For non-zero U, the Fourier integral solution consists of three distinct wave branches. Two of these branches correspond to the prior no-wind solution of Rotunno, except with Doppler shifting and associated wave dispersion. The third branch exists only for non-zero U and is shown to be broadly similar to flow past a steady heat source or a topographic obstacle. The relative importance of this third branch is determined largely by the parameter combination U/L. For sufficiently large U/L, the third branch becomes the dominant part of the solution.
The spatial structures of the three branches are described in terms of group velocity arguments combined with a desingularized quadrature method.