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EARTH 135 Lecture Notes - Lecture 7: Bending, Boundary Value Problem, BathymetryExam

Earth Science
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Figure 1: Figure 1: Seamount loading. xpis the distance to the bulge.
Lecture 7 : Plate Flexure due to a Seamount
1. Buoyancy: The equation for plate flexure by seamounts is slightly different from the
one for bending plates. This difference is due to the buoyancy force. The new
equation is
DW ′′′′ + (ρmρw)gW =q(x) (1)
where the second term on the lefthand side is the buoyancy term. ρmis density of
mantle and ρwis density of sea water. We assume a uniform half-space for the solid
earth, ignoring the crust.
2. The vertical load by the seamount is treated as a point force at x= 0 and will be
included in the boundary condition. The equation is then
DW ′′′′ + (ρmρw)gW = 0 (2)
3. General solutions: For a more detailed derivation, see the later section in pages 3
and 4. Here we note that an equation of the form
W′′′′ +4
α4W= 0 (3)
has four independent solutions that can be written
ex/α cos(x/α), ex/α sin(x/α), ex/α cos(x/α), and ex/α sin(x/α).
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