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The coupled equations governing whistler waves propagating along a duct with cylindrical cross section are derived and then solved numerically. These equations are expressed in terms of magnetic and current flux functions and show that it is possible to have a solution where the waves are finite in the duct and decay exponentially outside the duct. This solution has the property of having zero radial Poynting flux everywhere, so, as required for whistler waves to bounce back and forth losslessly between magnetically conjugate terrestrial hemispheres, no wave power leaks from the duct. The coupled equations are solved numerically for a tangible realistic situation by dividing the radial domain into an inner and an outer region, where the interface between these regions is at a mode conversion location, where fast and slow modes inside the duct merge and effectively reflect. The result of this effective reflection is that there are fast and slow standing waves in the duct. In the region external to the duct, the wave solutions are also a form of standing waves, but with a strong exponential decay and a radial wavelength that is intermediate between that of the fast and slow waves in the duct. The numerical solution is shown to be in good quantitative agreement with estimates made from analytic models. Detailed examination of the solutions in the vicinity of the mode conversion location shows that the classic plane wave assumption fails to describe the true nature of the modes.