A method and apparatus for nondestructive evaluation (NDE) of structures and materials using a highly nonlinear medium for the generation and detection of one or multiple highly nonlinear pulses (or highly nonlinear waves) impinging on a material or structure. The apparatus includes pulse exciters that induce the propagation of highly nonlinear, weakly nonlinear or linear stress waves in the material, system, or structure to be inspected and/or detectors for the observation and the detection of the output waves from the material/structure being tested. The NDE method includes the use of the tunable highly nonlinear apparatus as impulse exciter alone, or in combination with an accelerometer or a nonlinear sensor to detect the outgoing pulse.