A wave function expansion method for out-of-plane scattering of SH waves by two symmetrical circular cavities in two bonded exponentially graded half spaces is presented by using the plane elastic complex variable theory. The medium is composed of a semi-infinite homogeneous space with a circular cavity and a bonded exponentially graded half-space with a symmetrical circular cavity along the boundary interface. In terms of Helmholtz decomposition to the wave propagation equation, the stresses and displacements of the medium are expressed by the displacement potentials. The scattered waves by the boundary interface are assumed to transmit from the images of the origins of the respective two cavities. A conformal mapping function is introduced to convert the physical plane of two bonded semi-infinite spaces into two jointed concentric annular regions. The boundary value problem is formulated as a series of infinite algebraic equations. The convergence of the present solution is examined by investigating the variations of the solution results with the truncation of the series number. Parametric study shows that the interface displacements are almost independent to the inhomogeneous coefficient βR of the right medium. The amplification of the interface displacement becomes great for high incident frequency and with the decreasing of the distance-to-radius ratio h/R. The distribution of stress concentration of the right cavity depends significantly on the inhomogeneous coefficient βR, the distance-radius ratio h/R, the incident angle and frequency of the excitations.