  An important and widely neglected aspect of the interaction between an
accretion disc and a massive companion with a coplanar orbit is the vertical
component of the tidal force. As shown by Lubow, the response of the disc to
vertical forcing is resonant at certain radii, at which a localized torque is
exerted, and from which a compressive wave (p mode) may be emitted. Although
these vertical resonances are weaker than the corresponding Lindblad
resonances, the m=2 inner vertical resonance in a binary star is typically
located within the tidal truncation radius of a circumstellar disc.
  In this paper I develop a general theory of vertical resonances, allowing for
non-linearity of the response, and dissipation by radiative damping and
turbulent viscosity. The problem is reduced to a universal, non-linear ordinary
differential equation with two real parameters. Solutions of the complex
non-linear Airy equation are presented to illustrate the non-linear saturation
of the resonance and the effects of dissipation. It is argued that the m=2
inner vertical resonance is unlikely to truncate the disc in cataclysmic
variable stars, but contributes to angular momentum transport and produces a
potentially observable non-axisymmetric structure.
