Factorized One-Way Wave Equations

: The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical ﬁnite element (FE) and ﬁnite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the deﬁnition of the wave propagation direction by the geometric position of the transmitter and receiver.


Introduction
Wave propagation in all three states of aggregation-solid, fluid and gas-is calculated according to Cauchy's equation of motion. For this 2nd-order partial differential Equation (PDE), two independent solutions exist and result in-according D'Alembert's separation approach-two wave equations representing two opposite traveling waves.
Unfortunately, D'Alembert's approach cannot be used to pre-select the task-relevant wave to identify the irrelevant wave. For analytical calculations, both solutions can be distinguished by their sign. However, in numerical finite element (FE) and finite difference (FD) calculations, the ambiguity causes irregular phantom effects known as "one-way/twoway wave" problem. Further related information with comprehensive lists of references is given in [1,2]. Several recently published patent applications [3][4][5] confirm that the problem still persists.
Factorization is a well-known method to solve reducible PDEs [6][7][8][9][10]. Specifically for homogeneous media, the second-order PDE wave equation can be factorized into two first-order PDEs, resulting in opposite traveling waves according to D Alembert's solution. Generally, for wave propagation in homogeneous media, analytical solutions exist and calculation efforts for seconnd or first-order PDE do not differ significantly. However, in inhomogeneous media, waves can only be numerically calculated [11]. Seismic FE and FD calculations of geological structures require nodes in a Giga to Tera range and CPU computing times of days to weeks [12]. Thus, the primary task is the factorization of the wave equation for longitudinal waves in inhomogeneous media. Additionally, the factorization method for waves in homogeneous media is expanded to other wave types.

Method of PDE Factorization
The ambiguity of the two solutions is problematic in the context of FE and FD calculations. Therefore, the well-known 2nd-order wave Equation (1) was repeatedly separated according to the expression (a 2 − b 2 ) = (a − b)(a + b) into two PDEs of the 1st order [13]: In contrast with Equation (1), the resulting partial Equations (4) and (6) have the solutions (5) and (7) with the defined wave propagation direction +c and −c, respectively.
The factorization of a 2nd-order PDE into two independent first-order PDEs corresponds to the separation of a standing wave Equation (1) into two oppositely traveling wave Equations (4) and (6) and a change from force equilibrium to impulse equilibrium. Table 1 shows the solutions of Equation (1) and both first-order wave Equations (4) and (6). Table 1. The 2nd-order (two-way) wave equation has two solutions. There are two standing waves, sin(ωt) cos(kx) and cos(ωt) sin(kx). According to 2 sin(α) cos(β) = sin(α− β)+sin(α+ β), two propagating waves sin(ωt ± kx) in opposite directions are the result, i.e., the solutions are ambiguous, and, after calculation, the appropriate solution has to be chosen. In contrast, the firstorder (one-way) wave equation has one propagating wave as the solution, with a pre-defined direction depending on the choice of the wave velocity c = {+c, −c}.
Two Solutions: Unique Solution: Unique Solution: Standing Waves Propagating Wave +x-dir.

PDE Factorization-Waves in Inhomogeneous Media
An inhomogeneous, isotropic medium with coordinates x = {xi, yj, zk} has a local density ρ = ρ(x) [kg/m 3 ], a longitudinal elasticity modulus E = E(x) [Pa], and the wave velocity To factorize this second-order PDE, s is replaced by a new field variable Es. With the material law ρ = E/c 2 and E ∂ 2 s/∂t 2 Analogously to Equations (3), (4) and (6), this second-order wave equation for inhomogeneous media can be factorized into two first-order PDEs: Longitudinal wave displacement s and wave velocity c have the same direction and can be turned in the direction of the xi-axis (s = si, c = ci). Hence, the scalar form results: The Helmholtz approach ∂(Es)/∂t = iω(Es) and settlement (Es) = ∂(Es)/∂x lead to As can be controlled by insertion, the analytical solution (15)

PDE Factorization/Transversal Waves
In case of homogeneity, a transversal wave with vectorial deflection s and wave velocity c is given by tensorial Equation (18) A transversal wave consists of equal parts rotary and deviatoric portion, i.e., ∇s = (∇s) Rot + (∇s) Dev = 2(∇s) Rot = U × rot s Inserting this expression into Equation (19) and with c · U = c, the transversal one-way wave Equation (21) follows as [14] (unit tensor U = ii+jj+kk)

PDE Factorization/Further Mechanical and Electromagnetic Waves
According to PDE factorization (3) with the solutions for homogeneous media (4) and (6) further physical second-and fourth-order PDEs with the same structure as Equation (1) can also be factorized into first-order one-way wave equations in full analogy. Table 2 lists classical equations for longitudinal, transversal, and string, Moens/ Korteweg, bending, and electromagnetic wave equations and their factorized pendants with the respective solutions. Additionally, governing formulas for wave velocity c are given. The exclusion of asymmetrical tensor parts for mechanical waves and symmetrical tensor parts for electromagnetic waves are not carried over. Table 2. Formal mathematical factorization of 2nd and 4th order partial differential wave equations into 1st order PDEs for homogeneous media. Wave-mechanically this corresponds to the separation of a standing wave into two complementary, in opposite directions travelling waves (here the wave propagating in +xi-direction is given). The electromagnetic longitudinal wave is an expansion of the longitudinal telegraph wave. Note: Wave propagation direction of the scalar factorized wave equations has to be defined by selection of the wave velocity c = {+c, −c}.

Discussion
The main result is the factorization of the longitudinal wave equation for inhomogeneous media. Hereby, the elastic displacement s = s(x, t) was replaced by the new field variable Es. Table 3 compares Cauchy's first Equation of Motion with the field variable s and the factorized wave Equation (11) with the new field variable Es.  (15) When inserting a constant elasticity module (E =const.) into factorized Equation (11), the one-way wave equation for homogeneous media results:ṡ − c∇s = 0, with the scalar formṡ − cs = 0. This wave equation obtained by (a) PDE factorization can also be derived from (b) the impedance theorem [15] and (c) the tensorial impulse flow equilibrium [16] of kinetic impulse flow ρcṡ and potential impulse flow E∇s (Table 4).

Mathematical/Physical
Starting Conversion/ Scalar One-Way Traveling Wave Approach Equation

Conclusions
The classical approach of calculating wave propagation via 2nd order PDE or "Twoway wave equation" is standardely used, although it primarily describes a standing wave field and ambiguous solutions result. If the calculation of wave propagation only in a given direction is of interest, such in seismics, the 1st order PDE or "One-way wave equation" is necessary. Because there exists no analytical solution for the one-way wave equation in inhomogeneous media approximations have to be used. However, this gap has been closed by the analytical solution presented in this article. Moreover, the method of factorization can be analogously tranferred to other standard wave forms and electromagnetic waves. This creates opportunities for simpler and faster wave propagation calculations in this field of research and also in other technical areas.

Conflicts of Interest:
The authors declare no conflict of interest.