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

Furthermore,
[√ ( √ )]
βφt x, βη, t =√ ( √ ) ( √ )
βφ tx x, βη, t +βφtz x, βη, t ηx ,
x
= √ ( √ )
βφ tx x, βη, t + O(β),
= √ βφ tx (x, 0, t)+O(β),
where a Taylor expansion about z=0 was performed.
Therefore,
P x =− ρ 2
ρ 1
(
ηx + √ βφ tx (x, 0, t) ) + O(β). (2.21)
As in the flat bottom case [8], from Eq. (2.21) it is clear that it is sufficient to
find the horizontal velocityφ x at z=0 in order to obtain P x at the interface. Due
to the presence of the small parameter √ β in problem (2.20),φ x (x, 0, t) can be
approximated by the horizontal velocity at z=0 that comes from the linearized
problem around z=0,
⎧
φ xx +φ zz = 0, on− h 2
L + h 2h(Lx/l)
L
≤ z≤0,
⎪⎨
φ z =η t ,
at z=0,
(2.22)
⎪⎩
− h 2
l h′ (Lx/l)φ x +φ z = 0, at z=− h 2
L + h 2h(Lx/l)
.
L
In this systematic reduction we use Taylor expansion to ensure that
φ z (x, 0, t)=φ z
(
x,
√
βη, t
)
+ O
(√
β
)
,
=η t + √ βη x φ x + O (√ β ) ,
20