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We present a theoretical and numerical framework -- which we dub the DirectIntegration Method (DIM) -- for simple, efficient and accurate evaluation ofsurface wave models allowing presence of a current of arbitrary depthdependence, and where bathymetry and ambient currents may vary slowly inhorizontal directions. On horizontally constant water depth and shear currentthe DIM numerically evaluates the dispersion relation of linear surface wavesto arbitrary accuracy, and we argue that for this purpose it is superior to twoexisting numerical procedures: the piecewise-linear approximation and a methoddue to \textit{Dong \& Kirby} [2012]. The DIM moreover yields the fulllinearized flow field at little extra cost. We implement the DIM numericallywith iterations of standard numerical methods. The wide applicability of theDIM in an oceanographic setting in four aspects is shown. Firstly, we show howthe DIM allows practical implementation of the wave action conservationequation recently derived by \textit{Quinn et al.} [2017]. Secondly, wedemonstrate how the DIM handles with ease cases where existing methodsstruggle, i.e.\ velocity profiles $\mathbf{U}(z)$ changing direction withvertical coordinate $z$, and strongly sheared profiles. Thirdly, we use the DIMto calculate and analyse the full linear flow field beneath a 2D ring wave upona near--surface wind--driven exponential shear current, revealing strikingqualitative differences compared to no shear. Finally we demonstrate that theDIM can be a real competitor to analytical dispersion relation approximationssuch as that of \textit{Kirby \& Chen} [1989] even for wave/ocean modelling.

A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters