This produces an ordinary differential equation for the spatial part f (x) : ∂ ∂ 2 u ω ω ∂ ∂ t 2 = ∂ ∂ 2 ∂ ∂ t 2 ( e − − i ω ω t f ( x ) ) = − − ω ω 2 e − − i ω ω t f ( x ) = c 2 ∂ ∂ 2 ∂ ∂ x 2 ( e − − i ω ω t f ( x ) ) .