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Consider the socalled shallow water limit,

(1141) 
in which the depth, , of the water is much less than the wavelength,
, of the wave.
In this limit, the gravity wave dispersion relation (1127) reduces to

(1142) 
since
as
. It follows that the phase velocities and group velocities of gravity waves in shallow
water
all take the fixed value

(1143) 
irrespective of wave number.
We conclude thatunlike deep water wavesshallow water gravity waves are nondispersive in nature: i.e.,
waves pulses and plane waves all propagate at the same speed. Note, also, that the velocity (1143) increases with increasing water depth.
For a plane wave of wave number
, in the limit ,
Equation (1125) yields

(1144) 
Hence, Equations (1113) and (1133) give [cf., Equations (1151)(1154)]



(1145) 



(1146) 



(1147) 



(1148) 
Here, is again the amplitude of the vertical oscillation at the water's surface.
According to
the above expressions, the passage of a shallow water gravity wave causes a water particle located a depth below the surface to execute an
elliptical orbit, of horizontal radius , and vertical radius , about its equilibrium position. Note that the orbit is
greatly elongated in the horizontal direction. Furthermore, its vertical radius decreases linearly with increasing depth such that
it becomes zero at the bottom (i.e., at ). As before, whenever the particle's vertical displacement attains a maximum value the particle is
moving horizontally in the same direction as the wave, and vice versa.
Next: Energy of Gravity Waves
Up: Waves in Incompressible Fluids
Previous: Gravity Waves in Deep
Richard Fitzpatrick
20120427