Study of backward shear waves in piezoelectric plates of crystal class 2mm by asymptotic expansion of the secular equations

It follows from the expansion of the secular equations near the points of the higher-mode origin that the dispersion relations for waves under study can be represented as a sum of two terms corresponding to two different mechanisms of the appearance of backward waves. One of them is proportional to the curvature of the slowness curve in the vertical direction. The second one is proportional to the negative displacement of the wave rays upon reflection. The competition of these two mechanisms explains and describes quantitatively the spectrum and the existence of the backward shear waves in the X-cut case, or their absence in the Y-cut case in plates of KNbO$_3$ crystals of 2mm class.