Surface Tension

(11.112) |

where is small. Thus, the unperturbed interface corresponds to the plane . The unit normal to the interface is

(11.113) |

It follows that

The Young-Laplace Equation (see Section 3.2) yields

where is the jump in pressure seen crossing the interface in the opposite direction to . However, from Equations (11.114) and (11.115), we have

(11.117) |

Hence, Equation (11.116) gives

This expression is the generalization of Equation (11.99) that takes surface tension into account.

Suppose that the interface in question is that between a body of water, of density and depth , and the atmosphere. Let the unperturbed water lie between and , and let the unperturbed atmosphere occupy the region . In the limit in which the density of the atmosphere is neglected, the pressure in the atmosphere takes the fixed value , whereas the pressure just below the surface of the water is . Here, is the pressure perturbation due to the wave. The relation (11.118) yields

(11.119) |

where is the surface tension at an air/water interface. However, , where is the perturbed velocity potential of the water. Moreover, from Equation (11.9), . Hence, the previous expression gives

(11.120) |

This relation, which is a generalization of Equation (11.15), is the condition satisfied at a free surface in the presence of non-negligible surface tension. Applying this boundary condition to the general solution, (11.19) (which already satisfies the boundary condition at the bottom), we obtain the dispersion relation

which is a generalization of Equation (11.21) that takes surface tension into account.