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

expression above as
Let us express the quantities w (0)
1 z
expanding the incompressibility equation to obtain
p (1)
1 z =−D 1w (0)
1
− w (0)
1 z w(0) 1 . (3.4)
and w(0)
1
in terms ofηand u 1 . We begin by
w (0)
1 z =−u(0) 1
(x, t). (3.5)
x
Integrating Eq. (3.5) fromηto z≤1 and taking into account the z-independence
expressed by Eq. (2.12) we have that
w (0)
1
(x, z, t)=−u(0)
1
(x, t)(z−η)+w(0) ∣
x 1 z=η(x,t)
.
Now, from the kinematic condition in Eq. (2.2), to leading order
w (0)
1
∣ z=η(x,t)
=η t + u (0)
1 η x
so
that is,
w (0)
1
(x, z, t)=−u(0)
1 (x, t)(z−η)+η x t+ u (0)
1 η x,
w (0)
1
(x, z, t)=−u(0)
1 (x, t)(z−η)+ D x 1η. (3.6)
Substituting Eqs. (3.6) and (3.5) in Eq. (3.4):
p (1)
1 z = (z−η) (
D 1 (u (0)
1 x )−u(0)
2
1 x
)
− D 2 1 (η).
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