Oceanography

Dispersion Relation

Wave frequency ω  is related to wave number K by the dispersion relation

                             ω ² = g k tanh (kd)

Where d is the water depth and g is the acceleration of gravity

Two approximations are especially useful.

1. Deep-water approximation is valid if the water depth d is much greater than the  wave-length L. In this case, d d>>L, kd>>1,and tanh (kd)=1 

2. Shallow-water approximation is valid if the water depth is much less than a wavelength. In this case,

 d<<1,and tanh (kd) =kd

For these two limits of water depth compared with wavelength the dispersion relation reduces to:

?² = g k                 Deep-water dispersion relation   

d> L/4                                    

?² = g k ²              Shallow-water dispersion relation

 d  < L/11

The stated limits for d/L give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.