Small-load approximation	$\frac{\Delta \tilde{f}}{f_0}=\frac{\mathrm{i}}{\pi Z_{\mathrm{q}}}{\tilde{Z}}_{\mathrm{L}}=\frac{\mathrm{i}}{\pi Z_{\mathrm{q}}}\frac{\langle -{\widehat{\sigma }}_{\mathrm{S}}{\rangle }_{\mathrm{area}}}{{\widehat{v}}_{\mathrm{S}}}$
Gordon-Kanazawa equation, semi-infinite viscoelastic medium	$\frac{\Delta \tilde{f}}{f_0}=\frac{\mathrm{i}}{\pi Z_{\mathrm{q}}}{\tilde{Z}}_{\mathrm{L}}=\frac{-1+\mathrm{i}}{\sqrt{2}\pi Z_{\mathrm{q}}}\sqrt{\mathsf{\omega }}\sqrt{\mathsf{\rho }\tilde{\mathsf{\eta }}} \mathrm{i}\mathsf{\omega }\mathsf{\rho }\tilde{\mathsf{\eta }}=\mathsf{\rho }\tilde{G}=-{\left(\pi Z_{\mathrm{q}}\frac{\Delta \tilde{f}}{f_0}\right)}^2$
Sauerbrey equation, rigid sample	$\frac{\Delta \tilde{f}}{f_0}\approx -\frac{2n{f}_0}{Z_{\mathrm{q}}}{m}_{\mathrm{f}}=-n\frac{\langle m_{\mathrm{f}}\rangle {}_{\mathrm{area}}}{m_{\mathrm{q}}} \frac{\Delta f}{f_{\mathrm{ref}}}=-\frac{m_{\mathrm{f}}}{m_{\mathrm{q}}}$
Viscoelastic film in air	$\frac{\Delta \tilde{f}}{f_0}=\frac{-1}{\pi Z_{\mathrm{q}}}{\tilde{Z}}_{\mathrm{f}}\tan \left({\tilde{k}}_{\mathrm{f}}{d}_{\mathrm{f}}\right)$
Thin film in air	$\frac{\Delta \tilde{f}}{f_0}=\frac{-\mathsf{\omega }{m}_{\mathrm{f}}}{\pi Z_{\mathrm{q}}}\left[1+\frac{{(n\pi )}^2}{3}\left(\frac{{\tilde{J}}_{\mathrm{f}}}{{\mathsf{\rho }}_{\mathrm{f}}}{Z}_{\mathrm{q}}^2-1\right){\left(\frac{m_{\mathrm{f}}}{m_{\mathrm{q}}}\right)}^2\right]$
Viscoelastic film in liquid	$\frac{\Delta \tilde{f}}{f_0}=\frac{-{\tilde{Z}}_f}{\pi Z_{\mathrm{q}}}\frac{{\tilde{Z}}_{\mathrm{f}}\tan \left({\tilde{k}}_{\mathrm{f}}{d}_{\mathrm{f}}\right)-\mathrm{i}{\tilde{Z}}_{\mathrm{bulk}}}{{\tilde{Z}}_{\mathrm{f}}+\mathrm{i}{\tilde{Z}}_{\mathrm{bulk}}\tan \left({\tilde{k}}_{\mathrm{f}}{d}_{\mathrm{f}}\right)}$
Thin film in liquid	$\begin{array}{c}\frac{\Delta \tilde{f}}{f_0}\approx \frac{-\mathsf{\omega }{m}_{\mathrm{f}}}{\pi Z_{\mathrm{q}}}\left[1-n{\tilde{J}}_{\mathrm{f}}(\mathsf{\omega })2\pi \mathrm{i}\frac{1}{{\mathsf{\rho }}_{\mathrm{f}}}{f}_0{\mathsf{\rho }}_{\mathrm{bulk}}{\mathsf{\eta }}_{\mathrm{bulk}}\right]\\ \frac{\Delta \mathsf{\Gamma }}{-\Delta f}\approx {J_{\mathrm{f}}}^{\prime }\mathsf{\omega }{\mathsf{\eta }}_{\mathrm{bulk}}\end{array}$
Thin adsorbate in liquid	$\frac{\Delta \tilde{f}}{f_0}=\frac{-\mathsf{\omega }}{{\mathsf{\pi }Z}_{\mathrm{q}}}{\mathsf{\rho }}_{\mathrm{bulk}}{{\displaystyle \int }}_0^{\infty }\left[\frac{\mathsf{\rho }\left(z\right)}{{\mathsf{\rho }}_{\mathrm{bulk}}}-\frac{{\mathsf{\eta }}_{\mathrm{bulk}}}{\tilde{\mathsf{\eta }}\left(z\right)}\right] \mathrm{d}z$
Point contacts	$\Delta \tilde{f}=\frac{1}{n}\frac{n_{\mathrm{P}}}{A_{\mathrm{eff}}}\frac{1}{2{\pi }^2{Z}_{\mathrm{q}}}{\tilde{\mathsf{\kappa }}}_{\mathrm{P}}$
Time averaging, large amplitudes, nonlinear response	$\frac{\Delta \tilde{f}}{f_0}=\frac{\mathrm{i}}{\pi Z_{\mathrm{q}}}\frac{2{\langle -{\widehat{\sigma }}_{\mathrm{S}}(t)\exp (\mathrm{i}\mathsf{\omega }t)\rangle }_{\mathrm{area},\mathrm{time}}}{{\widehat{v}}_{\mathrm{S}}}$
Small-load approximation in tensor form	$\Delta \tilde{f} \approx \frac{\mathrm{i}}{{4\mathsf{\pi }\mathsf{\rho }}_{\mathrm{q}}}\frac{{\sum }_{ijk}{\int }_{\mathrm{Surface}}{\widehat{u}}_{\mathrm{S},\mathrm{i}}^{\left(0\right)}\left({\mathbf{r}}_{\mathrm{S}}\right)n_j{\tilde{Z}}_{\mathrm{L},ijk}\left({\mathbf{r}}_{\mathrm{S}}\right){\widehat{u}}_{\mathrm{S},\mathrm{k}}^{\left(0\right)}\left({\mathbf{r}}_{\mathrm{S}}\right) {\mathrm{d}}^2{\mathbf{r}}_{\mathrm{S}}}{{\sum }_{\mathrm{i}}{\int }_{\mathrm{Volume}}{\widehat{u}}_{\mathrm{i}}^{\left(0\right)}\left(\mathbf{r}\right) {\widehat{u}}_{\mathrm{i}}^{\left(0\right)}\left(\mathbf{r}\right){\mathrm{d}}^3\mathbf{r}}$
Piezoelectric stiffening	$\frac{\Delta {\tilde{f}}_{\mathrm{PE}}}{f_0}=\frac{\mathrm{i}}{\pi Z_{\mathrm{q}}}\left[\frac{4e_{26}^2{A}_{\mathrm{eff}}}{d_{\mathrm{q}}^2}\right]\left({\left(\mathrm{i}\mathsf{\omega }{C}_0+{\tilde{Z}}_{\mathrm{ext}}^{-1}\right)}^{-1}-{\left(\mathrm{i}\mathsf{\omega }{C}_0+{\tilde{Z}}_{\mathrm{ext},\mathrm{ref}}^{-1}\right)}^{-1}\right)$