Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 58

1/Q = - E/2 E (2.28)
where E is the peak strain energy stored in the volume, and - E is the energy
lost per cycle due to material imperfections (Aki and Richards, 1980; Toksoz
and Johnston, 1981; Kennett, 1983; Bourbie, Coussy and Zinszner, 1987).
This definition is rarely used: it is difficult to drive a material element with
stress waves having constant amplitude and period. More commonly, we
measure the timewise decay of amplitude in standing waves at fixed
wavenumber, or the spatial decay in a propagating waves at fixed frequency. In
either case, for a medium with a linear stress-strain relationship, the wave
amplitude A is proportional to E 1/2 . Hence, the foregoing definition for Q can be
written in the form
1/Q = -

A/ A (2.29)

or equivalently as
A/A = - /Q = - (2 )/2Q (2.30)

2.2.2 Relating Q to amplitude decay in space.
We wish to relate Q and A in Equation 2.30 to wave amplitude decay in
space. In modeling wave propagation, recall that we had introduced the
complex frequency in Chapter 1,
= r + i i (2.31)
Equation 2.31, used in “exp {i(kx- t)},” led to an amplitude factor “exp ( i t)”
that affected the propagating part “exp {i(kx- r t)}.” What is the same
amplitude factor in terms of Q? First, let us establish differential (versus per
cycle ) laws describing attenuation. In the limit of infinitesimal changes,

A/A

tends to dA/A, whereas the 2 term per cycle in Equation 2.30 becomes

r dt.

We therefore replace Equation 2.30 with the more general statement
dA/A = -

r dt/2Q

(2.32)

Equation 2.32 can be rewritten as
dA/A = - ( r /2Q) (dt/dx) dx = - ( r /2cQ) dx (2.33)
where we have not yet defined the speed c in dx/dt = c (this is addressed later).
Integration yields log e A = -

r x/2cQ to within a constant, so that we may write

A(x) = A 0 exp (-

r x/2cQ)

(2.34)

where A 0 is the initial wave amplitude at x = 0.
In summary, Equation 2.34 describes, as required, the spatial amplitude
decay of a wave, when A 0 , r , c, and Q are given; it is consistent with results
obtained in earthquake seismology (Aki and Richards, 1980).