1. Introduction to Seismology
Geology 5640/6640
Introduction to Seismology
8 Feb 2017
Last time: The solution to the Wave Equation!
• Solution of the wave equation is made easier by expressing
displacement in terms of scalar & vector displacement
potentials:
In that case:
These can be solved by plugging in eigenfunctions of the
form:
and solutions are:
Read for Fri 10 Feb: S&W (§2.5)
© A.R. Lowry 2017

2. Introduction to Seismology
Geology 5640/6640
Introduction to Seismology
Last time cont’d: Harmonic wave motion
• Harmonic wave motion can be described entirely by
specifying a period T (= 1/f = 2/), wavelength  (= 2/k),
and a phase  (= x/c = tan-1[a1/a2]).

3. Polarizations of P- and S-waves:
The polarization (or “polarity”) of a wave is a way of
describing the direction of initial particle motion of a wave
(i.e., the “first motion”).
For example, a seismologist might say “The first motion on
the North-South component was down.” This implies a
reference frame however…
First consider a P-wave (scalar displacement potential) with
equation: solution: We could write this as:
where the first part (1) travels in the +x direction, and the
second (2) travels in the –x direction.
Displacement is related to the potential  by the gradient:

4. Here our y, z components disappeared because there’s
no y, z dependence for a plane wave traveling in the
x-direction:
So, u = (–ik, 0, 0), and particle displacement is always
in ± the direction of wave propagation, i.e.,
perpendicular to the
wave front.
So if first motion is
compressive, all points
on the wavefront move
in the direction of the
black arrows (“positive”
or “upward” polarity); if
dilatational, they all move in the direction of the red
arrows. This is also called longitudinal displacement…

5. Now consider the S-wave (vector displacement potential)…
Like before we’ll consider a plane wave propagating
only in the x-direction:
equation: solution has three components:
To write this in terms of displacement we take the curl:
Since it’s propagating only in the x-direction, the y- and
z-derivatives again vanish, leaving:

6. So for an S-wave propagating in the x-direction, particle
motion is only in the y- and z-directions, and not the
x-direction.
We further decompose
the wavefield using a
convention that:
SH = horizontally
polarized shear wave
SV = vertically
Both of these are always perpendicular to the propagation
direction, with SH in the horizontal plane and SV in the
vertical plane orthogonal to SH.

7. Thus for our example of
a shear wave
propagating only in
the x-direction:
Different texts illustrate
this differently; one
way is to think of a
series of blocks:

8. To visualize SH and SV a little better, consider a point source
deep within a constant-velocity medium:
The ray paths must be straight lines; the wavefronts are
spheres.

9. Now imagine the same source-receiver pair for a medium
in which velocity increases with depth:
The definition’s the same, but now the angle between
direction of propagation & the horizontal is changing
(so angle of SV motion WRT horizontal also changes).

10. First, a quick look at some data and phases
Seismic Data and Analysis:
First, a quick look at some
data and phases

11. Ray Naming Conventions
P, S - P or S wave segments in the mantle
p, s - up-going energy from an earthquake
(not shown here)
K - P wave segment in the fluid outer core
I - P wave segment in the solid inner core
J - S wave segment in the solid inner core
c - a reflection off the core-mantle boundary
i - a reflection off the inner core boundary
Example: SKS—S-wave that converts to a P-wave in the outer core, then back to an S-wave when it leaves the outer core and travels up towards the seismometer.

12. (More like what it really looks like:)
A ray is the normal to a propagating wavefront.

13. More
ray
paths…

14. Ray paths in the upper part of the Earth (the “lithosphere”)