2 Seismic Wave Propagation and Earth models

2. Seismic Wave Propagation and Earth models
Note, however, that the velocity vtp(p) determined from dx/dt at distance x = D does always
relate to the respective depth half way between source and station! Nevertheless, practically
all one-dimensional Earth models have been derived this way assuming that lateral variations
of velocity are negligible as compared to the vertical velocity variations.
2.5.4 Amplitudes and phases
2.5.4.1 Energy of seismic waves
The energy density E contained in a seismic wave may be expressed as the sum of kinetic
(Ekin) and potential (Epot) energy densities :
E = Ekin + Epot . (2.23)
The potential energy results from the distortion of the material (strain; see. Figs. 2.2 and 2.5)
working against the elastic restoring force (stress) while the kinetic energy density is
Ekin = ½ ρ av 2 , (2.24)
where ρ is the density of the material, av = A ω cos(ωt – kx) is the ground-motion particle
velocity, with A - wave amplitude, ω - angular frequency 2πf and k - wavenumber. Since the
mean value of cos 2 is ½ it follows for the average kinetic energy density⎺Ekin = ¼ ρ A 2 ω 2 ,
and with Ekin = Epot in case of an isotropic stress-strain relationship in a non-dispersive
(closed) system for the average energy density
⎺E = ½ ρ A 2 ω 2 . (2.25)
The energy-flux density per unit of time in the direction of wave propagation with velocity v
is then
Eflux = ½ v ρ A 2 ω 2 (2.26)
and the total energy-flux density Eflux through a small surface area dS of the wavefront
bounded by neighboring rays which form a ray tube
Eflux = ½ v ρ A 2 ω 2 dS. (2.27)
When considering only waves with wavelengths being small as compared to the
inhomogeneities of the medium of wave propagation (high-frequency approximation), then
we can assume that the seismic energy only travels along the rays. According to the energy
conservation law, the energy flux within a considered ray tube must remain constant although
the surface area dS of the wavefront related to this ray tube may vary along the propagation
path due to focusing or defocusing of the seismic rays (compare Fig. 2.28). Considering at
different times two surface patches of the propagating wavefront dS1 ≠ dS2, which are
bounded by the same ray tube, and assuming that v and ρ are the same at these two locations
then
A1/A2 = (dS2/dS1) 1/2 , (2.28)
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