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Gravity Waves at an Interface

Consider a layer of fluid of density $\rho'$, depth $d'$, and uniform horizontal velocity $V'$, situated on top of a layer of another fluid of density $\rho$, depth $d$, and uniform horizontal velocity $V$. Suppose that the fluids are bounded from above and below by rigid horizontal planes. Let these planes lie at $z=-d$ and $z=d'$, and let the unperturbed interface between the two fluids lie at $z=0$. See Figure 93.

Figure 93: Gravity waves at an interface between two immiscible fluids.
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Consider a gravity wave of angular frequency $\omega$, and wavenumber $k$, propagating through both fluids in the $x$-direction. Let

\begin{displaymath}
\zeta(x,t)= \zeta_0\,\sin(\omega\,t-k\,x)
\end{displaymath} (1200)

be the small vertical displacement of the interface due to the wave. In the lower fluid, the perturbed velocity potential must be of the form (1197), with the constants $A$ and $B$ chosen such that $\left.v_z\right\vert _{z=-d}=0$ and $\xi_z(x,0,t)=\zeta(x,t)$. It follows that $A=-(V-c)\,\zeta_0/\tanh(k\,d)$ and $B=-(V-c)\,\zeta_0$, so that
\begin{displaymath}
\phi(x,z,t) = -(V-c)\,\zeta_0\,\frac{\cosh[k\,(z+d)]}{\sinh(k\,d)}\,\cos(\omega\,t-k\,x).
\end{displaymath} (1201)

In the upper fluid, the perturbed velocity potential must again be of the form (1197), with the constants $A$ and $B$ chosen such that $\left.v_z\right\vert _{z=d'}=0$ and $\xi_z(x,0,t)=\zeta(x,t)$. It follows that $A=(V'-c)\,\zeta_0/\tanh(k\,d')$ and $B=-(V'-c)\,\zeta_0$, so that
\begin{displaymath}
\phi(x,z,t) = (V'-c)\,\zeta_0\,\frac{\cosh[k\,(z-d')]}{\sinh(k\,d')}\,\cos(\omega\,t-k\,x).
\end{displaymath} (1202)

Here, $c=\omega/k$ is the phase velocity of the wave. From Equations (1191) and (1198), the fluid pressure just below the interface is
$\displaystyle p(x,0_-,t)$ $\textstyle =$ $\displaystyle p_0-\rho\,g\,\zeta + p_1(x,0_-,t)$  
  $\textstyle =$ $\displaystyle p_0 -\rho\,g\,\zeta_0\,\sin(\omega\,t-k\,x)-\rho\,k\,(V-c)\,A\,\sin(\omega\,t-k\,x)$  
  $\textstyle =$ $\displaystyle p_0-\left[\rho\,g-\frac{\rho\,k\,(V-c)^2}{\tanh(k\,d)}\right]\zeta_0\,\sin(\omega\,t-k\,x).$ (1203)

Likewise, the fluid pressure just above the interface is
$\displaystyle p(x,0_+,t)$ $\textstyle =$ $\displaystyle p_0-\rho'\,g\,\zeta + p_1(x,0_+,t)$  
  $\textstyle =$ $\displaystyle p_0 -\rho'\,g\,\zeta_0\,\sin(\omega\,t-k\,x)-\rho'\,k\,(V'-c)\,A\,\sin(\omega\,t-k\,x)$  
  $\textstyle =$ $\displaystyle p_0-\left[\rho'\,g+\frac{\rho'\,k\,(V'-c)^2}{\tanh(k\,d')}\right]\zeta_0\,\sin(\omega\,t-k\,x).$ (1204)

Now, in the absence of surface tension at the interface, these two pressure must equal one another: i.e.,
\begin{displaymath}
\left[p\right]_{z=0_-}^{z=0_+}=0.
\end{displaymath} (1205)

Hence, we obtain the dispersion relation
\begin{displaymath}
(\rho-\rho')\,g = \frac{k\,\rho\,(V-c)^2}{\tanh(k\,d)}+ \frac{k\,\rho'\,(V'-c)^2}{\tanh(k\,d')},
\end{displaymath} (1206)

which takes the form of a quadratic equation for the phase velocity, $c$, of the wave. We can see that:
  1. If $\rho'=0$ and $V=0$ then the dispersion relation reduces to (1149) (with $v_p=c$).
  2. If the two fluids are of infinite depth then the dispersion relation simplifies to
    \begin{displaymath}
(\rho-\rho')\,g = k\,\rho\,(V-c)^2+k\,\rho'\,(V'-c)^2.
\end{displaymath} (1207)

  3. In general, there are two values of $c$ that satisfy the quadratic equation (1206). These are either both real, or form a complex conjugate pair.
  4. The condition for stability is that $c$ is real. The alternative is that $c$ is complex, which implies that $\omega$ is also complex, and, hence, that the perturbation grows or decays exponentially in time. Since the complex roots of a quadratic equation occur in complex conjugate pairs, one of the roots always corresponds to an exponentially growing mode: i.e., an instability.
  5. If both fluids are at rest (i.e., $V=V'=0$), and of infinite depth, then the dispersion reduces to
    \begin{displaymath}
c^2 = \frac{g\,(\rho-\rho')}{k\,(\rho+\rho')}.
\end{displaymath} (1208)

    It follows that the configuration is only stable when $\rho>\rho'$: i.e., when the heavier fluid is underneath.

As a particular example, suppose that the lower fluid is water, and the upper fluid is the atmosphere. Let $s=\rho'/\rho=1.225\times 10^{-3}$ be the specific density of air at s.t.p. (relative to water). Putting $V=V'=0$, $d'\rightarrow \infty$, and making use of the fact that $s$ is small, the dispersion relation (1206) yields

\begin{displaymath}
c\simeq (g\,d)^{1/2}\left[\frac{\tanh(k\,d)}{k\,d}\right]^{1/2}\left\{1-\frac{1}{2}\,s\,[1+\tanh(k\,d)]\right\}.
\end{displaymath} (1209)

Comparing this with (1149), we can see that the presence of the atmosphere tends to slightly diminish the phase velocities of gravity waves propagating over the surface of a body of water.


next up previous
Next: Steady Flow over a Up: Waves in Incompressible Fluids Previous: Gravity Waves in a
Richard Fitzpatrick 2012-04-27