux vector known in geometry.