An extension of the Airy theory for linear waves into shallow water

As a fully developed (Airy) wave propagates from deep into shallow water, its crest becomes more peaked while the trough flattens out. The median crest diameter MCD, defined as the distance between the wave flanks under the crest at a level halfway between the crest and trough, therefore decreases relative to the similarly defined median trough diameter MTD, which remains constant up to the breaking point. The MCD is directly related to other wave characteristics, which enables water particle velocities to be calculated for any water depth without having to recur to more complex, higher-order Stokes, cnoidal or Fenton theories. Over a nearly horizontal bottom, most fully developed wave characteristics can be expressed as functions of the wave period T_w. It is shown that the horizontal particle velocity at the bottom under the breaker crest is at least 9 times faster than under the breaker trough, which explains why sediment is transported landward under fair weather conditions. The proposed equations also shed new light on the formation of spilling, plunging and surging/collapsing breakers.