This paper contains theoretical and experimental results on the relative motion of
two pulsating spherical bubbles along their line of centres, in a liquid subjected to an
acoustic field. The motion is caused only by the secondary Bjerknes forces. The linear
theory for the secondary Bjerknes forces is modified by introducing a model for the
coupling between the pulsations of the interfaces. The secondary effects introduced by
this model are determined by the frequency indices of the bubbles, defined as the ratio
of the forcing frequency to the resonance frequency of each bubble. The equations of
motion are set up with the conservative Lagrangian formalism. This approach allows
an analytical study of all the possible patterns of motion and the identification of
the set of governing parameters: total energy and interaction coefficient. A pair of
bubbles driven far from their resonance frequencies may attract or repel, depending
on whether their frequency indices are respectively on the same side or on either side
of unity. For forcing frequencies close to resonance, the proposed model predicts a
new pattern of relative motion, namely a periodic motion (oscillations) around an
equilibrium bubble separation. The experimental study identifies this new periodic
pattern of motion, for acoustically levitated bubbles of nearly equal sizes, forced
near their resonance frequency. A quantitative study on the variation of the relative
velocity with the separation between the bubbles shows that the conservative model
for the motion holds for large and moderate separations. The following information is
reported: (a) a classification of the pairs of bubbles, based upon their phase difference
in oscillations; (b) a model for the coupling of the pulsations of two bubbles; (c)
formulas for the interaction force field of two pulsating bubbles, for all of the
categories; (d) a study of all possible patterns of relative motion (collisions, scattering
and oscillations), with their conditions of occurrence; (e) experimental data for two
attracting bubbles; (f) experimental data for two oscillating bubbles.