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 can be seen that there are two waves travelling in opposite directions + c and − − 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 , 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 . For ∇ ∇ c = 0 , this simplifies to ( ∂ ∂ 2 ∂ ∂ t 2 + c 2 Δ Δ ) u = 0 . Therefore, the vectorial 1st-order one-way wave equation with waves travelling in a pre-defined propagation direction c results as ∂ ∂ u ∂ ∂ t − − c ⋅ ⋅ ∇ ∇ u = 0 .