For an experience surfer they generally ride along the curving/Bowling face of the wave (known as the wall) from which they can perform numerous manoeuvres, while occasionally riding the bottom of the wave to pass a unmake-able section in the tube and even onto of the lip of the wave performing a 'floater'.
For beginner surfers they usually ride the 'white-wash' (the broken wave) where it is easy to catch and normally longer that a unbroken wave breaking.
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A beginner surfer usually rides the white water of the wave, after it has broken. A more experienced surfer would get on the wave just before it breaks, riding it sideways through the tube that the breaking wave creates.