The Helmholtz instability of a vortex sheet separating two fluids in relative motion is unbounded in a simple linear model of the interaction of sound with the sheet. This paper presents a model which limits the amplitude of a harmonic wave in a physically realistic way but remains mathematically tractable. It is based on the idea that growth is limited by the onset of turbulence between the fluids when the Helmholtz wave reaches a critical size. An important consequence of the theory is a strong enhancement of the sound scattered upstream, which is significant both in the context of forward noise produced by a jet and possibly also of jet screech. The requirement of causality is of central importance in determining the correct solution, and detailed general results on the theory of zero ultradistributions are presented to establish an analytic definition of causality for the class of solutions encountered.