Multi-Scaled Short-Wave Asymptotic Solution of the Cauchy Problem to One-Dimensional Wave Equation with Smoothed Jump of the Velocity

The paper studies a wave equation whose velocity has a perturbation localized at a point $$x_0$$ . The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable to the scale of the inhomogeneity. Specifically, the length of the initial wave is of the order of $$\varepsilon$$ and the width of the localized inhomogeneity is of the order of $$\varepsilon^{\alpha}$$ , where $$\varepsilon$$ is a small parameter tending to $$0$$ and $$\alpha$$ is any positive number. The cases $$\alpha<1$$ and $$\alpha>1$$ are considered separately.