• Wave propagation• Hooke’s law• Newton’s law• � wave equation

• Wavefronts and Rays• Interfaces • Reflection and Transmission coefficients

body wavesP-waves (longitudinal, compressional)

S-waves (shear, transverse)SV-waveSH-wave

Body waves:

Different kind of waves

• Transversal waves(S-waves)

• Longitudinal waves (P-waves)

1 3 542 2 41 3 5

Examples of different waves

Elektromagnetic spectrum

Frequency

AM, FM, Georadar, Visible, X-ray

Acoustic spectrumEarthquake, audible + seismic106100 1019 Hz

Surface waves

Rayleigh-waves

Love-waves

�z

z

�

�

z

2

2 z� �

�

�z

� z�2

� 2�

�

� �z � z

z z �

�

z z �

z ��

�

� z�2

� 2�

�

z ��

�

� z�2

� 2�

�

z � ��

�

�(z) is the Dirac function

�2

� 2�2

� 22 �

� �2

� 2

�� -1/2

Propagation of seismic waves

(Roth et al., 1998)

Object detection using WAVES:

Object detection using WAVES

BSource

OReceiver

• 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 ReceiverSource

Rays are perpendicular to the wavefronts,

�1 �2v1

v2

Angle of incidence = angle of reflection

�1 = �2

�1

v1

v2

�1sin�2sin------- -----

v1v2-----=

Interface: Refraction

v1v2 >

�2 v1v2 <

2�

�1

v1

v2�1sin90�sin---------- �1sin

v1v2-----=

�2 90�=

Special case: critical angle

=

�2

�1

vp1,vs1

�1sin�1sin

-------- ---vp1

vs1

--------=

�1

�2�1sin�2sin

------- ---vp1

vs2

--------=

vp2,vs2

Interface: Conversion from P wave to S wave

�1sinvp1

------ = --�1sin

vs1

------ = --�2sin

vp2

-------- = -�2sin

vs2

-------------- = �nsin

vsn

--------- = p = constant

p = Slowness

�1

�2 �2

�3�3

�1

vp1vs1

vp2vs2

vp3vs3

Refraction caused by place dependent propagation velocity

Propagation of seismic waves

(Roth et al., 1998)

E ER

ET

v1���1

v2���2

E = ER + ET

R + T = 1

R = Reflection coefficientT = Transmission coefficient

E = Energy

Transmission- and Reflection coefficients

R

T

v1���1

v2���2

Reflection coefficient

Transmission coefficient

Rv2�2 v1�1–v2�2 v1�1+--------------- = ------------

Z2 Z1–Z2 Z1+--------------------=

T2v1�1

v2�2 v1�1+---------------- = -----------2Z 1

Z2 Z1+--------------------=

with Z = v� = acoustic Impedance

Zoeppritz’s equations at normal incidence

(Sheriff and Geldart, 1995)

Angle-dependent reflection- and transmission-coefficients