linguistic:translation
Vectorial wave equation in three space dimensions

The vectorial wave equation (from which the scalar wave equation can be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. In a homogeneous continuum (cartesian coordinate x {\displaystyle \mathbf {x} }) with a constant modulus of elasticity E {\displaystyle E} a vectorial, elastic deflection u (x, t) {\displaystyle \mathbf {u} (\mathbf {x},t)} causes the stress tensor T = E ∇ u {\displaystyle \mathbf {T} =E\nabla \mathbf {u} }. The local equilibrium of a) the tension force div T = ∇ ⋅ (E ∇ u) = E Δ u {\displaystyle \operatorname {div} \mathbf {T} =\nabla \cdot (E\nabla \mathbf {u})=E\Delta \mathbf {u} } due to deflection u {\displaystyle \mathbf {u} } and b) the inertial force ρ ∂ 2 u / ∂ t 2 {\displaystyle \rho \partial ^{2}\mathbf {u} /\partial t^{2}} caused by the local acceleration ∂ 2 u / ∂ t 2 {\displaystyle \partial ^{2}\mathbf {u} /\partial t^{2}} can be written as ρ ∂ 2 u ∂ t 2 − E Δ u = 0. {\displaystyle \rho {\frac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}-E\Delta \mathbf {u} =\mathbf {0}.} By merging density ρ {\displaystyle \rho } and elasticity module E, {\displaystyle E,} the sound velocity c = E / ρ {\displaystyle c={\sqrt {E/\rho }}} results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium: ∂ 2 u ∂ t 2 − c 2 Δ u = 0. {\displaystyle {\frac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}-c^{2}\Delta \mathbf {u} ={\boldsymbol {0}}.} (Note: Instead of vectorial u (x, t), {\displaystyle \mathbf {u} (\mathbf {x},t),} only scalar u (x, t) {\displaystyle u(x,t)} can be used, i.e. waves are travelling only along the x {\displaystyle x} axis, and the scalar wave equation follows as ∂ 2 u ∂ t 2 − c 2 ∂ 2 u ∂ x 2 = 0 {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}-c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}=0}.)

The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term c 2 = (+ c) 2 = (− c) 2 {\displaystyle c^{2}=(+c)^{2}=(-c)^{2}} can be seen that there are two waves travelling in opposite directions + c {\displaystyle +c} and − c {\displaystyle -c} are possible, hence results the designation “two-way wave equation”.It can be shown for plane longitudinal wave propagation that the synthesis of two one-way wave equations leads to a general two-way wave equation. For ∇ c = 0, {\displaystyle \nabla \mathbf {c} =\mathbf {0},} special two-wave equation with the d'Alembert operator results: (∂ ∂ t − c ⋅ ∇) (∂ ∂ t + c ⋅ ∇) u = (∂ 2 ∂ t 2 + (c ⋅ ∇) c ⋅ ∇) u = (∂ 2 ∂ t 2 + (c ⋅ ∇) 2) u = 0. {\displaystyle \left({\frac {\partial }{\partial t}}-\mathbf {c} \cdot \nabla \right)\left({\frac {\partial }{\partial t}}+\mathbf {c} \cdot \nabla \right)\mathbf {u} =\left({\frac {\partial ^{2}}{\partial t^{2}}}+(\mathbf {c} \cdot \nabla)\mathbf {c} \cdot \nabla \right)\mathbf {u} =\left({\frac {\partial ^{2}}{\partial t^{2}}}+(\mathbf {c} \cdot \nabla)^{2}\right)\mathbf {u} =\mathbf {0}.} For ∇ c = 0, {\displaystyle \nabla \mathbf {c} =\mathbf {0},} this simplifies to (∂ 2 ∂ t 2 + c 2 Δ) u = 0. {\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}+c^{2}\Delta \right)\mathbf {u} =\mathbf {0}.} Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction c {\displaystyle \mathbf {c} } results as ∂ u ∂ t − c ⋅ ∇ u = 0. {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}-\mathbf {c} \cdot \nabla \mathbf {u} =\mathbf {0}.}