Geoscience Reference
In-Depth Information
Like in the linear problem, we shall consider a perturbation of the atmospheric
pressure (deviation from a certain standard value), which propagates with a constant
velocity V in the positive direction of axis 0 x ,
p atm ( x , t )= p ( x
Vt ) .
(4.29)
We shall assume the response of the water column to represent an induced wave,
travelling with the velocity V in the same direction,
u ( x , t )= u ( x
Vt ) ,
(4.30)
ξ
( x , t )=
ξ
( x
Vt ) .
(4.31)
Successive differentiation with respect to time and integration over space of func-
tions with arguments of the form ( x
Vt ) is similar to multiplication by the
quantity
V ,
f ( x
Vt ) d x =
Vf ( x
Vt ) .
t
Making use of this fact, we pass from differential equations (4.27) and (4.28) to
the algebraic relations [Pelinovsky et al. (2001)]
Vu + u 2
2
+
ξ
=
p atm ,
(4.32)
V
ξ
+(1 +
ξ
) u = 0 .
(4.33)
Generally speaking, expressions (4.32) and (4.33) are correct with an accuracy
up to certain integration constants. We have chosen the values of these constants so
as to have zero flow velocities and zero free-surface displacements u = 0,
ξ
= 0,
respectively, to correspond to zero perturbation of the atmospheric pressure.
Excluding the quantity u from the set of equations (4.32) and (4.33), we ob-
tain the relationship between the moving perturbation of atmospheric pressure and
the free water surface response, corresponding to it,
p atm = V 2 ξ
1 +
) 2
2
ξ
ξ
ξ
.
(4.34)
2 (1 +
ξ
When
1, the problem considered reduces to the linear problem. In this case
formula (4.34) can be written in the form
ξ
p atm
V 2
ξ
=
1 ,
(4.35)
which fully corresponds to the first term in the analytical solution of the linear prob-
lem (4.23).
The free-surface displacement
can also be expressed explicitly in the non-
linear case via the perturbation of pressure p atm . Equation (4.34) has three solu-
tions, and some of them have no physical sense in the case of certain values of
quantities p atm and V . Moreover, the form of the solutions is determined by quite
ξ
 
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