A conformal-mapping technique for topographic-wave problems: semi-infinite channels and elongated basins

The basis of the conformal-mapping method for topographic-wave problems of Johnson (1985) is discussed in greater detail by considering the invariance under conformal mapping of the linear, barotropic, potential-vorticity equation, noted in Davis (1983). A method is presented for using this property to construct further solutions for waves propagating over varying topography. Results are given for semi-infinite channels and elongated basins. A coordinate system is constructed that approaches a Cartesian system exponentially fast with distance from end-walls. For exponentially sloping topography the solutions for infinite channels, semi-infinite channels, and basins have the same structure and dispersion relation as waves in an elliptical basin, discussed in Johnson (1987). The structures presented there can thus be considered as in some sense universal for exponentially sloping topography.