As a critical theoretical advance, Understanding Plane Waves discusses the acoustic Goos-HAnchen effect. The important applications of this effect are discussed, including plane wave propagating inside transversely isotropic elastic-solids, reflection/refraction at interface between two anisotropic rocks, and acoustical applications to petroleum logging and seismic exploration. Next, the authors explore a newly developed acoustic-measurement system with emphasis on measurement process and recent improvements that make an acoustic-measurement more accurate. Three approaches which are used to analyze the evolution of the plane longitudinal and transverse waves that are propagated in a nonlinear hyperelastic medium are discussed: the method of successive approximations, the method of slowly varying amplitudes and the method of restriction on the displacement gradient. Lastly, the subject of relativistic quantum particles interacting with classical plane wave fields is examined from the standpoint of space-time symmetries which have been found to be encoded in the solutions of relativistic equations.