There can exist waves that propagate along a free surface of helium II, but are damped in the direction going into the liquid. This phenomenon is completely analogous to capillary waves on the surface of a classical liquid. Let us choose the z-axis along the normal to the surface, and denote by ζ the deviation of the coordinates of points on the surface from their equilibrium values. If effects linked to the presence of vapor are neglected, then at the surface of the liquid the following boundary conditions should be satisfied

The formal flux of liquid across the surface should be equal to zero () ρ s v sz + ρ n v nz − ρ ζ ˙ = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429502897/a3547081-d1fe-4ca8-9f19-7cf8caff9ba3/content/eq428.tif"/>

The sum of the pressure forces and the surface tension should vanish (the surface is in the xy-plane) () p   −   γ ( ∂ 2 ζ ∂ x 2 + ∂ 2 ζ ∂ y 2 ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429502897/a3547081-d1fe-4ca8-9f19-7cf8caff9ba3/content/eq429.tif"/>

The entropy flux across the surface should vanish () v nz − ζ ˙ = 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429502897/a3547081-d1fe-4ca8-9f19-7cf8caff9ba3/content/eq430.tif"/>