A localized wave propagating horizontally at speedcthrough a fluid of arbitrary depthHwhich consists of a lower vertical layer (thicknessL) with a weak‐depth‐dependent density below an upper layer of constant density is considered. The flow is characterized by Long’s equation for the stream function &psgr;. Exact analytic results are obtained for &psgr; throughO[1/(h−1)] and the wave speed through orderO[1/(h−1)2],h=H/L. The terms in &psgr; ofO[1/(h−1)2] are obtained numerically. Using these results the range of validity of such a perturbative approach is examined.