Page 308 - Physical Principles of Sedimentary Basin Analysis

290 Flexure of the lithosphere

The load is divided by 2 because the integral is over only half the plate. The integral (9.32)
shows that V 0 is the surface part of the load, which is the part that is above z = 0. The
rest of the load, which is below z = 0, is the part that is filling the deflection. The solution
for the deflection tells us how deep the deflection becomes when it is filled with rock of
density c .Itisshown in Note 9.2 that the solution to equation (9.31), with these four
boundary conditions, is

x
x
x
w(ˆ) = w max e −ˆ x (cos ˆ + sin ˆ) (9.33)
in terms of the dimensionless coordinate ˆ = x/α, where α is the characteristic length
x

1/4
4D
α = , (9.34)
g
and the maximum deflection at x = 0is

V 0
w max = . (9.35)
2 gα
Notice that the length scale α is independent of the load. The deflection (9.33) is plotted
in Figure 9.7. The half-width ˆ 0 of the depression is given by w(ˆx 0 ) = 0, which has the
x
solution
3π
ˆ x 0 = . (9.36)
4
Figure 9.7 shows that the point load also creates a forebulge at the distance x 1 given by
dw/d ˆ = 0. We have
x
dw x
x
(ˆ 1 ) =−w max e −ˆ 1 sin ˆ 1 = 0 (9.37)
x
d ˆ x
which has the solution ˆ 1 = π, and the height of the forebulge then becomes
x
w 1 = w max e −π .
The distance to the forebulge is found to be roughly x 1 ≈ 250 km for the Hawaiian island
chain, see Figure 9.6, a distance that can be used to estimate the elastic thickness of the
lithosphere below the islands. The characteristic length scale becomes α = x 1 /π = 80 km.
The parameters E = 100 GPa, ν = 0.25, = 600 kg m −3 give that the elastic thickness
of the lithosphere is h = 30 km. Such an estimate is denoted the effective elastic thickness.
It is the thickness an homogeneous plate would have, a plate with the same linear elastic
properties everywhere. The effective thickness is not a property of a plate that can be
directly observed, and it is normally used as a means of comparing elastic properties of
different plates.
Observations of the maximum deflection w max can in combination with the characteristic
length scale α be used to estimate the size of the point load. It follows from (9.35) that the
point load is