In the process of acoustooptic refraction of an incident-finite- aperture light beam on a standing acoustic wave, there arises a cubic phase modulation of the beam in passing through an acoustooptic cell. After the Fourier transformation of the transmitted field, an Airy-type beam is formed with varying in time parameters. The propagation of such a beam is described explicitly with diffraction integral. By using the numerical simulation, it is established that the transverse intensity profile, acceleration and additional modulation of oscillations of a generated incomplete Airy beam are dependent on the frequency and power of an acoustic wave.