On the use of Schrödinger’s equation in the analytic determination of horn reflectance

Abstract

The flaring horn has traditionally been modeled in one dimension using piecewise conical or cylindrical elements. Acoustic properties within each element are known, and scattering between the elements is computed. Under the piecewise model, a shape for the wavefront of the acoustic disturbance within the horn is implicitly assumed (planar for cylindrical elements, spherical for conical elements). For horns of significant flare, the true wavefront shape will be neither planar nor spherical. A more general model is thus desirable. Here an alternate model is presented: The flaring horn is modeled according to Webster’s equation. A change of variables transforms the equation into the form of the Schrödinger wave equation using in one-dimensional particle scattering. Boundary conditions can be derived directly from the physical dimensions of the horn, and the solution of the equation gives estimates of acoustic properties in terms of frequency dependent reflection and transmission coefficients. Here, Webster’s equation is solved along the entire length of the horn, with no lumped scattering. Advantages over piecewise modeling techniques include the ability to specify arbitrary axisymmetric wavefront shapes for the acoustic disturbance within the horn. Under appropriate assumptions for wavefront shapes, results converge to those obtained with traditional piecewise models.