Flexural support ratio ($\chi =D{k}^4/(\Delta \rho g)$)

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 isFor a sinusoidal Fourier mode of horizontal wavenumber magnitude $k$, applying the fourth spatial derivative multiplies its amplitude by $k^4$. HenceThe 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.