Linear crystal acoustics and piezoelectricity of incommensurate dielectrics

We present a linear theory of the long-wavelength, low-frequency vibrations of incommensurate dielectrics which allow pyroelectricity or piezoelectricity or both. An important consideration in our theory is the rotational invariance of the Lagrangian density. We find the general equations of motion for the coupled sound-wave and phason displacement fields in the quasistatic approximation. We then display an asymmetry, in the low-frequency limit, of the attenuation constants between transverse sound waves under interchange of their polarization and propagation directions. As an example of this asymmetry we examine the incommensurate phase of ${\mathrm{BaMnF}}_4$.