Chemistry Reference
In-Depth Information
f
¦
M
E
(
r
/
r
e
kz
i
Z
t
C
(
x
)
J
(
k
r
r
)
cos
n
-
(
r
r
)
(10)
1
n
1
n
1
1
n
0
where -
(r-r
1
)
is the angle between the vector
r-r
1
and the
x
axis.
The scattering amplitude can be defined by the following equations
e
ikr
1
i
~
f
¦
T
(
0
)
o
f
(
-
)
;
f
(
-
)
H
A
cos
n
-
(11)
n
n
r
o
f
r
S
k
n
0
H
~
.
Under the conditions of scattering the phase velocity of capillary waves
will deviate from the value for the medium without scatterers. Therefore, it
is natural to introduce an effective wave number K for the former case and
to choose the following trial functions for the coefficients
C
n
n
where H
0
=1,
H
n
=2
at
n=1,2,...
;
AAi
A
2
n
n
2
n
C
(
1
x
)
v
e
iKx
(12)
1
n
Eqs. (9)-(12) allow us to obtain the following expression for the com-
plex effective wave number after rather cumbersome transformations
(Noskov 1998)
S
S
K
2
[
k
(
i
)
n
f
(
0
)]
2
[(
1
i
)
n
f
(
S
)]
2
(13)
0
0
k
k
where
f(0)
and
f(
S
)
are the forward and the backscattered amplitudes
1
i
f
~
1
i
f
~
¦
¦
j
f
(
0
)
H
A
;
f
(
S
)
(
1
H
A
(14)
j
2
j
j
2
j
S
k
S
k
j
0
j
0
Relation (13) is the main result. It allows us to determine the effective
wave number at the propagation of surface waves along the interface with
two-dimensional scatterers on the basis of the scattering characteristics on
a single particle. As it could be expected, the difference from the three-
dimensional case only consists in the form of coefficients
f(0)
and
f(ʌ)
and
also in the form of the expressions for the scattered amplitudes in them-
selves.
If the condition of weak scattering is satisfied and scatterers are statisti-
cally independent, Eq. (13) holds for two-dimensional particles of all sizes.
The condition of “two-dimensionality” of scatterers is not too rigid. It is
only important that the scattered wave approximately satisfied the Eq. (8)
and it was possible to define the scattered amplitude. Probably a rather
large group of particles meets these conditions, for example, three-dimen-
