Find the separation of variables solution for the longitudinal displacement, \(u(x, t)\), of the equation derived in Problem 8.21. Leave the answer in terms of constants, which must be determined from the boundary conditions and the initial conditions.

Problem 8.21

Longitudinal vibration of an isotropic, particle-reinforced composite bar may be modeled by using the 1D wave equation (Equation 8.99) if the material is linear elastic. Derive the equation of motion for longitudinal vibration of the bar if it can be assumed to be a Kelvin-Voigt linear viscoelastic material having the stress-strain relationship given by Equation 8.39.

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dx (8.99) t