Sources and related components to support high power electromagnetic (HPEM) ultra-wideband systems are of growing importance. The ultra-wideband domain is characterized by a bandwidth ratio, br=(f high/flow)>10 with a nominal center frequency in the GHz range. Shielded microstrips and coplanar waveguides may find increasing application for developing advanced ultra-wideband systems. Dielectric covered ground planes are often part of microstrip or coplanar waveguides and properties of their propagation modes need to be better understood. Over the years, several different techniques and theories have been used to analyze the propagation modes of dielectrics covered planes. In this paper, expressions for the transverse wave numbers within the dielectric and above it are derived for two types of modes, denoted as transverse magnetic and transverse electric (TM and TE). In contrast to earlier work our fields are transverse to the direction normal to the dielectric interface, not to the direction of propagation. In addition to modes with real transverse wave numbers, modes with complex transverse wave numbers are shown to occur in conjugate pairs whose members can be combined into patterns with sinusoidal variation in the direction normal to the surface of the dielectric. Some of the results are presented in tables in which the propagation constants, computed from the wave equation, are also given