Table 1 Dispersion relations for different combinations of compressibility, elasticity and gravity effects in the water column.

(1)

✓

✓

✓

\(r\,\tanh (r)=\frac{{\omega }^{2}\left({\varepsilon }_{1}+{\varepsilon }_{2}\right)}{{\varepsilon }_{1}+{\varepsilon }_{2}{\omega }^{4}/{r}^{2}+\beta {\gamma }_{l}/{r}^{2}}\)

\({r}^{2}+2{\gamma }_{l}{\omega }^{2}-{\gamma }_{l}^{2}\)

(2)

✓

✓

⨯

\(r\,\tanh (r)=\frac{{\omega }^{2}\left({\varepsilon }_{1}+{\varepsilon }_{2}\right)}{{\varepsilon }_{1}+{\varepsilon }_{2}{\omega }^{4}/{r}^{2}}\quad {\gamma }_{i}\to 0\)

\({r}^{2}+2{\gamma }_{l}{\omega }^{2}\)

(3)

✓

⨯

✓

\(r\,\tanh (r)=\frac{{\omega }^{2}}{1+{\gamma }_{l}/{r}^{2}({\omega }^{2}-{\gamma }_{l})}\)

\({r}^{2}+2{\gamma }_{l}{\omega }^{2}-{\gamma }_{l}^{2}\)

(4)

⨯

⨯

✓

\(r\,\tanh (r)=\frac{{\omega }^{2}}{1+{\gamma }_{l}/{r}^{2}({\omega }^{2}-{\gamma }_{l})}\)

\({r}^{2}-{\gamma }_{l}^{2}\)

(5)

✓

⨯

⨯

\(r\,\tanh (r)={\omega }^{2}\)

\({r}^{2}+2{\gamma }_{l}{\omega }^{2}\)

(6)

⨯

⨯

⨯

\(r\,\tanh (r)={\omega }^{2}\)

\({r}^{2}\)