There is a continuing debate about the role of linear instability waves in the prediction of jet noise. Parallel mean flow models, such as the one proposed by Lilley, usually neglect these waves because they cause the solution to become infinite. The result is then non-causal and can, therefore, be quite different from the more realistic causal solution, especially for the chaotic flows being considered here. The present paper solves the relevant acoustic equations for a non-parallel mean flow by using a vector Green's function approach and requiring that the mean flow be weakly non-parallel, i.e. requiring that the spread rate be small. It demonstrates that linear instability waves must be accounted for in order to construct a proper causal solution to the jet noise problem. Recent experimental results (e.g. see Tam, Golebiowski & Seiner 1996) show that the small-angle spectra radiated by supersonic jets are significantly different from those radiated at larger angles (say, at 90°) and even exhibit dissimilar frequency scalings (i.e. they scale with Helmholtz number as opposed to Strouhal number). The present solution is (among other things) able to explain this rather puzzling experimental result.