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

Therefore at z=η(x, t),
η 1 (u 1t + u 1 u 1x )=(η 1 u 1 ) t + u 1 η t + 1 2 η xu 2 1 + 1 2
( )
η 1 u 2 1
.
x
From the kinematic condition Eq. (2.2)
u 1 η t + 1 2 η xu 2 1 = u 1w 1 − 1 2 η xu 2 1 ,
and by substitution,
η 1 (u 1t + u 1 u 1x )=(η 1 u 1 ) t + u 1 w 1 − 1 2 η xu 2 1 + 1 2
( )
η 1 u 2 1
. (2.7)
x
On the other hand, integration by parts and incompressibility give
∫ 1 ∫ 1
η 1 w 1 u 1z =−w 1 u 1 − w 1z u 1 dz=−w 1 u 1 + u 1x u 1 dz.
η
η
From Eq. (2.6),
η 1 w 1 u 1z =−w 1 u 1 + 1 2 η xu 2 1 + 1 2
( )
η 1 u 2 1
. (2.8)
x
Substituting Eqs. (2.7) and (2.8) in Eq. (2.3), the following mean-layer equation
is derived
(η 1 u 1 ) t +
( )
η 1 u 2 1
=−η 1 p 1 x. (2.9)
x
The incompressibility condition gives w 1 =η 1 u 1x at z=η(x, t). This, together
with Eq. (2.5) shows that
w 1 = u 1 η x + (η 1 u 1 ) x .
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