i, j, k] is the impulse response of the scalar wave equation (scalar discrete Green's function) and can be achieved using multi dimensional z-transform as [6]:

i,j,k] is the solution of the second order central difference approximation of the scalar wave equation with Kronecker delta excitation expressed as (it has been considered as i' = j' = k' = n' = 0 due to the shifting capability of the Green's functions):

We have shown that such a scalar wave is purely electric and has no magnetic component.

41) that the scalar wave ([LAMBDA]) distribution induces a charge density, [[rho].

are exact NtD maps corresponding to the scalar problem, for each scalar wave equation with velocities [v.

The function f ([LAMBDA]) is the NtD map corresponding to the optimal PML for the scalar wave equation.

The resulting optical field is denoted by a

scalar wave function,

This description above of in-and out-waves is almost identical to the quantum waves of the electron that can be obtained rigorously using a scalar wave equation in Section H.

Wolff, (6,7) Mead, (8) and Haselhurst (13) explored the Scalar Wave Equation and found that its solutions form a quantum-wave structure, possessing all the electron's experimental properties, eliminating the paradoxes of quantum mechanics and cosmology.

The

scalar wave functions U discussed in this paper all denote the complex disturbance at a point of observation P(x, y, z) in the diffraction pattern of a perfectly conducting, infinitesimally thin, semi-infinite screen that occupies the half plane x > 0, z = 0 of a cartesian coordinate system, as depicted in Fig.

Quantum matter waves exist in space and are solutions of a

scalar wave equation.

In these equations, U is a

scalar wave function, S is a closed surface containing a plane aperture A located in the xy-plane of a cartesian coordinate system as indicated in Fig.

Gbur presents a textbook on optics where the intensity of the field is zero and the phase therefore indeterminate, for graduate and advanced undergraduate students in physics and optics who have a working knowledge of electromagnetic and

scalar waves, Fourier transforms, and ideally of Fourier optics.

Gomez R, Winicour J (1992) Asymptotics of gravitational collapse of

scalar waves.

In sections on acoustic waves in sonic crystals, elastic waves in phononic crystals, and wave phenomena in phononic crystals, he considers such topics as

scalar waves in periodic media, sonic crystals, phononic crystals for surface and plate waves, coupling acoustic and elastic waves in phononic crystals, evanescent Bloch waves, and spatial and temporal dispersion.