8W ^ 

 p __1 = -pa_^aj (1 + ^) cos(k^x -wt) , (A-49) 



o 



which integrated over depth gives: 



z 8W ^ 



p — ^ dz = - 4 P a w h CI + €-3 cos(k X -wt) . (A-50) 

 _j^ 8t 2 + h^ o 



o 



From the derivation of the pressure distribution (eq. A-13), the 

 term expressing the presence of the wave is 



p = p ga cos (k X -tot) , (A-51) 



and to justify the neglect of the term given by equation (A-50) we 

 must have that 



1 2 



^ p a U)''h << p ga , (A-52) 



2 + o f" 6 + ' 



which with the aid of equation (A-33) may be stated as 



^ (k h )^ << 1 , (A-53) 



2 o o 



i.e., a requirement of long waves as previously stated. 



Now, in the process of linearizing the governing equations it was 

 mentioned that this linearization was justified if 



N.I a^ 



« 1 , ' (A-54) 



h h 

 o o 



because the terms then omitted would be smaller than the terms retained 

 by the factor a /h. This in turn would be a consistent procedure only 

 if the pressure term given by equation (A-50) is greater than (a /h) 

 times the leading term given by equation (A-51), i.e., linearization 

 of the governing equations and taking the pressure distribution to be 

 hydrostatic is consistent only if 



1 2 ^+ 



2 p a^o) h^ » p ga^ j^ , (A-55) 



o 



114 



