Hello,
I don’t understand the general shape of your formulation and the RHS of your equation, since seems that f(w) is a vector. Lighhill’s analogy returns a scalar, through the double divergence of Lighthill Tensor. What you have is basically something that resembles a specific volume velocity distribution that is, indeed, a scalar.
What you should put in the wave equation has to be scalar and you should work always with scalar fields, derived from the vector fields.