Page 333 - Physical Principles of Sedimentary Basin Analysis

9.11 Further reading 315

4 2
˙
D v k Y − Fk Y + gY cos(kx) = 0 (9.150)
which can be written as
2
Y ˙ Fk − g 1
= = . (9.151)
Y D v k 4 t 0 (k)
This expression is slightly rewritten as
dY dt
= (9.152)
Y t 0
and integrated, which gives the function Y(t) = Y 0 e t/t 0 . The integration factor Y 0 is given
by the initial condition (at t = 0), and becomes Y 0 = w 0 .
Exercise 9.17 The characteristic time t 0 has a minimum in the regime λ<λ s (see
Figure 9.16). Show that the minimum is
4D v g λ s
t 0,min = for λ 0,min = √ . (9.153)
F 2 2
Exercise 9.18 Show that the characteristic time t 0 (k) as function of the wave number can
be approximated as follows:
⎧
D v k
⎪ 4
⎪
⎨ − , F
g
t 0 (k) ≈ g (9.154)
D v k
⎪ 2
⎪
⎩ , F
g.
F
Figure 9.16 shows this approximation by the straight lines at each side of wavelength λ s .
9.11 Further reading
Turcotte and Schubert (1982) has a chapter that covers the theory of plates and flexure,
where the equation for flexure of a plate is derived and solved. The solutions are applied to
a variety of cases.
Watts (2001) is dedicated to isostasy, flexure and gravity. The book begins with two
chapters on the history of how these concepts developed, before the theory is expanded,
applied and discussed in the next chapters. The book has a rich set of case studies from all
continents, and it is equipped with an extensive list of references to the work done in the
field.
Chapters 8 and 9 in Nadai (1963) are devoted to the mathematical treatment of flex-
ure of both elastic plates and viscoelastic plates. The mathematical treatment is complete,
with a large set of examples and applications. A number of the applications are geologi-
cal, like folding of viscoelastic layers under compression, and the viscoelastic rebound of
Fennoscandia after the last deglaciation.