text: Water waves of different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion. For the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation.
  The linear frequency dispersion characteristics for the above set A of equations are:
  with:
  * the phase speed, 
  * the wave number (, with  the wave length). 
  The relative error in the phase speed  for set A, as compared with linear theory for water waves, is less than 4% for a relative wave number . So, in engineering applications, set A is valid for wavelengths  larger than 4 times the water depth .
  The linear frequency dispersion characteristics of equation B are:
  The relative error in the phase speed for equation B is less than 4% for , equivalent to wave lengths  longer than 7 times the water depth , called fairly long waves. 
  For short waves with  equation B become physically meaningless, because there are no longer real-valued solutions of the phase speed. 
  The original set of two partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
  The shallow water equations have a relative error in the phase speed less than 4% for wave lengths  in excess of 13 times the water depth .