In this paper the author generalizes results obtained by V. H. Weston [J. Math. Phys. 13, 1952–1956 (1972; Zbl 0253.35054)] concerning the inverse problem for a particular type of hyperbolic partial differential equation. In particular the author considers the equation
\[ u_{xx} -u_{tt} + A(x)u_x +B(x)u_t +C(x)u = 0, \quad -\infty <x< \infty,\;-\infty<t<\infty, \]
where a) support \(A,B,C\subseteq [0,\ell]\), b) \(A,B\in C'(0,\ell)\) with jump discontinuities at \(x=0\), \(x= \ell\), c) \(C\in C^0[0,\ell]\) with jump discontinuities at \(x=0\), \(x=\ell\), d) \(u(x,t)\) is everywhere continuous and piecewise \(C^2\), e) there is a jump discontinuity in \(u_x\) across \(x=0\), \(x=\ell\). The coefficients \(A,B,C\) are unknown on \((0,\ell)\) and the author presents a method (based on the construction of a weak Riemann function and the generalization of the Gelfand-Levitan approach) for constructing \(B(x)\) and \(C(x)-A'(x)/2-A^2(x)/4\) given a certain set of scattering data. The problem is motivated by an example taken from electromagnetic scattering theory.