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NPTEL- GEOTECHNICAL EARTHQUAKE ENGINEERING
Module 2
WAVE PROPAGATION
(Lectures 7 to 9)
Lecture 8
Topics
2.3 WAVES IN A SEMI-INFINITE BODY
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.3.7
Rayleigh waves
Rayleigh wave velocity
Rayleigh wave displacement amplitude
Love waves
Higher-mode surface waves
Dispersion of surface waves
Phase and group velocities
2.3 WAVES IN A SEMI-INFINITE BODY
The earth is obviously not an infinite body-it is a very large sphere with an outer
surface on which stresses cannot exist. For near-surface earthquake engineering
problems, the earth is often idealized as a semi-infinite body with a planar free
surface (the effects of the earth’s curvature are neglected). The boundary conditions
associated with the free surface allow additional solutions to the equations of motion
to be obtained. These solutions describe waves whose motion is concentrated in a
shallow zone near the free surface (i.e., surface waves). Since earthquake,
engineering is concerned with the effects of earthquakes on humans and their
environment, which are located on or very near the earth’s surface, and since they
attenuate with distance more slowly than body waves, surface waves are very
important.
Two types of surface waves are of primary importance in earthquake engineering.
One, the Rayleigh wave, can be shown to exist in a homogeneous, elastic half-space.
The other surface wave, the Love wave, requires a surficial layer of lower s-wave
velocity than the underlying half-space. Other types of surface waves exist but are
much less significant from an earthquake engineering standpoint.
2.3.1 Rayleigh waves
Waves that exist near the surface of a homogeneous elastic half-space were first
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NPTEL- GEOTECHNICAL EARTHQUAKE ENGINEERING
investigated by Rayleigh (1885) and are known to this data as Rayleigh waves. To
describe Rayleigh waves, consider a plane wave (figure 2.8) that travels in the xdirection with zero particle displacement in the y-direction (
). The z-direction
is taken as positive downward; so all particle motion occurs in the x-z plane. Two
potential functions,
, can be defined to describe the displacements in the xand z-directions:
Figure 2.8 Motion induced by a typical plane wave that propagates in the xdirection. Wave motion does not vary in the y-direction
(2.35.a)
(2.35.b)
The volumetric strain, or dilatation, ,̅ of the wave is given by ̅
̅
(
)
The rotation in the
(
)
, or
(2.36)
plane is given (equation 2.19) by
(
)
(
)
(2.37)
Use of the potential functions allows separation of the effects of dilation and rotation
i.e., (equation 2.36 and 2.37) indicate that
are associated with dilation and
rotation, respectively. Therefore, Rayleigh waves can be thought of as combinations
of p- and s-waves (SV waves for this case, since the
plane is vertical) that
satisfy certain boundary conditions. Substitution of the expressions for u and w into
the equations of motion as written in (equation 2.26a and 2.26c) gives
(
)
(
)
(
)
(
)
Dept. of Civil Engg. Indian Institute of Technology, Kanpur
(
)
(2.38.a)
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NPTEL- GEOTECHNICAL EARTHQUAKE ENGINEERING
(
)
(
)
(
)
(
)
(
Solving (equations 2.38) simultaneously for
)
and
(2.38.b)
shows
(2.39.a)
(2.39.b)
If the wave is harmonic with frequency and wave number , so that it propagates
with Rayleigh wave velocity
, the potential functions can be expressed as
( )
(
)
( )
(
)
(2.40.a)
(2.40.b)
Were F and G are functions that describe the manner in which the amplitude of the
dilational and rotational components of the Rayleigh wave vary with depth.
Substituting these expressions for
into (equation 2.39) gives
( )
( )
( )
( )
( )
( )
(2.41.a)
(2.41.b)
Which can be rearranged to give the second-order differential equations
(
)
(2.42.a)
(
)
(2.42.b)
The general solution to these equations can be written in the form
( )
(2.43.a)
( )
(2.43.b)
Where
The second term of equations 9430 corresponds to a disturbance whose displacement
amplitude approaches infinity with increasing depth. Since this type of behavior is
not realistic,
must be zero, and the potential functions can finally be
written as
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(
)
(
)
(2.44.a)
(2.44.b)
Since neither shear nor normal stresses can exist at the free surface of the half-space,
and
when
. Therefore
̅
̅
(
(2.45.a)
)
(2.45.b)
Using the potential function definitions of u and w (equations 2.35) and the
solutions for the potential functions (equation 2.44) the free surface boundary
conditions can be written as
(
)
[(
)
(
]
(2.46.a)
)
(2.46.b)
Which can be rearranged to yield
(
)
(2.47.a)
(2.47.b)
With these results, the velocities and displacement pattern of Rayleigh waves can be
determined.
