The trajectory of a particle moving passively with an incompressible flow is, related to the streamfunction of that flow, such that the particle motion may be interpreted as a Hamiltonian system with the streamfunction as Hamiltonian. This is applied to linear Rossby waves superimposed on a shear flow in a two-dimensional incompressible and inviscid rotating shallow water system in \(\beta\)-plane approximation. The Rayleigh-Kuo criterion is invoked to avoid critical layer problems. The resulting Hamiltonian system is studied for chaotic motion, especially barriers to transport and conditions for their destruction. This is done numerically using Poincaré maps, and analytically. Results are compared with those of laboratory experiments.