It is shown that the "Webster" horn equation is an exact consequence of “one-parameter" or "1P" wave propagation. If a solution of the Helmholtz equation depends on a single spatial coordinate, that coordinate can be transformed to another coordinate, denoted by ƫ, which measures arc length along the orthogonal trajectories to the constant-e surfaces. Webster's equation, with ƫ as the axial coordinate, holds inside a tube of such orthogonal trajectories; the cross-sectional area in the equation is the area of a constant ƫ cross-section. This derivation of the horn equation makes no explicit assumption concerning the shape of the wavefronts. It is subsequently shown, however, that the wavefronts must be planar, circular-cylindrical or spherical, so that no new geometries for exact 1P acoustic waveguides remain to be discovered.
It is shown that if the linearized acoustic field equations are written in arbitrary curvilinear orthogonal coordinates and approximated by replacing all spatial derivatives by finite-difference quotients, the resulting equations can be interpreted as the nodal equations of a three-dimensional L-C network. This "finite-difference equivalent-circuit" or "FDEC" model can be truncated at the boundaries of the simulated region and terminated to represent a wide variety of boundary conditions. The presence of loosely-packed fibrous damping materials can be represented by using complex values for the density and ratio of specific heats of the medium. These complex quantities lead to additional components in the FDEC model.
Two examples of FDEC models are given. The first example predicts the frequency response of a moving-coil loudspeaker in a fiberglass-filled box, showing the effects of internal resonances. Variations of the model show how the properties of the fiberglass contribute to the damping of resonances and the shaping of the frequency response. It is found that viscosity, rather than heat conduction, is the dominant mechanism of damping. The second example addresses the classical problem of radiation from a circular rigid piston, and confirms that a free-air anechoic radiation condition with oblique incidence can be successfully represented in the FDEC model.