Another way to arrive at this result is to factor the wave equation using two first-order differential operators: [ ∂ ∂ ∂ ∂ t − − c ∂ ∂ ∂ ∂ x ] [ ∂ ∂ ∂ ∂ t + c ∂ ∂ ∂ ∂ x ] u = 0. Note that these operators are the ones used in the one-way wave equations : ∂ ∂ w ∂ ∂ t − − c ∂ ∂ w ∂ ∂ x = 0 , ∂ ∂ w ∂ ∂ t + c ∂ ∂ w ∂ ∂ x = 0.