This paper discusses solutions to the differential equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700045007/resource/name/S0004972700045007_eqnU1.gif?pub-status=live)
which governs the height
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700045007/resource/name/S0004972700045007_eqnU2.gif?pub-status=live)
of the non-dimensional water level inside a conventional tide-well when the corresponding height of the sea-level outside is X(τ). A perturbation solution correct to O(β4), for small β, for the particular case X = sinτ described previously has been extended to O(β6) for any function X(τ) differentiable at least three times. Two special cases, where X(τ) is a single sinusoid and the sum of two sinusoids, are treated in detail.