Formulas for detecting a spherical stiff inclusion from interior data: a sensitivity analysis for the Helmholtz equation

In this paper, we establish sensitivity results that are relevant for imaging stiffness in tissue but may also be useful in other contexts. The data are the displacement at a single frequency throughout the imaging domain. The goal is to determine how the quantities—(1) amplitude of displacement, or alternatively (2) the displacement itself, the average displacement, the phase or the phase gradient—change within a homogeneous stiff inclusion embedded within a homogeneous background. The results are easily interpreted formulas that show the dependence on the radius of the inclusion, the frequency and the stiffness contrast between the inclusion and the background. Our assumptions are: (1) the displacement satisfies the Helmholtz equation with the variable stiffness parameter; (2) the experiment produces a plane wave in the absence of any inclusions; (3) in 3D, the inclusion is spherical; (4) in 2D the inclusion is a circular disc; and alternatively in 3D the inclusion is an infinite circular cylinder. Our method of analysis is to use series expansions of the solution expanded about the center of the inclusion. (Some figures may appear in colour only in the online journal)