# Dimensionless Formulation

The dimensionless formulation of the tidal equations is almost identical to the corresponding formulation of the oscillation equations, differing only in the definition of a couple of variables and a single boundary condition.

## Variables

The definitions of the \(y_{3}\) and \(y_{4}\) dependent variables, given in eqn. (8), are replaced by

\[\begin{split}\begin{align}
y_{3} &= x^{2-\ell}\, \frac{\tPsi'}{gr}, \\
y_{4} &= x^{2-\ell}\, \frac{1}{g} \deriv{\tPsi'}{r}.
\end{align}\end{split}\]

## Boundary Conditions

The outer potential boundary condition, the second line of eqn. (9), is replaced by

\[\alphagrv U y_{1} + (\alphagrv \ell + 1) y_{3} + \alphagrv y_{4} = (2\ell+1) \yT,\]

where

(16)\[\yT \equiv x^{2 - \ell} \frac{\tPhiTlmk}{gr}.\]