Wave front propagation is a subject that has been widely studied and has several applications ranging from combustion to signal processing in fiber optics and to mathematical biology. We will discuss wave front propagation in narrow domains D(µ) of width µ<<1. In addition we will examine how the motion of the interface (wave front) depends on the behavior of the cross-sections of the domain D(1). In particular cases, a jump of the wave front occurs when the surface area to volume ratio of the cross-sections increases rapidly. This implies that the jumps of the wave front occur at places where the tube D(1) becomes thinner, i.e. when the volume of the cross-sections decreases significantly, at least when the tube D(1) retains its shape as x increases. This is a very interesting problem since one can predict jumps of the wave fronts in narrow tubes. This problem is important in applications in thin waveguides (e.g. thin tubes, quantum wires, photonic crystals, fiber optics, etc.). The mathematical formulation of the problem goes as follows: We consider second initial boundary problem in the narrow domain D(µ) of width µ<<1 for linear second order differential equations with nonlinear boundary conditions. Using probabilistic methods, we showed that the solution of this problem converges to the solution of a standard reaction-diffusion equation in a domain of reduced dimension, as µý0. This reduction allows obtaining the aforementioned results regarding wave front propagation.