Page 330 - Physical Principles of Sedimentary Basin Analysis

312 Flexure of the lithosphere

g gλ 2
F s = = (9.144)
k 2 (2π) 2
when the wavelength is fixed. The characteristic wavelength that gives an infinite t 0 when
the force is fixed is
"
F
λ s = 2π . (9.145)
g
How the characteristic time depends on the force and the wavelength is summarized as
follows:
t 0 < 0, F < F s or λ>λ s
t 0 =∞, F = F s or λ = λ s
t 0 > 0, F > F s or λ<λ s .

The initial deformations will decrease with time when t 0 < 0, and they will increase with
time when t 0 > 0. An infinite characteristic time t 0 means that the periodic buckling of the
plate is stable under the compressive force F s . The initial buckling becomes preserved. The
deformations will decrease with time for a compressive force less than F s , and increase for
a force larger than F s .
Wavelengths longer than λ s will die out, and the longer the wavelengths are the faster
will they die out. Wavelengths that are ∼λ s will be preserved, and wave lengths that
are less than λ s will increase with time. The wavelengths that can be observed in fold
belts should therefore be the wavelengths that are less than λ s . It is seen that the limit-
ing wavelength (9.145) does not depend on the viscosity of the layer, and that it is only
dependent on the compressive force F.
A periodic deflection of a lithospheric plate can be written as a Fourier series

w(x) = a n sin(k n x), with k n = πn/L (9.146)
n
where the sum is a superposition of deflections of different wavelengths. The development
of this deflection through time is the superposition of the solution (9.142):
t/t 0 (k n )
w(x, t) = a n e sin(k n x). (9.147)
n
Compressing the plate with a force F implies that wavelengths λ n = 2π/k n = 2L/n
larger than λ s in the Fourier decomposition will die out, and that wavelengths less than λ s
will increase. Figure 9.15 shows the time development of the deflection from Figures 9.8
and 9.14 in the case of a viscous plate. The initial state is just the deflection filled with rock
of density c when the surface load is absent. The compressive force is measured relative
to the average force from the isostatic pressure through the plate,

h 1
2
F u = c gz dz = c gh . (9.148)
0 2