## Exact solutions for reflection coefficients, in 1D and 2D

### Heather K. Hardeman-Vooys, Michael P. Lamoureux

In this paper, we solve the 1D and 2D elastic wave equation using two different velocity fields: a velocity jump and a velocity ramp. We require the density and modulus satisfy the relation established in (Lamoureux et al., 2012) and (Lamoureux et al., 2013) using some parameter . We find the reﬂection coefﬁcient for the 1D case of a velocity jump given general . Extending these velocities to the two dimensions, we compute the analytic solutions to the 2D elastic wave equation and ﬁnd the reﬂection coefficients for a plane wave hitting the jump and ramp at normal incidence. Finally, we conclude with discussion of the case where the plane wave hits the transition zone of the 2D velocity ramp at nonnormal incidence given varying density. The motivation for this paper is to extend the work of the authors in (Lamoureux et al., 2012) and (Lamoureux et al., 2013) and to demonstrate explicit reﬂection coefficients in a continuously varying velocity ﬁeld.