Wave Propagation in a Random Lattice. I

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Abstract
The small amplitude periodic classical motion of a lattice of particles about their equilibrium positions in a lattice is considered. The effect of random masses and random spring constants upon the coherent or mean motion is treated by using an equation for the coherent motion derived previously by Keller and others. From this equation the dispersion equation for coherent wave motion is determined. It is solved for the case in which the spring constants are not random but the masses are random. Explicit results are obtained in the one-dimensional case for both uncorrelated and exponentially correlated mass defects. They show an alteration of frequency or of wavelength and of phase velocity, as well as an attenuation due to scattering by the defects. In addition new highly attenuated modes are found. These results are utilized in Part II in which various reflection and Green's function problems are treated.