Electron transmission and wave functions through junctions with a pair of a pentagonal defect and a heptagonal defect connecting two metallic carbon nanotubes are analyzed by the analytical calculation with the effective-mass equation. The energy region $|E|<{E}_{c}$ is considered where the channel number is kept to two. Close relation between the transmission rate and the wave function is found; the transmission rate is given by the inverse squared absolute value of the wave function. The dependence of the transmission rates on the energy and on the size of the junction is clearly explained by the nature of the wave function. Though the wave function and the transmission rate calculated by the tight-binding model agree well with the corresponding analytical results by the effective-mass approximation, the discrepancy becomes considerable when $|E|\ensuremath{\simeq}{E}_{c}.$ To study the origin of this discrepancy, an efficient numerical calculation method is developed with a generalized transfer matrix for the tight-binding model. Their numerical results are compared with the corresponding analytical ones and the results show that the origin of the discrepancy comes from the evanescent waves with the longest decay length in the tube parts.