Page 314 - Physical Principles of Sedimentary Basin Analysis

296 Flexure of the lithosphere
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d w
D + ( m − c ) g w = q 0 cos(2πx/λ) (9.56)
dx 4
which can be solved by trying a solution of the same form as the load, w(x) =
w 0 cos(2πx/λ). When an expression of this form is inserted into equation (9.56)for the
deflection we get

4
D(2π/λ) + g w 0 = q 0 (9.57)
which gives the coefficient w 0 , and the deflection from the periodic load is
q 0 cos(2πx/λ)
w(x) = . (9.58)
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D(2π/λ) + g
Note 9.5 shows that solution (9.58) can be used to construct a Fourier series solution for
the deflection of nearly any load. An inspection of the deflection (9.58) shows that loads
with sufficiently long wavelengths to fulfill the condition
4
D(2π/λ)
g (9.59)
will experience a deflection that is close to isostatic subsidence
q 0 cos(2πx/λ)
w iso (x) ≈ . (9.60)
g
The wavelength of the load is then sufficiently large for the support from the elastic strength
of the plate to be unimportant, and it is the buoyancy of the plate that controls the deflection.
The condition (9.59) can be rewritten as λ
λ c , where the critical wavelength λ c is

1/4
D
λ c = 2π . (9.61)
g
The deflection (9.58) is almost zero in the opposite regime, λ
λ c , where the elastic
strength of the plate is supporting the load. The critical wavelength λ c therefore defines
a length scale where the periodic load is partly supported by the elastic strength of the
plate and partly by the buoyancy of the displaced mantle. The degree of isostatic equilib-
rium can be measured by the ratio of deflection over isostatic subsidence, w(x)/w iso (x),
which is
w(x) 1
c = = . (9.62)
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w iso (x) (λ c /λ) + 1
Figure 9.10ashowsaplotof c as a function of the scaled wavelength λ/λ c . An important
3
parameter in the critical wavelength is the flexural rigidity of the plate, D = Eh /(12(1 −
2
ν )), which is controlled by the effective elastic thickness. For example, the parameters
h = 30 km, E = 70 GPa, ν = 0.25, = 600 kg m −3 give that λ c = 46 km. Figure 9.10b
also shows this, which plots the critical wavelength λ c as a function of the plate thickness h.
The distance from the center of the Hawaiian island chain to the flexural arch is estimated to
be ∼250 km, which is less than λ c = 400 km. These islands are therefore to a large degree
supported by the elastic strength of the plate. On the other hand, large-scale structures