The propagation of dominant mode plane sound waves within a cylindrical tube having arbitrary cross section and extending indefinitely from an open end is studied theoretically. A quantity of special interest is the end correction, which characterizes the reflection coefficient at wave lengths large compared to the transverse tube dimensions. The reflection coefficient and end correction fit naturally into a boundary value formulation of the wave propagation problem, whose solution hinges on that of one or more integral equations. Since the integral equations do not in general admit of exact solution, practical techniques for approximately determining the physical quantities are important. Among these are the stationary or variational principles, which can be cast into a variety of independent forms, corresponding to the different nature of boundary distributions. Another technique is based on initial modification of an integral equation so as to utilize the possibility of explicit and rigorous solution. Approximate forms of the reflection coefficient and end correction are obtained in this manner, with the distinction of yielding exact results when the tube cross section is circular. A brief comparison of the different procedures is included.