Consider a gravity wave of angular frequency , and wavenumber , propagating through both fluids in the -direction. Let

(11.94) |

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 (11.91), with the constants and chosen such that and . It follows that and , so that

(11.95) |

In the upper fluid, the perturbed velocity potential must again be of the form (11.91), with the constants and chosen such that and . It follows that and , so that

(11.96) |

Here, is the phase velocity of the wave. From Equations (11.85) and (11.92), the fluid pressure just below the interface is

Likewise, the fluid pressure just above the interface is

In the absence of surface tension at the interface, these two pressure must equal one another: that is,

Hence, we obtain the dispersion relation

which takes the form of a quadratic equation for the phase velocity, , of the wave. We can see that:

- If and then the dispersion relation reduces to (11.43) (with ).
- If the two fluids are of infinite depth then the dispersion relation simplifies to
(11.101)

- In general, there are two values of that satisfy the quadratic equation (11.100). These are either both real, or form a complex conjugate pair.
- The condition for stability is that is real. The alternative is that is complex, which implies that is also complex, and, hence, that the perturbation grows or decays exponentially in time. Because the complex roots of a quadratic equation occur in complex conjugate pairs, one of the roots always corresponds to an exponentially growing mode. In other words, an instability.
- If both fluids are at rest (i.e.,
), and of infinite depth, then the dispersion reduces to
(11.102)

It follows that the configuration is only stable when : that is, 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 be the specific density of air at s.t.p. (relative to water). Putting , , and making use of the fact that is small, the dispersion relation (11.100) yields

(11.103) |

Comparing this with Equation (11.43), 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.