We derive solutions to an inhomogeneous Mathieu equation that describes the photoacoustic effect in a one-dimensional phononic structure whose acoustic properties vary sinusoidally in space. Solutions show splitting of resonances, the space equivalent of subharmonic generation, and spatial confinement. Properties of the photoacoustic effect including the damping of waves inside the band gaps, the dispersion relation, the positions and widths of the gaps, the frequencies of resonances, and the space dependence of the acoustic waves can be found in closed form from known properties of Mathieu functions.