This paper studies the interactions of two plane waves in elastic solids with quadratic nonlinearity. In particular, the necessary and sufficient conditions for resonant mixing of two plane waves are derived. It is shown that the conventional resonance condition for resonant mixing of plane waves is only a necessary condition, not sufficient. Based on the newly derived necessary and sufficient conditions, resonant mixing of various types of plane waves are investigated and specific conditions for generating a resonant mixed wave are obtained for each case. These results are useful for developing nonlinear ultrasonic nondestructive evaluation techniques using the wave mixing method.