98
V. Liapidevskii and N. Gavrilov
In view of (15) we have
P(𝜂, 𝜁) =
̄
Q
2
2𝜂 2 −
(Q
+
)
2
2𝜁 2 + ̄
b(H − 𝜁 ) = ̄
J − J
+
= ̄
b(h 0 + 𝜂 0 )
or
𝜂
2
=
̄
Q
2
2( ̄
J − J + ) + (Q + ) 2 ∕𝜁 2 − 2 ̄
b(H − 𝜁 )
.
(16)
It follows from (16)
𝜂 = 𝜂(𝜁 ), h = h(𝜁 ) = H − 𝜁 − 𝜂(𝜁 ).
(17)
We may find dependencies 𝜁 = 𝜁 1 (h), 𝜂 = 𝜂 1 (h) from (17) and rewrite (15) in the
form
hh
′′
−
1
2
(h
′
)
2
=
3
u 2 (J
−
−
1
2
u
2
− bh − ̄
b𝜂 +
1
2
w
2
− J
+
) = 𝛷(h).
(18)
Finally, (18) reduces to ODE
(h
−1∕2 h
′
)
′
= h
−3∕2
𝛷(h)
or
(h
′
)
2
= 2h𝛹 (h)
(19)
with
𝛹 (h) =
h
∫
h 0
𝛷(s)
s 2 ds.
(20)
The wave profile h = h(x) is calculated from (19), (20). Other unknown variables
may be found from (17)–(19). For given dimensionless parameters ̄
b∕b, h 0 ∕H, 𝜂 0 ∕H
solitary waves represent the one-parameter family depending on the Froude number
Fr =
u 0
√
bH
.
(21)
Remark 5 For ULM (𝛽
−
= 0, 𝛽
+
= 1) describing the bottom waves of elevation, the
one-parameter family of solitary waves may be constructed in a similar way or just
by “inversion” solutions of BLM relative to the midline of the channel.
Nonsymmetric Solitary Waves
In contrast to the models ULM, BLM, SM considered above, the solitary waves for
BM can be found only for special values of the dimensionless parameters ̄
b∕b, h 0 ∕H,