as it has been developed, is based on a series of assumptions and simplifications, but the resulting equations can still give useful information about the behavior. I will review the assumptions later, and discuss how well they hold up in practice. The problem of sound propagation in horns is a complicated one, and has not yet been rigorously solved analytically. Initially, it is a three-dimensional problem, but solving the wave equation in 3D is very complicated in all but the most elementary cases. The wave equation for three dimensions. This equation describes how sound waves of very small (infinitesimal) amplitudes behave in a three-dimensional medium. I will not discuss this equation, but only note that it is not easily solved in the case of horns. In 1919, Webster presented a solution to the problem by simplifying equation 1from a three-dimensional to a one-dimensional problem. He did this by assuming that the sound energy was uniformly distributed over a plane wave-front perpendicular to the horn axis, and considering only motion in the axial direction. The result of these simplications is the so-called “Webster’s Horn Equation,” which can be solved for a large number of cases: the wave number or spatial frequency (radians per meter),φ is the velocity potential, and S is the cross-sectional area of the horn as a function of x. The derivation of equation 2 is given in the appendix. You can use this equation to predict what is going on inside a horn, neglecting higher order effects, but it can’t say anything about what is going on outside the horn, so it can’t predict directivity.
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