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An effective surface equation, that encapsulates the detail of amicrostructure, is developed to model microstructured surfaces. The equationsdeduced accurately reproduce a key feature of surface wave phenomena, createdby periodic geometry, that are commonly called Rayleigh-Bloch waves, but whichalso go under other names such as Spoof Surface Plasmon Polaritons inphotonics. Several illustrative examples are considered and it is shown thatthe theory extends to similar waves that propagate along gratings. Line sourceexcitation is considered and an implicit long-scale wavelength is identifiedand compared to full numerical simulations. We also investigate non-periodicsituations where a long-scale geometric variation in the structure isintroduced and show that localised defect states emerge which the asymptotictheory explains.

Asymptotic theory of microstructured surfaces: An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces