EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 15B
Seismic Surface Waves: P-SV Case
In this lecture, we will look at the simplest P-SV case for the in-plane particle motion. As it turns out, a simple P-SV coupled surface wave can propagate in the presence of the free surface alone with no layering. This wave is called a free Rayleigh wave and is “nondispersive” (i.e., the horizontal velocity along the surface doesn’t depend on frequency). But, the presence of layering will result in a dispersive Rayleigh wave (the same as in the SH case). Rayleigh waves have elliptical particle motions and would be recorded on the vertical and radial particle motions away from the source.
For a free Rayleigh wave, consider a homogeneous elastic halfspace
with boundary conditions 
at the free surface. Now, both P and SV waves are possible. A downgoing reflected
P-wave at the free surface will have the form
or
and the downgoing SV wave
or
where from Snell’s law at the free surface
= constant
and
Now
consider the case where all waves interacting with the boundary are evanescent
in x3,
that is decaying exponentially with x3. Then 
and 
, or
and
Then,
The two waves are now exponentially decaying in the half
space. Also, they are coupled by the
free surface where at x3 = 0. Recall that 
is related to ui in
a linear isotropic solid by
Then for 
and 
,
and
The two boundary conditions 
and 
at x3 = 0 can be written,
Boundary condition (1): 
at x3 = 0
or
Boundary condition (2): 
at x3 = 0
Since this gives two equations for the amplitude ratio 
, then the determinant of the coefficients must vanish and p must satisfy R(p) = 0 where
The equation R(p) = 0 has one root for p
and it is real and positive. The resulting
value 
is slightly greater
than 
for all elastic solids
by about 4-14% (or 
).
Thus,it is possible to construct a P-SV coupled wave that will propagate along the surface of a half space without layering.
The surface wave is named for Rayleigh who described its properties in 1886 and R(p) is known as the Rayleigh function. Rayleigh waves are widely observed in seismology. It’s main properties are
1) For 
or 
= .25, where is the Poisson ratio, then
cR
= .9194
2) cR is independent of frequency. This means that the free Rayleigh is nondispersive since there are no length scales in the problem. Nonetheless, general Rayleigh waves will be dispersive for a layered media similar to Love waves.
3) The horizontal and vertical particle motions
for with cR =
.9194 are
where 
is the source spectrum.
4) At the free surface x3 = 0, then the particle motions are
where, again, is the source spectrum.
For a given x1 as 
, then,
Thus, as the wave passes a point x1, the particle displacement of the point traces out the
path of an ellipse. The ratio of the
axes is 
. This is referred to
as retrograde elliptical particle motion.
As we go deeper into the half space, the particle motion remains
elliptical until a depth of x3
~ 0.192
, where 
. Then, for greater
depths, the particle motion becomes prograde since u1 changes sign. This is shown in the figure below for one
horizontal position.
At a depth of 1 wavelength, the u1 component has decayed to about 10% of its surface
value and the u3 component
has decayed to 20% of its surface value.
Thus, the free Rayleigh wave has a “skin
depth” of about 1 wavelength. This
depends on the frequency since the wavelength .
Note that
any layering in the half space will result in a dispersive wave where 
becomes 
and this will also
change the particle motion as a function of depth.
Ex) Some anomalous sediment columns can result in prograde motions directly at the free surface.
For general dispersive Rayleigh waves, then
1) Earth structure can be obtained by measuring and
inverting 
since will depend upon
velocity depth structure.
2) Surface wave amplitudes at different frequencies depend on the size, depth, and orientation of the fault in the earthquake source region. Thus, measuring surface waves can be used to infer the earthquake source mechanism.
3) A decrease of surface wave amplitude with distance depends on the anelastic attenuation of the Earth. This can be used to infer the Earth’s attenuation structure.
Additional Comments:
1) Love waves have limiting phase velocities of
the shear velocity in the lowest half space in the layered medium. Rayleigh waves have
a limiting phase velocity of .92
, where is the shear velocity
in the lowest half space in the layered medium.
2) Love waves are seen only on a horizontal seismometer (on the SH out-of-plane component). Rayleigh waves (P-SV) are seen on the P-SV in-plane components; radial and vertical. But, a rotation of the horizontal components to radial and transverse is needed to separate the Love waves and Rayleigh waves on the horizontal components.