Transcript Love Waves

Seismology
Part VI:
Surface Waves: Love
Augustus Edward Hough Love
1863 - 1940
A Google result for “Love Wave”
Love Waves
Rayleigh waves propagate by
interference of P and SV waves, but
SH waves do not interact with either.
However, they can propagate in a low
wavespeed waveguide like the
Earth’s crust.
We consider a layer of thickness H
and wavespeed 1 over a half space
with wavespeed 2, and 2 >1 (so
that there is a critical refraction).
Because SH displacements satisfy the wave equation directly, we
needn’t resort to potentials:
V1  A exp(i( px1   1 x3  t)) B exp(i( px1   1 x3  t))
V2  C exp(i ( px1   2 x3  t))

Where V1 describes the up and down waves in the layer, and V2 is
the wave transmitted into the half space.

The boundary condition at the free surface is:
u2 u3 
u2 
V1 
 32  1 
 1  1  0
x3 x2 
x3 
x3 
or
Ai 1 exp(i( px1   1 x3  t))  Bi 1 exp(i( px1   1 x3  t))
evaluated at x3 = 0, this results in A = B.
At the bottom of the layer (x3 = H), the same stress condition gives:
A1 1 exp(i 1 H ) exp(i 1 H ) C 2 1 exp(i 2 H )
The continuity of displacement gives:
Aexp(i 1 H ) exp(i 1 H ) Cexp(i 2 H )
exp( i ) exp(  i )  cos( ) i sin( ) cos( ) i sin( )  2i sin( )
exp( i ) exp(  i )  cos( ) i sin( ) cos( ) i sin( )  2cos( )

so, taking the ratio of the above two boundary conditions gives:
i1 1 tan( 1 H )   2 2
or


 2
i 2
 2
ˆ
ˆ
tan( H ) 


i1
i1
1
2
2
2
1
1
1
1
Note that if we are past the critical angle, then the argument on the
right is real. Let’s rewrite the above using the definition of :


tan H 1/ 12  1/ c 2 
 2 1/ c 2  1/ 22
1 1/ 12  1/ c 2
The above is a relationship between two variables: frequency and
apparent horizontal wavespeed (c = 1/p). Solutions exist for 2 > c

> 1. We consider the solutions to the above equation graphically:
Here’s how to think about what’s going on in the top part of the figure:
Consider the argument to the tangent function on the left to be y(c).
Choose a frequency  = 2f and let c vary from its extremes of 1 and
2. When c = 1, y(c) = 0 and tan(0) = 0. The tangent repeats itself
every time the argument increments by . Repeat the same choices of
c for the right hand side of the equation, and it forms the dashed line
decreasing from infinity to zero. Solutions to the equation are found
where the lines intersect.
Note that there are a few
discrete solutions to the
equation. The one with
the lowest value of c is
called the fundamental
mode, and the ones to
the right are higher
modes or overtones.
The number of overtones depends on the frequency: the higher the
frequency, the more overtones. But because only a few discrete
solutions exist, this happens discretely, not continuously. We can see
from the above plot that a new overtone will be possible whenever c =
2 is a solution to the above equality. This happens when at the critical
frequencies c satisfy:


tan  c H 1/ 12  1/ 22  0
or

c H 1/ 12 1/ 22  m
c 

for the mth overtone.
m
H 1/ 12 1/ 22
You might imagine all the tangents in the figure getting increasingly
squished to the left as the frequency increases, with the value of c
corresponding to a given overtone gradually decreasing from 2 to 1
as the frequency increases. This behavior is shown in the bottom plot.
The mode idea is easily explained in terms of constructive
interference of plane waves traveling with a phase shift of 2m (i.e.
and integral multiple of 2.
Suppose the angle of incidence is j, then the distance traveled from
top to bottom (or bottom to top) is
H
X
cos( j)

The distance from the wavefront to the reflection point is
A  X cos(2 j)  X2cos2( j)1
Thus, the total distance is
2
2H
cos
( j)
2
2
X  X2cos ( j) 1  2X cos ( j) 
 2H cos( j)
cos( j)

2H cos( j)  2H 1 1
If  is the wavelength, then the total change in phase just due to
propagating this distance is:

4 H 1 1 4 H 1 1 2H 1 1
d 


 2H 1

c sin( j)
c sin( j)
where c is the horizontal (apparent) wavelength, and /c = 2/c.
The total phase change is this amount plus the phase changes due to
reflection
 at the reflection at the interface (because of the postcritical
reflection – there is no phase shift at the free surface).
According to the expressions for RSS in Table 3.1 of Lay and
Wallace:
1 1  i 2
ˆ 2
RSS 
 e i
1 1  i 2
ˆ 2
The denominator is the complex conjugate of the numerator, so
|Rss| = 1 and the phase angle  is

  2
 
ˆ 2
1
1 2 ˆ 2
 n  2tan
 2tan
1
1
1
1
(NB: think of how complex conjugates look in the complex plane –
the angles above and below the real axis are equal and opposite,
which is
the reason for the factor of “2” out front). The total phase
change is thus
 
1 2 ˆ 2
 d  n  2H 1  2tan
1
1
Constructive interference will occur whenever the above sum is a
multiple of 2. For the fundamental mode, we take the phase shift to
be zero, in which case:
 2
ˆ
2H  2tan
1
1
2
1
1
or

 2
ˆ
tanH  
1
2
1
1
which is the equation derived above. The higher orders
correspond to integral multiples of 2.

Notes:
1. Because Love waves are all SH,
they show up on the transverse
components of a seismogram (not
radial or vertical).
2. Love waves travel at the shear
waves speed of the upper mantle,
which means they travel faster than
Rayleigh waves and thus arrive
sooner.
3. Love waves are naturally
dispersive, with longer wavelengths
(lower frequencies) traveling faster.
Seismogram recored by a three-component long-period system (BNY) at the State
Univ. of New York, Binghamton approximately 380 km from an earthquake. Note
that the Rayleigh-type surface waves appear on the Z and East components, while
the Love-type surface waves appear on the North component (back azimuth is 260
deg.)