  The primary Bjerknes force is responsible for the quick translational motion
of radially oscillating bubbles in a sound field. The problem is classical in
the case of small-amplitude oscillations, for which an analytical expression of
the force can be easily obtained, and predicts attraction of sub- resonant
bubbles by pressure antinodes. But for high-amplitude sound fields, the bubbles
undergo large amplitude nonlinear oscillations, so that no analytical
expression of the force is available in this case. The bubble dynamics is
approximated on physical grounds, following the method of Hilgenfeldt et al.
[J. Fluid Mech., 365, 171 (1998)], but carefully accounting for surface
tension. The analytical expression of the maximum radius of the bubble is
recovered, the time of maximum expansion is noticeably refined, and an
estimation of the collapse-time is found. An analytical expression for the
time-varying bubble volume is deduced, and the Bjerknes force is obtained in
closed form. The result is valid for any shape of the sound field, including
purely standing or purely traveling waves, and is ready to use in a theoretical
model of bubble clouds evolution. Besides, the well-known sign inversion of the
Bjerknes force for large standing waves is recovered and the inversion
threshold in the parameter space is obtained analytically. The results are in
good agreement with numerical simulation and allow a quantitative assessment of
the physical parameters effect. It is found that either reducing surface
tension, or increasing the static pressure, should produce a widening of the
bubble-free region near high-amplitude pressure antinodes.
