The Elastic Wave Equation
Elastic waves in infinite homogeneous isotropic media
Numerical simulations for simple sources
Plane wave propagation in infinite media
Frequency, wavenumber, wavelength
Conditions at material discontinuities
Snell’s Law
Reflection coefficients
Free surface
Reflection Seismology: an example from the Gulf of Mexico
Equations of motion
What are the solutions to this equation? At first we look at infinite homogeneous isotropic media, then:
ij j i
i
t
u f
2
ij ij
ij
2
) (
i j j iij k k
ij
u u u
( )
2
i j j
i ij
k k j
i
i
f u u u
t
u
u u
u f
u
22
Equations of motion – homogeneous media
We can now simplify this equation using the curl and div operators
2 2
2 2
z y
x
i j
j i k
k i i
i
f u u u
u
jt
2
2
u -
u f
u
2( 2 )
t
i i
u u
and u u - u
… this holds in any coordinate system …
This equation can be further simplified, separating the wavefield into curl
free and div free parts
Equations of motion – P waves u -
u
u
2( 2 )
t
Let us apply the div operator to this equation, we obtain
2 ( 2 )
t
where
u
iu
i
or
2t 2
2
P wave velocity Acoustic wave
equation
Equations of motion – shear waves u -
u
u
2( 2 )
t
Let us apply the curl operator to this equation, we obtain
) (
)
2
(
i
i
u
t
u
we now make use of 0 and define to obtain
2 2t
i
u
i
Shear wave
velocity Wave equation for
shear waves
Elastodynamic Potentials
Any vector may be separated into scalar and vector potentials
u
Shear waves have no change in volume
2 2t
P-waves have no rotation
where F is the potential for P waves and Y the potential for shear waves
u
2 2t
Seismic Velocities
Material and Source P-wave velocity (m/s) shear wave velocity (m/s)
Water 1500 0
Loose sand 1800 500
Clay 1100-2500
Sandstone 1400-4300
Anhydrite, Gulf Coast 4100
Conglomerate 2400
Limestone 6030 3030
Granite 5640 2870
Granodiorite 4780 3100
Diorite 5780 3060
Basalt 6400 3200
Dunite 8000 4370
Gabbro 6450 3420
Solutions to the wave equation - general
Let us consider a region without sources
Where n could be either dilatation or the vector potential and c is either P- or shear-wave velocity. The general solution to this equation is:
2c
2t
Let us take a look at a 1-D example
) (
) ,
( x
it G a
jx
j ct
Solutions to the wave equation - harmonic
Let us consider a region without sources
2c
2t
The most appropriate choice for G is of course the use of harmonic functions:
)]
( exp[
) ,
( x t A ik a x ct
u
i i
i j j
Solutions to the wave equation - harmonic
… taking only the real part and considering only 1D we obtain
)]
( cos[
) ,
( x t A k x ct
u
T t x
t x
kct kx
ct x
k
2 2
) 2
(
c wave speed
k wavenumber
l wavelength
T period
w frequency
A amplitude
Spherical Waves
Let us assume that h is a function of the distance from the source
where we used the definition of the Laplace operator in spherical coordinates
let us define
to obtain
2c
2t
r r tc
r
2 22
2 1
r
r
2c
2t
with the known solution f ( r t )
Geometrical spreading
so a disturbance propagating away with spherical wavefronts decays like
... this is the geometrical spreading for spherical waves, the amplitude decays proportional to 1/r.
r
If we had looked at cylindrical waves the result would have been that the waves decay as (e.g. surface waves)
t r r
r f
) 1 1 (
r
1
Plane waves
... what can we say about the direction of displacement, the polarization of seismic waves?
u u P S
P S
... we now assume that the potentials have the well known form of plane harmonic waves
) (
exp i t
A
k x B exp i ( k x t ) )
(
exp i t
A
k k x
P S k B exp i ( k x t )
shear waves are transverse because S is normal to the wave vector k P waves are longitudinal as P is
parallel to k
Heterogeneities
.. What happens if we have heterogeneities ?
Depending on the kind of reflection part or all of the signal is reflected or transmitted.
What happens at a free surface?
Can a P wave be converted in an S wave
or vice versa?
How big are the amplitudes of the
reflected waves?
Wavenumber, slowness, phase velocity
Reflection at an interface
Boundary Conditions
... what happens when the material parameters change?
r
1v
1r
2v
2welded interface
At a material interface we require continuity of displacement and
traction
A special case is the free surface condition, where the surface tractions
are zero.
Reflection and Transmission – Snell’s Law
What happens at a (flat) material discontinuity?
Medium 1: v
1Medium 2: v
2i
1i
22 1 2
1
sin sin
v v i
i
But how much is reflected, how much transmitted?
Reflection and Transmission coefficients
Medium 1: r1,v1
Medium 2: r2,v2
T
Let’s take the most simple example: P-waves with normal incidence on a material interface
A R
1 1 2
2
1 1 2
2
A R
1 1 2
2
1
2 1
A T
At oblique angles conversions from S-P, P-S have to be
considered.
Reflection and Transmission – Ansatz
How can we calculate the amount of energy that is transmitted or reflected at a material discontinuity?
We know that in homogeneous media the displacement can be described by the corresponding potentials
u
in 2-D this yields
y x z
z
z x x
z y
z y x
x
u u u
an incoming P wave has the form
) (
0
exp i a x t
A
j j
Reflection and Transmission – Ansatz
... here a
iare the components of the vector normal to the wavefront : a
i=(cos e, 0, -sin e), where e is the angle between surface and ray direction, so that for the free surface
where
) ' tan
( ' exp
) tan
( exp )
tan (
exp
3 1
3 1
3 1
0
t c f
x x
ik B
ct e
x x
ik A
ct e
x x
ik A
e c k
c e
cos cos
f k
c f
cos '
' cos
P
P
rSV
re f
what we know is that
0 0
zz xz
Reflection and Transmission – Coefficients
... putting the equations for the potentials (displacements) into these equations leads to a relation between incident and reflected
(transmitted) amplitudes
2 2
2 0
2 2
2 2
0
) tan
1 ( tan
tan 4
) tan
1 ( tan
4
) tan
1 ( tan
tan 4
) tan
1 ( tan
tan 4
f f
e
f e
A R B
f f
e
f f
e A
R A
PS PP
