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

(1200) |

(1201) |

(1202) |

Likewise, the fluid pressure just above the interface is

Now, in the absence of surface tension at the interface, these two pressure must equal one another:

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 (1149) (with ).
- If the two fluids are of infinite depth then the dispersion relation simplifies to

(1207) - In general, there are two values of that satisfy the quadratic equation (1206). 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. 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. - If both fluids are at rest (
*i.e.*, ), and of infinite depth, then the dispersion reduces to

(1208) *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
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 (1206) yields

(1209) |