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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