The method of characteristics involves some fairly complex mathematics which may not be accessible to the non-mathematician. It is important to know that wave intensity analysis is founded on rigorous mathematics and so the details of the theory are given here {mathematical details}.
If you want to skip the mathematical development, the results are:
Any perturbations introduced into an artery will propagate as a wave with the speed U + c in the forward direction and speed U - c in the backward direction. U is the velocity of the blood and c is the wave speed which depends on the elastic properties of the artery. The wave speed is very important in wave intensity analysis and will be discussed next.