So what is going on inside the bore?
When a normal beach wave breaks, water rushes up the beach as a foaming mass. Waves 'break' because in water that is shallow relative to distance between waves, their speed is related to the depth to the bed. This means that in very shallow water the wave tops (crests) are moving much faster than the troughs between them**. The crest overtakes the trough infront and it crashes down. Waves are more like patterns superimposed on water than actual water movements. The water in them usually moves back and forth less than the distance between waves. However, when a wave breaks, there is enough momentum in the crest water for it to rush up the beach as a turbulent mass.
River bores form by similar processes1,2. Because they have more water in them, they look more sedate, but the turbulence below the surface allows the river water to interfinger with the tide water in complex structures. These structures also allow them to swallow up the water surface irregularities that would normally mess up waves, and they can therefore run huge distances up river**.