On Plane and Spherical Waves in Horns with Nonunif orm Flare: I. Theory of Radiation, Resonance Frequencies, and Mode Conversion

The relation between axially symmetric plane waves in a cylindrical duct, and spherical waves in a conical horn, is reviewed initially as a basis for a study of waves in a horn of rapidly varying taper. The following terminology is adopted: s-waves have no nodes between the horn axis and the walls; p-waves have one such node; higher order waves have more nodes. In straight sided horns the s-wave is of all-pass nature, but there are high-pass cut-off frequencies for p-waves etc. A modified “Webster” pressure wave equation for p(z,t) is derived for (axially symmetric) s-waves and generalized to waves of higher order, assuming the wave fronts to be curved with area S(z) while gas density and bulk modulus are functions of z. The “reduced” equation (of Schrodinger type) for ψ = pr, with πr ² ≡ S is obtained as a convenient means for discussing certain later results. A variational calculation by Weibel is joined with experiment to justify a spherical representation of the wave front shape. The s-wave radiation and resonance frequency properties of a horn in spherical and plane wave approximation are compared. The inevitability of s-to-p wave mode conversion in a flaring horn is shown, and an upper bound obtained for the conversion rate. Considerable conversion takes place, e.g., in the last few centimeters of a trombone- like Bessel horn. Storage of (non-propagating) p-Wave energy modifies the resonance frequencies of a horn. Storage, plus poor radiation ability of p-waves at the horn mouth reduces the net radiation below that expected from pure s-waves. Correction techniques for horn wall perturbations, air temperature variations and boundary layer effects are presented for use in the interpretation of experimental data of real horns, as dealt with in Part II of this report.