We apply the rigorous mathematical methods to investigate surface wave transformation in a basin with a variable bathymetry. The development of rigorous mathematical methods for the description of wave transformation in the coastal zone represents an important and topical problem both from the academic and practical points of view, especially in the application of the protection of the Australian coastline from the hazardous impacts of oceanic waves. We consider long linear wave transformations in a coastal zone where water depth can vary arbitrarily smoothly. The derived system of differential equations that describes the changes of wave speed and free surface elevation are solved exactly for the different types of bottom profile. New analytical solutions in terms of special functions for wave transformation were obtained. The transformation coefficients were derived as the functions of wave frequency and the total depth drop for three typical models: piece-linear, piecequadratic, and hyperbolic tangential depth profiles. For all these cases we have derived exactly solvable equations and obtained their solutions in terms of hypergeometric functions. The influence of the governing parameters on the type of wave transformation was analysed. We show that the results obtained are in agreement with the energy flux conservation and Lamb's formula in the limiting case of zero frequency.