A Legendre-Galerkin spectral method is applied to the construction of atmospheric acoustic normal modes above level ground, represented by a complex impedance. A search in the complex plane for modal eigenvalues is replaced by a complex symmetric matrix eigenvalue problem. The Legendre-Galerkin spectral method projects the acoustic normal modes onto an orthogonal basis of Legendre polynomials. The matrix eigenvalue problem can be solved by readily available, public domain, software. Prior knowledge of the location of a set of nearby real eigenvalues is unnecessary since the complex symmetric matrix formulation embodies an approximation of all of the physical constraints of the problem. The normal modes are used to compute the acoustic field, due to a harmonic point source in the atmosphere, including a group of discrete modes radiating into the upper atmosphere, usually associated with the continuous spectrum. The validity of the acoustic field calculation is tested in a comparison with the fast field program and interpreted, with aid of the normal modes, in a downward-refracting atmosphere.