A nuclei size measurement technique is developed, based on a dispersion relation for propagation of sound waves through a bubbly liquid. This is used to relate the attenuation and phase velocity of a sound wave to the bubble population, leading to two integral equations. These equations are ill posed, and require special treatment for solution. Algorithms based on a minimization method that imposes a number of physical constraints on the solution, rendering the equation well posed, are developed. The procedure is first tested on analytical data with varying artificial noise added, and found to be successful in recovering the bubble density function, and to perform much better than other published solution techniques. Then, bubbles were generated using electrolysis and air injection through porous tubes, and bubble populations measured. Short monochromatic bursts of sound at different frequencies were emitted and received using hydrophones. The received signals were then processed and analyzed to obtain the attenuation and phase velocity. The void fraction and known experimental errors were also obtained and were fed as constraints to the inverse problem solution procedure. This resulted in bubble populations which compare favorably to those obtained by microphotography. © 1998 Acoustical Society of America.