First, a new method for numerical inversion of the Laplace transform is proposed. The essential point of this method is to approximate the exponential function in the Bromwich integral by the function Eec(s, a) d= exp (a)/2 cosh (a – s). With this we can get an infinite series which gives a good approximation to the inverse transform. In practice, we accelerate the convergence of the series by the Euler transformation. Thus in ordinary cases, only twenty to thirty terms give satisfactory results, and even a $100 pocket calculator can solve practical problems which are not so easy by the usual method. Second, the wave propagation in dispersive media is studied by using the above method. This problem has been investigated analytically by many authors. The problem is so difficult that there has been no detailed solution available other than asymptotic expressions. Some detailed numerical results by our method are given for the transient phenomena in a waveguide and the medium discussed by Sommerfeld and Brillouin. Our numerical results show that Brillouin's results should not be taken quantitatively except the first forerunner whose approximate variation is predicted by Sommerfeld correctly.