a reduced model for internal waves interacting with submarine ...

and substituting in the asymptotic expansion Eq. (3.3) for p 1 z we have
p 1 z(x, z, t)=−1+β(z−1)G 1 (x, t)+O(β 2 ).
Integrating from z=η(x, t) to z≤1 we obtain
( )
(z−1)
2
p 1 (x, z, t)=P(x, t)−(z−η)+βG 1 (x, t) − (η−1)2 + O(β 2 ).
2 2
Differentiating once in x,
( ( ))
(z−1)
2
p 1 x=η x + P x (x, t)+β G 1 (x, t) − (η−1)2 + O(β 2 )
2 2
x
and taking means
p 1 x=η x + P x (x, t)− β ( ) 1
η 1 3 η3 1 G 1(x, t) + O(β 2 ).
x
Substituting in Eq. (3.2) we have
(
u 1t + u 1· u 1x =− η x + P x (x, t)− β ( ))
1
η 1 3 η3 1 G 1(x, t) + O(β 2 ). (3.8)
x
If the lower fluid layer is neglected and P is regarded as the external pressure
applied to the free surface, Eqs. (3.1) and (3.8) are the complete set of evolution
equations for one homogeneous layer derived by Su and Gardner in [27] and
independently by Green and Naghdi in [11].
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