  We consider a second-order nonlinear wave equation with a linear convolution
term. When the convolution operator is taken as the identity operator, our
equation reduces to the classical elasticity equation which can be written as a
$p$-system of first-order differential equations. We first establish the local
well-posedness of the Cauchy problem. We then investigate the behavior of
solutions to the Cauchy problem in the limit as the kernel function of the
convolution integral approaches to the Dirac delta function, that is, in the
vanishing dispersion limit. We consider two different types of the vanishing
dispersion limit behaviors for the convolution operator depending on the form
of the kernel function. In both cases, we show that the solutions converge
strongly to the corresponding solutions of the classical elasticity equation.
