A method is presented for manipulating ultrasonic data from a complete longitudinal wave scan of a body subject to a plane state of residual stress in such a way that the complete state of stress can be evaluated at each scan point. The basic theory is presented and discussed and a numerical implementation of the pertinent equations is demonstrated for an ideal stress state similar to that found in a quenched rod. The assumed input is corrupted by noise to examine the robustness of the algorithm in the presence of such noise. Prospects are discussed for applying the technique to evaluate actual residual stresses in light of several potential difficulties.