Integral expressions for mountain wave steepness

Expressions are derived for the wave steepness ηz of mountain waves, defined as the altitude (z) derivative of the vertical displacement η. The derivation begins with a known Fourier integral for η. The altitude derivative of the Fourier integral for η introduces a singularity in the integrand that is not absolutely integrable at any altitude. It is seen, nonetheless, that the altitude derivative can be moved under the integral sign for altitudes above the ground, but not at the ground, for which a special expression is found relating wave steepness to the mountain height function. It is shown that the upwind limit of wave steepness at the ground is zero, but the downwind limit is nonzero in general, and an expression for the downwind limit in terms of the mountain height function is given. It is also found that the imaginary part of complex wave steepness diverges approaching the ground, and an asymptotic formula is given. The results are applied to three mountain topography shapes: elliptical Gaussian, circular with algebraic decay, and infinite ridge.