Acoustics of a bubble rising in a gravitational field

The rising motion of a pulsating spherical bubble of gas and vapour in an isochoric, inviscid liquid is investigated. The motion is driven by a uniform and constant force field, as the gravitational one. The liquid is assumed at rest at the initial time, so that the flow is irrotational and is described in terms of a velocity potential, which is represented by means of a series of Legendre polynomials. After relating the bubble motion to the liquid flow, a nonlinear ODE system of two equations for the radius and the position of the center of mass of the bubble is written. The acoustic signal produced in the liquid by the bubble motion is evaluated by means of elliptic integrals. Moreover, an efficient approximation of it free from these special functions is proposed. The numerical integration of the ODE system allows to evaluate the acoustic signal and to compare it with the proposed approximation, in several sample flows.