Elastic-cover frequency ratio for water waves ($F(k{)}^2=[1+B{k}^4/(\rho g+\sigma k^2)]/[1+(a{\rho }_s/\rho )k]$)

In the specified thin-sheet deep-water model, bending stiffness adds restoring force while sheet mass adds inertia. Their ratio to an uncovered gravity-capillary wave gives the displayed expression. Added inertia first lowers frequency at small wavenumber; at formally large wavenumber bending produces a frequency ratio proportional to $k^{1/2}$. That formal asymptote must not be extrapolated past the range where a thin-sheet continuum description is valid.