Interfacial wave propagation parallel to a dielectric interface has been studied by considering an electric current line source present at the interface. The first order asymptotic evaluation of field components shows a null of the electric field at the interface. An amplitude null represents an unstable structure in the phase map and a phase front discontinuity across the interface. Higher order asymptotic evaluation has been employed to gain further insight into this propagation problem. The results show that the wavefronts need not be discontinuous. The continuity of the phase fronts is preserved with the help of interesting and stable structures such as saddle points and center points in the phase map of the electric field in both half spaces.