Sparse representation systems that encode signal structure have had a profound impact on sampling and compression paradigms. Remarkable examples are multi-scale directional systems, which, similar to our vision system, encode the underlying structure of natural images with sparse features. Inspired by this philosophy, we introduce a representation system for wave-based acoustic signal processing in 2D space–time, which has one spatial and one temporal dimension. We refer to this representation as the boostlet transform, which encodes sparse features of natural acoustic fields using the Poincaré group and isotropic dilations. Boostlets are spatiotemporal functions parametrized with dilations, Lorentz boosts, and translations in space–time. Physically speaking, boostlets are wave functions propagating with phase speeds other than the speed of sound, resulting in a peculiar scaling function. We formulate a discrete boostlet transform using a tensor product of Meyer wavelets and bump functions. An analysis with experimentally measured fields indicates that discrete boostlet coefficients decay significantly faster and attain superior reconstruction performance than wavelets, curvelets, shearlets, and wave atoms. The results suggest that boostlets offer a natural and compact representation of broadband acoustic waves in space–time.