= Flexural support ratio
{title2=$\chi=Dk^4/(\Delta\rho g)$}
In the linear flat-<elastic plate> model, let downward deflection be $w$, applied downward <pressure> be $q$, <flexural rigidity> be $D$, and the restoring <geophysical density contrast> be $\Delta\rho>0$. The static balance is
$$
D\nabla^4w+\Delta\rho g w=q.
$$
For a sinusoidal <Fourier mode> of horizontal <wavenumber> magnitude $k$, applying the fourth spatial derivative multiplies its amplitude by $k^4$. Hence
$$
\widehat w=\frac{\widehat q}{Dk^4+\Delta\rho g},\qquad
\frac{\widehat w}{\widehat w_{\rm local}}=\frac{1}{1+\chi},\qquad
\chi=\frac{Dk^4}{\Delta\rho g}.
$$
The displayed dimensionless ratio compares bending resistance with buoyant restoration. Large $\chi$ means substantial elastic support and suppressed deflection. Small $\chi$ gives nearly local <isostasy>. The associated length $L_f=(D/\Delta\rho g)^{1/4}$ makes the load-width dependence explicit; some conventions define a flexural parameter with an additional factor $4^{1/4}$. At zero <wavenumber>, bending contributes nothing: total support remains buoyant even when the spatial distribution of subsidence changes.
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