where the double dot on u ¨ ¨ denotes two time derivatives of u, ∇ is the nabla operator, and ∇ = ∇ · ∇ is the (spatial) Laplacian operator (not vector Laplacian): u ¨ ¨ = ∂ ∂ 2 u ∂ ∂ t 2 , ∇ ∇ = ( ∂ ∂ ∂ ∂ x , ∂ ∂ ∂ ∂ y , ∂ ∂ ∂ ∂ z ) , ∇ ∇ 2 = ∂ ∂ 2 ∂ ∂ x 2 + ∂ ∂ 2 ∂ ∂ y 2 + ∂ ∂ 2 ∂ ∂ z 2 .