On the simultanenous identification of two space dependent coefficients in a quasilinear wave equation

This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter $\kappa(x)$ often referred to as $B/A$ in the acoustics literature and the wave speed $c_0(x)$. The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. More precisely, we provide results on local uniqueness of $\kappa(x)$ from a single observation and on simultaneous identifiability of $\kappa(x)$ and $c_0(x)$ from two measurements. For a reformulation of the problem in terms of the squared slowness $\ssl=1/c_0^2$ and the combined coefficient $\nlc=\frac{\kappa}{c_0^2}$ we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples.