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Every One-Parameter Acoustic Field Obeys Webster's Horn Equation
The Helmholtz equation admits one-parameter (1P) solutions in u, that is, solutions depending o a single spatial coordinate u, if and only if ytuy and t2u are functions of u alone. The ytuy condition allows u to be transformed to another coordinate (symbol), which measures arc length. For a 1P field inside a tube of orthogonal trajectories to the surfaces of constant (symbol)f, the wave equation reduces exactly to Webster's horn equation, in which (symbol) is the axial coordinate of the horn and S(symbol) the area of a constant-(symbol) surface segment bounded by the tube. The 1P existence conditions can be expressed in terms of coordinate scale factors and used to determine whether a given coordinate system admits 1P waves. They can also be expressed in terms of the principal curvatures of the constant-(symbol) surfaces, leading to the unfortunate conclusion that the only coordinates meeting the conditions are those whose level surfaces are parallel planes, coaxial cylinders, or concentric spheres; that is, no new 1P horn geometries remain to be discovered.
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