[en] Many particle accelerators and colliders throughout the world make use of superconducting magnets to focus highly relativistic beams. These magnets are cooled to ∼4.2 degree K For practical reasons, the beam pipes, encircled by the magnets, also operate at these cryogenic temperatures. This paper presents a theoretical model for determining pressure profiles, in space and time, stemming from either hydrogen or helium gas leak into the cold-bore tube with appendage pumps located at periodic intervals. It is shown that a wave-like pressure gradient propagates from the leak source at a rate which is dependent on the leak magnitude, gas species, speed and location of appendage pumps, and the geometry and effective roughness of the cold-bore tube. Steady-state, linear pressure gradients eventually equilibrate between the appendage pumps in a magnitude commensurate with both the adsorption isotherm of the species and mass flow in the beam pipe. Results are given for a variety of conditions relevant to the Relativistic Heavy Ion Collider being constructed at Brookhaven, and a general procedure, with expressions, is provided for the making of similar calculations in other installations