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Abstract Parametric amplification of an elastic wave and a framework for using elastic waves that could enable a new generation of high performance, low noise acoustic amplifiers, mixers and circulators are presented. Using a novel approach with nonlinear materials produces highly desirable non-reciprocal characteristics. Parametric amplification of a weak elastic signal wave is achieved by an elastic pump wave of higher intensity. By careful selection of material orientation together with precise excitation of signal and pump waves, ‘up frequency conversion’ is suppressed and selective amplification of the elastic signal wave occurs at its original frequency. In addition, a general mathematical framework is developed and used for analytical studies of coupled wave equations in nonlinear anisotropic materials. The results obtained from the analytical studies are verified using a finite element implementation.