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where square brackets denote the discontinuity jump at the interface. According to
Eq. (4), condition (8) provides the continuity of pressure p everywhere in the flow
domain. Hence, we take into account all the nonlinearities from the exact Euler equations (1). Finally, we reformulate boundary condition (8) on the basis of conservation
of the total horizontal momentum in a steady two-layer flow,
h 2
∫
−h 1
(p + 𝜌 u
2
) dy = const.
Excluding the pressure p from this relation under the Bernoulli equation (4) leads to
the integral relation
𝜌 1
𝜂(x)
∫
−h 1
e
−
N 2
1
𝜓
gu 1
[
(𝜓
2
y
− 𝜓
2
x
+ u
2
1
+ 2g
( 𝜓
u 1
− y
)
−
2g 2
N
2
1
(
e
N 2
1
𝜓
gu 1 − 1
)]
dy + (9)
+ 𝜌 2
h 2
∫
𝜂(x)
e
−
N 2
2
𝜓
gu 2
[
(𝜓
2
y − 𝜓
2
x + u
2
2 + 2g
( 𝜓
u 2
− y
)
−
2g
2
N
2
2
(
e
N 2
1
𝜓
gu 2 − 1
)]
dy = C,
where, the constant C depends on parameters of upstream flow as follows:
C = 2𝜌 1 g
[(
e
N 2
1
h 1
g
− 1
) (
u
2
1
N
2
1
+
g
2
N
4
1
)
−
gh 1
N
2
1
]
+
+ 2𝜌 2 g
[(
1 − e
−
N 2
2
h 2
g
) (
u
2
2
N
2
2
+
g
2
N
4
2
)
−
gh 2
N
2
2
]
.
Indeed, it is not obvious that combersome integral relation (9) is equivalent to boundary condition (8), which is rather simple. However, this equivalence can be checked
immediately by differentiating the relation (9) with respect to variable x, so the integrals become to be evaluated explicitly due to Eq. (7). We note in this context that
Eq. (9) is used instead of (8) and produces more effectively the model differential
equation for the function 𝜂(x) describing strongly nonlinear waves.
Non-dimensional Formulation
Now we introduce scaled independent variables x, y and scaled unknown functions 𝜂, 𝜓 in order to reformulate the Eqs. (5), (7) and (9) in a dimensionless form.