2.3.2 Rayleigh wave velocity
The velocity at which Rayleigh waves travel is interest in geotechnical earthquake
engineering. As discussed in chapter 6, Rayleigh waves are often mechanically
generated and their velocities measured in the field to investigate the stiffness of
surficial soils. Adding (equation 2.47) and cross-multiplying gives
(
)[(
)
]
(2.48)
Which, upon introducing the definitions of q and s and factoring out a
yields
(
Defining
)(
)
(
) (
)
term,
(2.49)
as the ratio of the Rayleigh wave velocity to the s-waves velocity
Then
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Where
√
as
(
(
((
√(
)
)
√(
) (
)
) . Then (equation 2.49) can be rewritten
(
) (
)
(2.50)
Which can expanded and rearranged into the equation
(
)
(
)
(2.51)
This equation is cubic in
, and real solutions for
can be found for various
values of Poisson’s ratio. These allow evaluation of the ratios of the Rayleigh wave
velocity to both s- and p-wave velocities as functions of v. the solution shown in
(figure 2.9) shows that Rayleigh waves travel slightly than s-waves for all values of
Poisson’s ratio except 0.5.
Figure 2.9 Variation of Rayleigh wave and body wave propagation velocities with
Poisson’s ratio
2.3.3 Rayleigh wave displacement amplitude
Showed how the velocity of a Rayleigh wave compares with that of p- and s-waves.
Some of the intermediate results of that section can be used to illustrate the nature of
particle motion during the passage of Rayleigh waves. Substituting the solutions for
the potential functions
(equation 2.44) into the expression for u and w
(equation 2.35) and carrying out the necessary partial differentiations yields
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(
(
)
(
)
(
)
(
(2.52.a)
)
(2.52.b)
From (equation 2.47b),
Which, substituting into (equation 2.52), gives
(
(
)
)
(
(
)
)
(2.53.a)
(2.53.b)
Where the terms in parentheses described the variation of the amplitudes of u and w
with depth. These horizontal and vertical displacement amplitudes are illustrated for
several values of Poisson’s ratio in (figure 2.10). Examination of (equations 2.53)
indicates that the horizontal and vertical displacements are out of phase by
.
Hence the horizontal displacement will be zero when the vertical displacement
reaches its maximum (or minimum), and vice versa. The motion of a particle near
the surface of the half-space is in the form of a retrograde ellipse (as opposed to the
prograde ellipse particle motion observed at the surface of water waves). The general
nature of Rayleigh wave motion was illustrated earlier.
Figure 2.10 Horizontal and vertical motion of Rayleigh waves. A negative
amplitude ratio indicates that the displacement is the opposite direction of the
surface displacement. (Richart et al., 1970).
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The Rayleigh waves produced by earthquakes were once thought to appear only at
very large epicentral distance (several hundred km). It is not recognized, however,
that they can be significant at much shorter distance (a few tens of kilometers). The
ratio of minimum epicentral distance, R, to focal depth, h, at which Rayleigh waves
first appear in a homogeneous medium, is given by
(2.54)
)
√(
Where
are the wave propagation velocities of p-waves and Rayleigh
waves, respectively (Ewing et al., 1975).
2.3.4 Love waves
In a homogeneous elastic half-space, only p-waves, and Rayleigh waves can exist. If
the half-space is overlain by a layer of material with lower body wave velocity,
however, Love waves can develop (Love, 1972). Love waves essentially consist of
SH-waves that are trapped by multiple reflections within the surficial layer. Consider
the case of a homogeneous surficial layer of thickness H overlying a homogenous
half-space as shown in (figure 2.11). A Love wave traveling in the +x-direction
would involve particle displacements only in the y-direction (SH-wave motion), and
could be described by the equation.
