Measurement of air-gun bubble oscillations

In this paper, I provide a theoretical basis for a practical approach to measuring the pressure field of an air gun array and present an algorithm for computing its wavefield from pressure measurements made at known positions in the vicinity of the gun ports. The theory for the oscillations of a single bubble is essentially a straightforward extension of Lamb's original paper and provides a continuous, smooth transition from the oscillating wall of the bubble to the far-field, preserving both the fluid flow and the acoustic radiation, all to the same accuracy and valid for bubbles with initial pressures up to about 200 atm (3000 psi or 20 MPa). The simplifying assumption, based on an argument of Lamb, is that the particle velocity potential obeys the linear acoustic wave equation. This is used then in the basic dynamic and kinematic equations to lead, without further approximations, to the nonlinear equation of motion of the bubble wall and the wavefield in the water. Given the initial bubble radius, the initial bubble wall velocity, and the pressure variation at any point inside or outside the bubble, the algorithm can be used to calculate the bubble motion and the acoustic wavefield. The interaction among air-gun bubbles and the resultant total wavefield is formulated using the notional source concept, in which each bubble is replaced by an equivalent notional bubble obeying the same equation of motion but oscillating in water of hydrostatic pressure, thus allowing the wavefields of the notional bubbles to be superposed. A separate calibration experiment using the same pressure transducers and firing the guns individually allows the initial values of the bubble radius and bubble wall velocity to be determined for each gun. An appendix to the paper provides a test of the algorithm on real data from a single gun.