Position dependence of the standing-wave radiation pressure quadrupole projection on a sphere applied to drop shape

There have been decades of interest in using the ultrasonic radiation pressure of standing waves to deform nearly spherical objects. An analytical approach sometimes associated with the present author involves approximating projections of the radiation pressure on spheres small in comparison with the wavelength and calculating the response to that projection. In 1981, for small fluid spheres, some terms in the quadrupole projection were published along with the dependence on the size and location of the sphere. An associated application was the flattening of levitated drops in air which are attracted toward velocity antinodes of a standing wave having horizontal equiphase surfaces. In subsequent applications of those results, the predicted analytical dependence on the location of the drop is frequently neglected. For the case of small weakly deformed drops in air in normal gravity, that omission is shown to result in an overestimation of the deformation and of the magnitude of the quadrupole radiation pressure projection. The present discussion simplifies the early results when applied to oblate drops and illustrates the consequence of including the position dependence on the modified small deformation. For large trapped oblate bubbles in water (also reviewed), the shape and location depend on the size.