Figure 2.11 Schematic illustration of softer surficial layer (
overlying elastic half-space
(
)
( )
(
)
)
(2.55)
Where v is the particle displacement in the y-direction, ( )describes the
variation of v with depth, and
is the wave number of the Love wave. The
Love wave must satisfy the wave equations for s-waves in both the surficial layer
and the half-space
(
(
)
)
(
)
(
)
{
The amplitude can be shown (
and Richards, 1980) to vary with depth according to
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( )
(
(
{
)
)
Where the A and B coefficients describe the amplitudes of down going and up going
waves, respectively, and
√
√
(2.58)
Since the half-spaces extends to infinite depth,
must be zero (no energy can be
supplied or reflected at infinite depth to produce an upgoing wave). The
requirements that all stresses vanish at the ground surface is satisfied if in other
words, if
( )
(
)
(
) (
)
. The amplitudes can now be rewritten in terms of the two remaining
unknown amplitudes as
( )
(
(
{
At the
)
)
interface, continuity of stresses requires that
(
)
(2.60)
And compatibility of displacements requires that
(
)
(2.61)
Using (equations 2.60 and 2.61),
(
)
(2.62)
can be expressed in terms of
by
Substituting (equations 2.59 and 2.60 into 2.55) gives
(
{
)
[ (
)
* (
)
]
+
(
*
)
(
) (
)+
(
)
Where
are the shear wave velocities of materials 1 and 2, respectively,
and
is the velocity of the Love wave. (Equation 2.63) shows, as illustrated in
(figure 2.12) that the Love wave displacement amplitude varies sinusoidally with
depth in the surficial layer and decays exponentially with depth in the underlying
half-space. Because of this, Love waves are often described as SH-waves that are
trapped in the surficial layer. The general nature of Love wave displacement was
shown in (figure2.1.a) from Module 2.
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Figure 2.12: Variation of particle displacement amplitude with depth for Love
waves
The Love wave velocity is given by the solution of which indicates, as illustrated in
figure 2.13 that Love wave velocities range from the s-wave velocity of the halfspace (at very low frequencies) to the s-wave velocity of the surficial layer (at very
high frequencies).
Figure 2.13: Variation of Love wave velocity with frequency
(
)
√
(2.64)
This frequency dependence indicates that Love waves are dispersive.
2.3.5 Higher-mode surface waves
Any surface wave must (1) satisfy the equation of motion. (2) Produce zero stress at
the ground surface, and (3) produce zero displacement at infinite depth. Nontrivial
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solutions do not exist for arbitrary of frequency and wave number; rather, a set of
discrete and unique wave numbers exist for a given frequency. Each wave number
describes a different displacement pattern, or mode, of the surface wave. The
preceding derivations have been limited to the fundamental modes of Rayleigh and
Love waves, which are the most important for earthquake engineering applications.
Detailed treatment of higher-mode surface waves can be found in most advanced
seismology texts.
2.3.6 Dispersion of surface waves
For a homogeneous half-space, the Rayleigh wave velocity was shown to be related
to the body wave velocities by Poisson’s ratio. Since the body wave velocities are
constant with depth, the Rayleigh wave velocity in a homogeneous half-space is
independent of frequency. The velocity of the Love wave, on the other hand, varies
with frequency between an upper and a lower limit.
Dispersion is a phenomenon in which waves of different frequency (and different
wavelength) propagate at different velocities. Hence Love waves are clearly
dispersive, and Rayleigh waves in a homogeneous half-space are non dispersive.
Near the earth’s surface however, soil and rock stiffness’s usually increase with
depth. Since the depth to which a Rayleigh waves causes significant displacement
increases with increasing wavelength (figure 2.10), Rayleigh waves of long
wavelength (low frequency) can propagate faster than Rayleigh waves of short
wavelength (high frequency). Therefore, in the real world of heterogeneous
materials. Rayleigh waves are also dispersive. The dispersion of Rayleigh waves can
be used to evaluate subsurface stiffness profiles by field testing techniques.
Since the velocities of both Rayleigh waves and Love waves decrease with
increasing frequency, the low-frequency components of surface waves produced by
earthquakes can be expected to arrive at a particular site before their high frequency
counterparts. This tendency to spread the seismic energy over time is an important
effect of dispersion.
2.3.7 Phase and group velocities
The solutions for Rayleigh wave velocity,
, and Love wave velocity, , were
based on the assumption of harmonic loading which produces an infinite wave train.
These velocities describe the rate at which points of constant phase (e.g., peaks,
troughs, or zero points) travel through the medium and are called phase velocities. A
transient disturbance may produce a packet of waves with similar frequencies. This
packet of waves travels at the group velocity, , given by where c is the phase
velocity (equal to
, depending on which type of wave is being considered)
and k is the wave number (equal to
).
(2.65)
In a nondispersive material,
, so the group velocity is equal to the phase
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velocity. Since both
generally decrease with increasing frequency in
geologic materials,
is less than zero and the group velocity is lower than the
phase velocity. Consequently, a wave packet would appear to consist of a series of
individual peaks that appear at the back end of the packet, move through the packet
to the front, and then disappear. The opposite behavior can be observed by (for
) dropping a rock into a calm pond of water and watching the resulting ripples
carefully.
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