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Theoretical amplification

For the most general case, it's to say with surface tension and gravity, the equations become :

$\displaystyle \begin{array}{c}
kB_1+(\sigma + ik \frac{U}{2})A=0\\
...ho_2(\sigma- ik \frac{U}{2})B_2+((\rho_2-\rho_1)g
+\gamma k^2)A

The determinant give the compatibility equation. Finaly we have :

$\displaystyle (\sigma + ik \frac{U}{2} \frac{\rho_2-\rho_1}{\rho_2+\rho_1})^2=
\frac{\rho_2-\rho_1}{\rho_2+\rho_1}gk-\frac{\gamma k^3}{\rho_2+\rho_1}

So if the condition of instability is verified, the real part of $ \sigma$ is positive.

Without surface tension we have a curve of the amplification coefficient which depends on the wave number on figure 4.6

Figure 4.6: Amplification coefficient for $ U=0.2$, with surface tension and without
\includegraphics [scale=0.5]{ampli.eps} \includegraphics [scale=0.5]{amplit.eps}

Stephanie Terrade
Julien Delbove