Time-delay modulation of finite-distance organization and orbital motion in a three-bubble acoustic cluster

A time-delayed pulsation-translation model is developed to investigate finite-distance organization and orbital dynamics in a three-bubble acoustic cluster. The results show that the time-delayed interaction generally selects a larger mean inter-bubble spacing and tends to reduce spacing uniformity. Pressure and initial-radius scans further show that this spacing enlargement is robust in the examined finite-distance regime, although its magnitude depends on the driving pressure and bubble-size combination. A centroid-frame acceleration decomposition further reveals that the radial component governing spacing regulation remains of the same order of magnitude, whereas the tangential component responsible for sustained orbital motion is strongly weakened. Consequently, the cluster remains spatially bounded, but its orbital angular velocity is markedly reduced. These results show that finite sound-speed propagation mainly enlarges the selected spacing while suppressing the tangential component of the collective motion.