Abstract
The equation of motion for a one-dimensional inhomogeneous medium is derived using the Lagrangian description of a continuous medium. The equations of motion are determined from the Euler–Lagrange equations derived from the Lagrangian density. A general form of the solution is obtained and the specific solution for the special case of a medium whose elastic stiffness changes linearly with distance in the propagation direction is determined. The displacement field is calculated for this case showing the effects of the inhomogeneity on shape and signal velocity of elastic pulses.
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