Classic horn comparisons
Classic horns are those whose performance can be predicted from the one-dimensional wave equation.
\begin{align} \label{eq:10801a} E \, \frac{\partial^2 u}{\partial x^2} = \rho \, \frac{\partial^2 u}{\partial t^2} \end{align}
or
\begin{align} \label{eq:10801a} {c_o}^2 \, \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \end{align}
The one-dimensional wave equation can only be explicitly solved for a few resonator shapes. Among these are cylindrical horns, exponential horns, and catenoidal horns.
The one-dimensional wave equation assumes that the displacement depends only on the location along the center axis of the resonator (i.e., it does not depend on the transverse location). This means that plane sections remain plane during vibration. This assumption is completely true if Poisson's ratio is zero. It is approximately true for slender resonators whose cross-sectional area changes only gradually along the length of the resonator.
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