Minnaert Frequency and Simultaneous Reconstruction of the Density, Bulk and Source in the Time-Domain Wave Equation

Abstract We deal with the inverse problem of reconstructing acoustic material properties or/and external sources for the time-domain acoustic wave model. The traditional measurements consist of repeated active (or passive) interrogations, such as the Dirichlet-Neumann map, or point sources with source points varying outside of the domain of interest. It is reported in the existing literature that based on such measurements, one can recover some (but not all) of the three parameters: mass density, bulk modulus or the external source term. In this work, we first inject isolated small-scales bubbles into the region of interest and then measure the generated pressure field at a single point outside, or at the boundary, of this region. Then we repeat such measurements by moving the bubble to scan the region of interest. Using such measurements, we show that If either the mass density or the bulk modulus is known then we can simultaneously reconstruct the other one and the source term. If the source term is known at the initial time, precisely we assume to know its first non vanishing time-derivative, at the initial time, then we reconstruct simultaneously the two parameters, namely the mass density with the bulk modulus and eventually the source function. Here, the source term, which is space-time dependent, can be active (and hence known) or passive (and unknown). It is worth mentioning that in the induced inverse problem, we use measurements with $$4=3+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dimensions (3 in space and 1 in time) to recover 2 coefficients of 3 spatial dimensions, i.e. the mass density and the bulk modulus and the 4 = 3 + 1 dimensional source function. In addition, the result is local, meaning that we do reconstruction in any subpart, of the domain of interest, we want.