24
A. M. Antonov et al.
dimensionless form: both values are related to the square of the shear wave velocity
c
2
2 . Surface wave speed calculated for Poisson’s coefficient ν = 0.5. From the above
graph, it can be concluded that the velocity of the surface wave cannot exceed the
phase velocity of the shear wave, but can reach it at certain values of the wave number
α.
From (2.14) and (2.15), it follows that displacements u and v can be written in
the form:
u = −i Ake
ζ y+i(ωt−kx)
+ B 1 η 1 e
η 1 y+i(ωt−kx)
+ B 2 η 2 e
η 2 y+i(ωt−kx)
=
−i Ake
ζ y
+ B 1 η 1 e
η 1 y
+ B 2 η 2 e
η 2 y
e
i(ωt−kx)
,
v = Aζ e
ζ y+i(ωt−kx)
+ i B 1 ke
η 1 y+i(ωt−kx)
+ i B 2 ke
η 2 y+i(ωt−kx)
=
Aζ e
ζ y
+ i B 1 ke
η 1 y
+ i B 2 ke
η 2 y
e
i(ωt−kx)
.
(2.21)
The system of equations (2.16) allows us to express the constant B 1 and A through
B 2 :
B 1 = −B 2
η 2
η 1
k
2
− η
2
2
k 2 − η
2
1
, A =
2i B 2 η 2 kμ
λζ 2 − λk 2 + 2μζ 2
k
2
− η
2
2
k 2 − η
2
1
− 1
. (2.22)
Relations (2.22), after taking the real parts (2.21), allow us to write displacements
in the form:
u = B 2 η 2
2k
2
μ
λζ 2 − λk 2 + 2μζ 2
f
2
− η
2
2
f 2 − η
2
1
− 1
e
−ζ y
−
f
2
− η
2
2
f 2 − η
2
1
e
−η 1 y
+ e
−η 2 y
cos(kx − ωt),
v = −B 2
2k
2
μη 2 ζ
λζ 2 − λk 2 + 2μζ 2
f
2
− η
2
2
f 2 − η
2
1
− 1
e
−ζ y
−k
η 2
η 1
f
2
− η
2
2
f 2 − η
2
1
e
−η 1 y
+ ke
−η 2 y
sin(kx − ωt).
(2.23)
Figure 2.3 presents the dependences of the amplitudes of displacements u and
v in the surface wave on the depth. The curves are represented in dimensionless
form: the amplitudes of the displacements are related to the amplitude of the normal
displacement on the surface v y=0 . Depth is plotted in fractions of wavelength. It can
be seen from the graph that the displacement normal to the surface first increases,
reaching its maximum at approximately y = 0.1λ, and then decreases monotonically
with depth, whereas the displacement parallel to the surface changes sign at a depth
of approximately y = 0.15λ.