From the definition of the delta function, u may also be written as u ( t , x , y , z ) = t 4 π π ∬ ∬ S φ φ ( x + c t α α , y + c t β β , z + c t γ γ ) d ω ω , where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u (t, x) is t times the mean value of φ on a sphere of radius ct centered at x : u ( t , x , y , z ) = t M c t [ ϕ ϕ ] .