Inverse problems for the wave equation with an impulse source of unknown form (Q957771)

The paper deals with an inverse problem more general then the the inverse dynamic problem of acoustics (see, e.g., [\textit{V. G. Romanov}, Stability in Inverse Problems. Moscow: Nauchnyĭ\ Mir. (2005; Zbl 1171.35125)]). Let \(u(x,y,t)\) be a function in the domain \(G=\mathbb R^2_+\times \mathbb R\), where \(\mathbb R^2_+= \{ (x,y) \in \mathbb R^2\mid y>0 \}\). The differential equation \[ u_{tt}-(c^2 u_x)_x-(c^2 u_y)_y=0, \quad (x,y,t) \in G, \] with the positive coefficient \(c(y) \in C^2(\mathbb R_+)\) is considered. In addition, \(u(x,y,t)\) satisfies the initial and boundary conditions \[ u|_{t<0}\equiv 0, \quad (cu_y)_{y=0}=f(t) \delta(x); \quad f(t)\equiv 0, t<0. \] The displacement of medium points measured on the boundary of the domain is given: \[ u\mid_{y=0}=F(x,t), \quad (x,t) \in \mathbb R^2_+. \] The inverse problem under study is to determine two functions \(c(y)\) and \(f(t)\) from the given data \(F(x,t)\). It is clear that one cannot do it without some natural a priori assumptions on the structure of \(f(t)\). So, in the present paper it is assumed that \(f(t)\) has the structure \(f(t)=\alpha \delta(t)+\widehat{f}(t) \theta(t)\), where \(\theta(t)\) is the Heaviside function and \(\widehat{f}(t) \in C^1[0,T]\) for \(T>0\). It is also assumed that \(c(y)\) is known in a sufficiently thin layer \(y \in [0, y_0]\). Under the assumptions made above, it is proved that, given \(F(x,t)\) on the set \((x,t) \in \mathbb R \times [0,T]\) with \(T>0\), one can uniquely determine the function \(f(t)\) for all \(t \leq T\) and the velocity of wave propagation \(c(y)\) on some finite interval \([y_0, y_1]\). A stability estimate for the solution to this problem is obtained.