Inverse problem for an integro-differential equation of acoustics (Q1649472)

The authors consider the following initial value problem \[ \frac{1}{c^2(z)}v_{tt}=\rho_z-\frac{\partial \ln \rho(z)}{\partial z}\rho,~~ z>0, ~t>0, \] \[ v|_{t\leq 0}\equiv 0,~\rho(+0,t)=\delta'(t), \] where \[ \rho(z,t)=v_z(z,t)+\int_0^t k(t-\tau)v_z(z,\tau) d\tau. \] They study the inverse problem, which consists in determining the one-dimensional kernel of the integral term from the known solution of the direct problem. This problem reduces to solving a system of integral equations in unknown functions.