An approximate Wiener-Hopf (WH) technique is developed for solving pro
blems involving fine spatial structure. As an example of the applicati
on of this method we investigate the atomisation of a liquid jet. The
jet exits from a nozzle into an ambient fluid. Short interfacial waves
become unstable and break into small particles. This problem is treat
ed as a potential flow under the influence of capillary effects at the
interface and the pressure fluctuation at the nozzle wall. Two simult
aneous WH equations are obtained. To solve them, the singular parts in
each equation are separated from the regular ones, that leads to a li
near system of algebraic equations for the residues. The response-wave
amplitudes are evaluated numerically and the instability diagram is p
resented. It is found that resonance occurs at double roots of the dis
persion relation. For a given azimuthal number m, the double roots for
m two curves parametrised by the Weber number beta. They merge at a ce
rtain critical point, where an even stronger resonance occurs. This fi
nally selects the dominant modes. By gauging one parameter, namely the
velocity ratio U, the theoretical prediction agrees quite well with e
xperimental results of the jet atomisation.