Summary: The dynamics of an infinite string on an elastic foundation subjected to a moving load is under investigation in this paper. The load is modelled by a moving concentrated force. Both analytical and numerical methods are used. Non-stationary problems are analyzed. In particular the wave process caused by the accelerating load passing through the sonic speed is investigated. It is shown tht the load at the moment when its speed is equal to the critical velocity gives rise to a wave front travelling at the sonic speed along the string. The asymptotical solution describing this front for large values of time is obtained. It allows the investigation of the qualitative properties of the solution. This solution and the results obtained in the numerical simulation carried out are in a good agreement.