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This thesis is primarily concerned with a new method for the approximate evaluation of Fourier sine and cosine transforms. A problem of linear surface waves, discussed by Forbes, initially gave rise to a singular integrodifferential equation over the real line. We have been able to transform this integrodifferential equation into a linear second order differential equation. The solution of this differential equation has been found explicitly in terms of Fourier sine and cosine transforms of simple rational functions. However, the integrands of these integrals decay algebraically rather than exponentially and this leads to problems with their approximate evaluation. This is what has motivated the major part of this thesis. A commonly used technique of quadrature involves transforming the integral to one over the entire real line and then using the trapezoidal rule in order to approximate the transformed integral. These methods are characterised as to whether the transformed integrand has single exponential decay or double exponential decay. After a discussion of the literature, we have developed and analysed a new quadrature rule for Fourier sine and cosine transforms. A complete error analysis is made using contour integration and several examples are examined in detail. In particular, we consider an open problem posed by Ooura and Mori. The method we have developed is characterised by its simplicity. We conclude by considering again the linear surface waves problem.