Complete Solutions of the “Webster” Horn Equation

Suppose that one solution of the “Webster” equation for any given horn is known, valid for particular boundary conditions and at one frequency. Expressions for the other linearly independent solution and for the derivatives of both solutions with respect to frequency are deduced and given in computable form. This allows the behavior of the horn for any loading, and in a wide range of frequencies, to be derived from a solution with one loading, at one frequency. The originally known solution may be computed, or experimental, or it may be the initial wavefunction in the inverse method of horn design (where the horn contour is derived from the initial wavefunction, which is constructed to satisfy the design requirements). With the new results, it is straightforward to compute the behavior of horns that were designed by the inverse method (for example, the “Terminated Gaussian” and the “Fourier”) in conditions not envisaged in their design. Further, the inverse method can now be used to design new horns with specified characteristics at a number of loads (including dissipative toads) and frequencies. The impedances of horns are discussed in the light of the relation between the solutions of Webster's equation, with special reference to the ideal transformer. The general impedance relations are thereby considerably simplified. Some basic results of the theory of horns are briefly reviewed and an annotated bibliography of over 200 references is given. Attention is drawn to very important work on the horn equation and its solution—long before Webster—in particular to brilliant papers by Daniel Bernoulli, by Lagrange, and by Euler in the 18th century.