Summary: We study elastic wave propagation in cracked, functionally graded materials (FGM) with elastic parameters that are exponential functions of a single spatial co-ordinate. Conditions of plane strain are assumed to hold as the material is swept by time-harmonic, incident waves. The FGM has a fixed Poisson’s ratio of 0.25, while both shear modulus and density profiles vary proportionally to each other. More specifically, the shear modulus of the FGM is given as \(\mu (x)=\mu _{0}\) exp (\(2ax _{2}\)), where \(\mu _{0}\) is a reference value for isotropic and homogeneous material. The method of solution is the boundary integral equation method (BIEM), an essential component of which is the Green’s function for infinite inhomogeneous plane. This solution is derived here in closed form, along with its spatial derivatives and asymptotic form for small argument, using functional transformation methods. Finally, a non-hypersingular, traction-type BIEM is developed employing quadratic boundary elements, supplemented with special edge-type elements for handling crack tips. The proposed methodology is first validated against benchmark problems, and then is used to study wave scattering around a crack in an infinitely extending FGM under incident, time-harmonic pressure and vertically polarized shear waves. The parametric study demonstrates that both far-field displacements and near-field stress intensity factors at the crack-tip are sensitive to this type of inhomogeneity, as gauged against results obtained for the reference homogeneous material.