Probability density function for waves propagating in a straight rough wall tunnel

The radio channel places fundamental limitations on the performance of wireless communication systems in tunnels and caves. The field propagating in caves or tunnels is a complex-valued Gaussian random process, by the central limit theorem. We assume that the phase probability density function is uniformly distributed over a range of extent 2π. Under this assumption, the phase and amplitude of the field joint probability density function are independent and uncorrelated. We show that the probability density function for the amplitude of a random field propagating in a straight rough wall tunnel or cave is Rician. This tells us that there is a dominant signal component, such as a line-of-sight propagation path. In such a situation, random components arriving at different angles are superimposed on a stationary signal. At the output of an envelope detector, this has the effect of adding a DC component to the random multipath signal (Gibson, J.D., "The Mobile Communications Handbook", CRC Press, 1996). Since both expected value and standard deviation depend only on radial position, the probability density function for the amplitude of a random field propagating in a straight rough wall tunnel or cave is a radially dependent function.