Editing Isostasy (section)

=== Vening Meinesz / flexural ===
[[File:Local-regional isostasy - flexure, elastic thickness.jpg|thumb|Cartoon showing the isostatic vertical motions of the lithosphere (grey) in response to a vertical load (in green)]]
This hypothesis was suggested to explain how large topographic loads such as [[seamounts]] (e.g. [[Hawaiian Islands]]) could be compensated by regional rather than local displacement of the lithosphere. This is the more general solution for [[lithospheric flexure]], as it approaches the locally compensated models above as the load becomes much larger than a flexural wavelength or the flexural rigidity of the lithosphere approaches zero.<ref name=Watts2001/>{{sfn|Kearey|Klepeis|Vine|2009|pp=44-45}}

For example, the vertical displacement ''z'' of a region of ocean crust would be described by the [[differential equation]]

:<math>D\frac{d^4z}{dx^4}+(\rho_m-\rho_w)zg = P(x)</math>

where <math>\rho_m</math> and <math>\rho_w</math> are the densities of the aesthenosphere and ocean water, ''g'' is the acceleration due to gravity, and <math>P(x)</math> is the load on the ocean crust. The parameter ''D'' is the ''flexural rigidity'', defined as

:<math>D=ET^3_c/12(1-\sigma^2)</math>

where ''E'' is [[Young's modulus]], <math>\sigma</math> is [[Poisson's ratio]], and <math>T_c</math> is the thickness of the lithosphere.  Solutions to this equation have a characteristic wave number

:<math>\kappa=\sqrt[4]{(\rho_m-\rho_w)g/4D}</math>

As the rigid layer becomes weaker, <math>\kappa</math> approaches infinity, and the behavior approaches the pure hydrostatic balance of the Airy-Heiskanen hypothesis.{{sfn|Kearey|Klepeis|Vine|2009|p=45}}