Substitute $A_\pm$ into the dynamic boundary condition, starting from $$ \rho_1\bigl(-ikcA_{-}+ikU_1A_{-}+g\zeta_0\bigr) = \rho_2\bigl(-ikcA_{+}+ikU_2A_{+}+g\zeta_0\bigr), $$ and using $$ kA_{-}=-(ikU_1-ikc)\zeta_0, \qquad kA_{+}=(ikU_2-ikc)\zeta_0, $$ divide by the common factor $\zeta_0$ $$ \rho_1\!\Bigl(( -ikc+ikU_1 )^{2}+gk\Bigr) = \rho_2\!\Bigl(( -ikc+ikU_2 )^{2}+gk\Bigr) $$

\(\rightarrow\)Two solutions $$ c=\frac{\rho_2U_2+\rho_1U_1}{\rho_2+\rho_1} \pm \left[ \left(\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\right)\frac{g}{k} - \frac{\rho_2\rho_1}{(\rho_2+\rho_1)^2}(U_2-U_1)^2 \right]^{1/2} $$

Neutral stability occurs when the term inside the square root is non-negative. Instability (exponential growth, $c_i>0$) occurs when $$ \left(\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\right)\frac{g}{k} < \frac{\rho_2\rho_1}{(\rho_2+\rho_1)^{2}}(U_2-U_1)^{2} $$ or equivalently $$ g(\rho_2^{2}-\rho_1^{2}) < k\rho_1\rho_2(U_2-U_1)^{2} $$ This occurs when the free-stream velocity difference is high enough the density difference is small enough or the wave number $k$ is large enough