  We extend unsteady thin aerofoil theory to model aerofoils with generalised
chordwise porosity distributions. The analysis considers a linearised porosity
boundary condition where the seepage velocity through the aerofoil is related
to the local pressure jump across the aerofoil surface and to the unsteady
characteristics of the porous medium. Application of the Plemelj formulae to
the resulting boundary value problem yields a singular Fredholm--Volterra
integral equation which does not admit an analytic solution. Accordingly, we
develop a numerical solution scheme by expanding the bound vorticity
distribution in terms of appropriate basis functions. Asymptotic analysis at
the leading- and trailing-edges reveals that the appropriate basis functions
are weighted Jacobi polynomials whose parameters are related to the porosity
distribution. The Jacobi polynomial basis enables the construction of a
numerical scheme that is accurate and rapid, in contrast to the standard choice
of Chebyshev basis functions that are shown to {be unsuitable} for porous
aerofoils. Applications of the numerical solution scheme to discontinuous
porosity profiles, quasi-static problems, and the separation of circulatory and
non-circulatory contributions are presented. Analogues to the classical
Theodorsen and Sears functions are computed numerically, which show that an
effect of trailing-edge porosity is to reduce the amount of vorticity shed into
the wake, thereby reducing the magnitude of the circulatory lift. {Fourier
transform inversion of these frequency-domain functions produces porous
extensions to the Wagner and K\"{u}ssner functions for transient aerofoil
motions or gust encounters, respectively.}
