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Solutions to Long's equation for a stably stratified incompressible fluid traversing a mountain range are obtained for various terrain shapes and amplitudes when the horizontal scale is large compared to the vertical wavelength. Nonlinear lower and upper (radiative) boundary conditions are utilized and found to have a strong influence on the wave structure at large amplitudes. The results for symmetric and asymmetric mountain profiles reveal that the wave amplitude and wave drag are significantly enhanced for mountains with gentle windward and steep leeward slopes. These results confirm and explain those obtained by Raymond (1972) using a different solution method. Several results obtained by Smith (1977) from perturbation analysis are also confirmed and extended to large amplitudes. The methods are also applied to investigate the nonlinear nature of the singularity predicted by linear theory for flow over a step.
Lilly et al. (Wed,) studied this question.