Seismic waves Wave propagation Hooke’s law Newton’s law  wave equation Wavefronts and Rays Interfaces Reflection and Transmission coefficients

Seismic Waves body waves P-waves (longitudinal, compressional) S-waves (shear, transverse) S V -wave S H

Body waves:

Different kind of waves Transversal waves (S-waves) Longitudinal waves (P-waves)

Examples of different waves Elektromagnetic spectrum Frequency AM, FM, Georadar, Visible, X-ray Acoustic spectrum Earthquake, audible + seismic Hz

Surface waves Rayleigh-waves Love-waves

Newton’s law P(z) P(z+  z) UzUz P is the acoustic pressure U z is the displacement

Newton’s law P(z) P(z+  z) P(z+  z) - P(z) = UzUz d2d2 dt 2 UzUz - z- z  is the massdensity

Newton’s law P(z) P(z+  z) UzUz -- UzUz 22 t2t2 P =  zz

Hooke’s law P -   z P U z (z+  z) - U z (z) = U z (z) U z (z+  z)  is the compressibility

Hooke’s law P U z (z) U z (z+  z) U z = -  P  zz

Acoustic Wave equation 22 z2z2 P 22 t2t2 P 1 c2c2 = -w(t)  (z)w(t) =  q(t) (sourcesignal) 22 t2t2 c = (  ) -1/2 (wavespeed)

Propagation of seismic waves (Roth et al., 1998)

Object detection using WAVES:

Object detection using WAVES B Source O Receiver

Wavefronts versus Rays 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 SourceReceiverSource Rays are perpendicular to the wavefronts,

 1  2 v 1 v 2 Angle of incidence = angle of reflection  1 =  2 Interface: reflection

 1 v 1 v 2  1 sin  v 1 v = Interface: Refraction v 1 v 2 >  2 v 1 v 2 < 2 

 1 v 1 v 2  1 sin 90  sin  1 sin v 1 v =  2 90  = Special case: critical angle =  2

 1 v p1, v s1  1 sin  v p 1 v s =  1  2  1 sin  v p 1 v s = v p2, v s2 Interface: Conversion from P wave to S wave

 1 sin v p = --  1 sin v s = --  2 sin v p = -  2 sin v s =  n sin v s n = p = constant p = Slowness  1  2  2  3  3  1 Snell’s law v p 1 v s 1 v p 2 v s 2 v p 3 v s 3

Propagation of seismic waves (Roth et al., 1998)

EE R E T v 1  1 v 2 2 E = E R + E T R + T = 1 R = Reflection coefficient T = Transmission coefficient E = Energy Transmission- and Reflection coefficients

R T v 1  1 v 2 2 Reflection coefficient Transmission coefficient R v 2  2 v 1  1 – v 2  2 v 1  = Z 2 Z 1 – Z 2 Z = T 2v 1  1 v 2  2 v 1  = Z 1 Z 2 Z = with Z = v  = acoustic Impedance Zoeppritz’s equations at normal incidence