Acoustic modes of a spherical thin-walled tank for liquid propellant mass gauging: Theory and experiment

Acoustic response of a thin-walled spherical flight tank filled with water is explored theoretically and experimentally as a testbed for an application of Weyl's law to the problem of mass-gauging propellants in zero-gravity in space. Weyl's law relates the mode counting function of a resonator to its volume and can be used to infer the volume of liquid in a tank from the tank's acoustic response. One of the challenges of applying Weyl's law to real tanks is to account for the boundary conditions which are neither Neumann nor Dirichlet. We show that the liquid modes in a thin-walled spherical tank correspond to the spectrum of a slightly larger spherical tank with infinitely compliant wall (Dirichlet boundary condition), where Weyl's law can be applied directly. The mass of the liquid enclosed by this "effective" tank's wall is found to equal the actual mass of the liquid plus the mass of the wall. This finding is generalized to thin-walled tanks and liquid configurations of arbitrary shapes and thus provides a calculable correction factor for the propellant mass inferred using Weyl's law with Dirichlet boundary conditions.