AbstractDensity fronts are common features of ocean and atmosphere boundary layers. Field observations and numerical simulations have shown that the sharpening of frontal gradients, or frontogenesis, can spontaneously generate inertia–gravity waves (IGWs). Although significant progress has been made in describing frontogenesis using approximations such as quasi-geostrophy (Stone, J. Atmos. Sci., vol. 23, 1966, pp. 455–565, Williams & Plotkin J. Atmos. Sci., vol. 25, 1968, pp. 201–206) semi-geostrophy (Hoskins, Annu. Rev. Fluid Mech., vol. 14, 1982, pp. 131–151), these models omit waves. Here, we further develop the analytical model of Shakespeare & Taylor (J. Fluid Mech., vol. 736, 2013, pp. 366–413) to describe the spontaneous emission of IGWs from an initially geostrophically balanced front subjected to a time-varying horizontal strain. The model uses the idealised configuration of an infinitely long, straight front and uniform potential vorticity (PV) fluid, with a uniform imposed convergent strain across the front, similar to Hoskins & Bretherton (J. Atmos. Sci., vol. 29, 1972, pp. 11–37). Inertia–gravity waves are generated via two distinct mechanisms: acceleration of the large-scale flow and frontal collapse. Wave emission via frontal collapse is predicted to be exponentially small for small values of strain but significant for larger strains. Time-varying strain can also generate finite-amplitude waves by accelerating the cross-front flow and disrupting geostrophic balance. In both cases waves are trapped by the oncoming strain flow and can only propagate away from the frontal zone when the strain field weakens sufficiently, leading to wave emission that is strongly localised in both time and space.