EOSC 252
Lab6: Elastic Properties Part 1: Dry
1
Introduction
Seismic
waves
provide
1.1
an
P- and S-waves are two forms of seismic body
incomparable
waves. This means that they travel through
means of revealing the complexity of the the
the body of a material (like your rock sam-
Earth's interior. They are used in a variety
ple), rather than on its surface. We often talk
of applications such as oil exploration and
engineering investigation.
Background
about sound waves, which travel through the
Many researchers
air; these waves travel at the speed of sound,
record seismic waves to carry out fundamen-
which actually varies a bit depending on air
tal research on the structure of the Earth,
pressure, but is around
from the crust to the core.
340
m/s at sea-level.
Primary waves, are longitudinal (or compres-
If you've ever been close to a construction
sional) waves. This means that the material
site, or anywhere outside with heavy equip-
is displaced in the direction of propagation
ment, you might have noticed something in-
(the same direction as the waves themselves
teresting about vibrations from the equip-
are moving). Secondary waves are transverse
ment. When something hits the ground hard
(or shear) waves, meaning the material they
enough for you to feel the impact in your legs,
travel through is displaced perpendicular to
you actually feel that impact a split second
the direction of propagation. Figure 1 illus-
before you hear it. Why is that?
trates the dierence between P- and S-waves.
The short (and correct!) answer is that the
The velocity of seismic body waves varies
vibrations carried through the ground move
from rock to rock, and for any other material
faster than the vibrations carried through the
you can imagine.
air. So, what controls how fast these vibra-
Furthermore, the velocity
of P-waves is dierent than that of S-waves.
tions move?
These velocities depend on three parameters:
During this experiment you will use seismic
waves to recover two important characteristics of your rock sample: the bulk modulus
and the shear modulus
µ.
κ
To do so, you will
κ
The bulk modulus
µ
The shear modulus
measure the primary and secondary wave (of-
ρ
ten shortened to P-wave and S-wave) travel
The density
times, and calculate their respective veloci-
1
ties .
1 Just
locity,
t
as a reminder,
Bulk and shear modulus are both measured
2
in Pascals (Pa, N/m , kg/(s· m)), and den3
sity is measured in kg/m . For a dry rock
V = l/t, where V is the vel the distance from
is the travel time and
sample, these values aren't going to change -
source to receiver
1
Figure 2:
An example of the oscilloscope
readout, showing a seismic wave.
The hor-
izontal axis represents time, and the vertical
axis amplitude.
Figure
1:
Illustration
showing
oscillation
modes for a P- and S-waves
Vp =
v
u
uκ + 4µ
t
3
ρ
hence, the velocity of P-waves and S-waves in
your rock is something we can (and will) nd
s
uniquely.
Vs =
In fact, we know that waves can travel in lots
of dierent directions.
For example, if you
µ
ρ
(1)
(2)
Immediately, we can see something interest-
shout in a crowded lab everyone in the room
ing about these: Since
can hear you, not just someone in front of
positive, the P-wave velocity
you.
than the S-wave velocity. This is important
When you shout, you sometimes hear
echos that come to your ears after the main
κ, µ,
and
has
ρ are always
to be faster
to remember.
part of the signal does.
When we work through the lab, you will be
oscilloscope
In the same way, signals travelling through
using an
your rock can take dierent amounts of time
record. The readout will resemble Figure 2,
to reach the other end (for example by bounc-
with two signals (one above the other). Fig-
ing o the sides), so we'll just measure the
ure 3 shows how all of the equipment is con-
fastest signal, and we'll call this the rstarrival traveltime or just the traveltime.
to view the signal we
nected.
In order to nd the bulk and shear moduli, we
will need to measure the times of the P-wave
1.2
P-waves and S-waves
and S-wave arrivals, and solve Equations 1
and 2 for
The P-wave velocity,
locity,
Vs
Vp
and the S-wave ve-
are expressed by Equations 1 and 2
below.
2
κ
and
µ.
Figure 3: Experimental setup
Fun Fact
We will nd out in the second part of this set of labs that
µ
can be zero (as in a uid such
as air), in which case the S-wave doesn't propagate, and the expression for an
is just the P-wave expression without the contribution from the
s
Va =
shear
acoustic
wave
component:
κ
ρ
Knowing this, and that the bulk modulus for air is approximately 1.4 times its pressure (for
adiabatic conditions) we can write:
Vair,ST P =
v
u
u 1.4 × 100
t
1.2754
= 331
kPa
3
kg/m
m/s
o
for air at Standard Temperature and Pressure (100 C, 100 kPa).
3
Table 1: Sample dimensions
1
Diameter (
2
3
4
5
Average
)
±
Length (
)
±
Table 2: First-arrival traveltime measurements
Travel Time (s)
t0p
tp
t0s
ts
2
Experimental Procedure
1. Measure the length and the diameter of your sample. Repeat 5 times.
2. Take an initial reading of the P-wave rst-arrival time without the sample in the sample
holder. Ensure that the sample holder is under a pressure of 120 psi, which is reached
when the voltmeter reads 5 mV. In the view screen of the oscilloscope, use the cursor
to determine the time of the rst break. Record your reading in Table 2.
3. Place your sample into the sample holder and measure the P-wave arrival time again.
Record this in Table 2.
In your analysis, you will subtract
t0p
from
tp
to nd the
traveltime through the sample itself.
4. Switch the pulse generator to S-wave mode, and repeat steps 4 and 5.
3
Analysis
1. Calculate
Vp
and
Vs
using the following relations:
ls
,
tp − t0p
ls
Vs =
,
ts − t0s
Vp =
where ls is the length of your sample.
2. Calculate the bulk and shear moduli (κ and
3. Would
µ
respectively).
κ, and µ be dierent if the experiment was carried out using a pressure of 500psi
instead of 120psi?
4