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In contrast to the traditional elastodynamic equations, a more comprehensiveformulation of one dimensional (1D) elastodynamic equations is given forinhomogeneous media by using the coordinate transformation method. Thesemodified equations consider the gradient of pre-stresses so that they areform-invariant and can retain time synchronization under spatial coordinatetransformation, which comply with the principle of general invariance. Anumerical example is conducted to compare the distributions of wave speedscalculated by the modified equations and the traditional equations. Itdemonstrates that the traditional equations are good approximations of themodified equations only when the wave frequency is sufficiently high.

Modified 1D elastic wave equations that retain time synchronization under spatial coordinate transformations