Wave Kernels

We begin with taking the partial Fourier transform of u with respect to x and get 

∂2uˆ ∂t2 (ξ, t) + |ξ|2uˆ(ξ, t) = 0, ξ ∈ Rn, t > 0, uˆ(ξ, 0) = fˆ(ξ), ξ ∈ Rn, ∂uˆ ∂t (ξ, 0) = gˆ(ξ), ξ ∈ Rn.

Thus, we get

uˆ(ξ, t) = C1cos(|ξ|t) + C2sin(|ξ|t), ξ ∈ Rn, t > 0.