This paper presents an investigation into the fields radiated into air by ultrasonic phased arrays under transient excitation. In particular, it includes a theoretical prediction of spatial variations in amplitude throughout the both the near-field and far-field of such arrays. The approach has been used to predict the result of phasing to produce a focus in air, which can be seen to be particularly effective in the near-field of the array. Interesting features are observed, which are then described in terms of the performance of both individual elements and the resulting array. It is shown how some elements of design can be used to improve performance in focussing. The predictions are compared to the results of experiments in air using electrostatic arrays, where good focussing could be achieved provided the appropriate design principles were followed. The approach has been developed specifically for use in air, but the results would also hold for modelling in certain medical arrays where a focussing requirement might be needed close to the array itself.