  In contrast to the traditional elastodynamic equations, a more comprehensive
formulation of one dimensional (1D) elastodynamic equations is given for
inhomogeneous media by using the coordinate transformation method. These
modified equations consider the gradient of pre-stresses so that they are
form-invariant and can retain time synchronization under spatial coordinate
transformation, which comply with the principle of general invariance. A
numerical example is conducted to compare the distributions of wave speeds
calculated by the modified equations and the traditional equations. It
demonstrates that the traditional equations are good approximations of the
modified equations only when the wave frequency is sufficiently high.
