This paper deals with the effects of traffic bottlenecks using an extended Lighthill-Whitham-Richards (LWR) model.;The solution structure is  analytically indicated by the study of the Riemann problem characterized by a discontinuous flux. This leads to a typical solution describing a queue upstream of the bottleneck and its width and height, and informs the design of a $\delta$-mapping algorithm. More significantly, it is found that the kinetic model is able to reproduce stop-and-go waves for a triangular fundamental diagram. Some simulation examples, which are in agreement with the analytical solutions, are given to support these conclusions.