One-Way Wave Operator

Abstract

The second-order partial differential wave Equation (Cauchy’s first equation of motion), derived from Newton’s force equilibrium, describes a standing wave field consisting of two waves propagating in opposite directions, and is, therefore, a “two-way wave equation”. Due to the second order differentials analytical solutions only exist in a few cases. The “binomial factorization” of the linear second-order two-way wave operator into two first-order one-way wave operatoras has been known for decades and used in geophysics. When the binomial factorization approach is applied to the spatial second-order wave operator, this results in complex mathematical terms containing the so-called “Dirac operator” for which only particular solutions exist. In 2014, a hypothetical “impulse flow equilibrium” led to a spatial first-order “one-way wave equation” which, due to its first order differentials, can be more easily solved than the spatial two-way wave equation. To date the conversion of the spatial two-way wave operator into spatial one-way wave operators is unsolved. By considering the one-way wave operator containing a vector wave velocity, a “synthesis” approach leads to a “general vector two-way wave operator” and the “general one-way/two-way equivalence”. For a constant vector wave velocity the equivalence with the d’Alembert operator can be achieved. The findings are transferred to commonly used mechanical and electromagnetic wave types. The one-way wave theory and the spatial one-way wave operators offer new opportunities in science and engineering for advanced wave and wave field calculations.