• Wave propagation
• Hooke’s law
• Newton’s law
• Þ wave equation
• Wavefronts and Rays
• Interfaces
• Reflection and Transmission coefficients
body waves
P-waves (longitudinal, compressional)
S-waves (shear, transverse)
SV-wave
SH-wave
Body waves:
Different kind of waves
• Longitudinal waves
(P-waves)
12 3 4 5
• Transversal waves
(S-waves)
1 2 34 5
Examples of different waves
Elektromagnetic spectrum
100
Earthquake, audible + seismic
AM, FM, Georadar,
106
Visible,
Acoustic spectrum
Frequency
X-ray
1019
Hz
Surface waves
Rayleigh-waves
Love-waves
z
z
D
D
z
D
r
rD
2
2
z
D
z
¶
¶
¶2
r 2
¶
z
z
z
z
k
D
z
kD
D
z
z
¶
¶
z
k
D
¶
¶
¶
¶
¶2
r 2
¶
z
k
z
¶
¶
¶
¶
¶2
r 2
¶
z
k
z
d
d(z) is the Dirac function
¶2
¶ 2
2
¶2
¶2
2
¶
r 2
¶
rk
-1/2
d
Propagation of seismic waves
(Roth et al., 1998)
Object detection using WAVES:
Object detection using WAVES
Source
B
Receiver
O
• Wavefronts indicate the boundary of the material
which already moves and the material which is
still undisturbed.
• Rays are plotted perpendicular with respect to the
wavefronts and describe the dominant propagation
of the seismic energy between two locations
Geometrical Wave propagation
Source
Source
Receiver
Rays are perpendicular to the wavefronts,
a1 a2
v1
v2
Angle of incidence =
angle of reflection
a1 = a 2
Interface: Refraction
a1
v1
v2
a2
a2
sin a1 v1
----------- =----sin a2 v2
v2 > v1
v2 < v1
Special case: critical angle
a1
a2 = 90°
v1
v2
a2
sin a1
v1
---------=sin a1= ----sin 90°
v2
Interface: Conversion from P wave to S wave
b1
a1
vp1,vs1
vp
sin a 1
1
--------------- = --sin b1
vs
1
vp2,vs2
b2
vp
sin a 1
1
-------------- = --sin b2
vs
2
vp
a1
vp
a2
a3
1
vs
1
2
vs
2
vp
vs
3
3
a1
a2
a3
sin a
sin b
sin a2
sin b2
sin bn
------ 1= --------1 = ---------= -------------= --------= p = constant
vp
vs
vp
vs
vs
1
1
2
2
n
p = Slowness
Refraction caused by place dependent
propagation velocity
Propagation of seismic waves
(Roth et al., 1998)
Transmission- and Reflection coefficients
E
ER
E = ER + ET
v1, r1
v2, r2
ET
E = Energy
R = Reflection coefficient
T = Transmission coefficient
R+T=1
Zoeppritz’s equations at normal incidence
Reflection coefficient
R
v1 , r1
v 2 r 2 – v1 r 1
Z2 – Z1
=
--------------=
------------------------------R
v2 r2 + v 1 r1
Z2 + Z1
v 2 , r2
T
Transmission coefficient
2v 1 r1
2Z 1
-------------------T = ---------------- = ----------v 2r 2 + v1 r1
Z 2 + Z1
with Z = v r = acoustic Impedance
Angle-dependent reflection- and transmission-coefficients
(Sheriff and Geldart, 1995)