Theory of the Propagation of Finite Amplitude Ultrasonic Waves in Pure Mode Directions in Hexagonal and Trigonal Crystals

摘要

The nonlinear theory describing the propagation of finite amplitude ultrasonic waves in pure mode directions in hexagonal and trigonal crystals is developed. By evaluating the coefficient of the term which couples the longitudinal wave of finite amplitude to the transverse modes, one finds the pure mode directions by requiring this coefficient to vanish. The result is that for hexagonal crystals the directions along the symmetry axis and in the basal plane are pure longitudinal mode directions. Pure mode directions are also found for longitudinal waves propagating tangential to a cone whose apex is centered about the symmetry axis and whose apex angle is a function of the second-order elastic constants of the sample under consideration. The third-order elastic constants which determine the magnitude of the second harmonic of an initially sinusoidal ultrasonic wave are evaluated for these additional directions. For finite amplitude longitudinal ultrasonic waves in trigonal crystals we have specialized to nonpiezoelectric crystals and have evaluated the third-order elastic constants which determine the magnitude of the second harmonic of an initially sinusoidal wave propagating in the following pure mode directions: (1) along the symmetry axis (the c-direction); (2) along the a-direction in the basal plane; (3) along the direction in the basal plane that makes an angle of 60 deg with the a-direction; and (4) along a direction in the b-c plane whose angle with respect to the b-axis is a function of the second-order elastic constants of the crystal under consideration.