Mori equation
The half wavelength for a thin wire is —
\begin{align} \label{eq:14501a} \Gamma_{tw} = \frac{c_{tw}}{2 \, f} \end{align}
where —
| \( \Gamma_{tw} \) | = thin-wire half wavelength |
| \( c_{tw} \) | = thin-wire wave speed |
| \( f \) | = frequency |
However, as the lateral dimensions of the resonator increase, the effective wave speed decreases. This occurs because of the additional stored energy due to the radial "breathing" due to Poisson's coupling. This means that stout resonators will tune shorter than thin resonators. The Mori equation attempts to account for this phenomena for a prismatic cylinder.
\begin{align} \label{eq:14502a} \Gamma_D = \Gamma_{tw} \, \left[ \frac{1 - B_1 \, \Delta}{1 - B_2 \, \Delta} \right]^{1/2} \end{align}
where —
| \( \Gamma_D \) | = half wavelength at diameter D |
| \( \Gamma_{tw} \) | = thin-wire half wavelength |
| \( D \) | = resonator diameter |
| \( B_1 \) | = \( 1 - \nu^2 \) |
| \( B_2 \) | = \( 1 - 3 \, \nu^2 - 2 \, \nu^3 \) |
| \( \nu \) | = Poisson's ratio |
| \( \Delta \) | = \( \left[ \frac{\pi \, \Large\frac{D}{\Gamma_{tw}}}{2 \, \alpha} \right]^{2} \) |
| \( \alpha \) | \( \approx 1.84 + 0.68 \, \nu \) [Derks, p. 42, eqn. 5.8] |
| \( \alpha \) | \( \approx 1.85 \left(1 + 0.386 \, \nu - 0.146 \, \nu^2 + 0.115 \, \nu^3 \right) \) [Gladwell (1), p. 345] |
The quantity in brackets in equation \eqref{eq:14502a} is always \( \leqslant 1.0\) so \( \Gamma_D \leqslant \Gamma_{tw} \).
Note that \( \alpha \) is approximate. Derks and Gladwell give somewhat different equations. Derks takes his linear equation from a graph by Kleesattel (2), p. 3, figure 1, curve 1. Kleesattel's graph of this curve is not quite linear so Gladwell's approximation may be somewhat better. However, the difference is small — for \( \nu \) = 0.33 (approximately typical for many acoustic materials), Gladwell's equation gives 2.0666 whereas Dirks' equation gives 2.0639. Figure 1 shows that the results are nearly indistinguishable.
From a physical standpoint, when the cylinder is infinitely thin (i.e., a thin wire) there is no "breathing" and no correction is needed. As the cylinder diameter increases more breathing occurs and the cylinder tunes progressively shorter. At the extreme the cylinder is reduced to a thin flat disk. Then the breating is simply the fundamental radial resonance of the disk.
From its method of derivation, Mori's results are exact at the two extreme diameters — when \( D \) = 0 and when \( D \) is that of a the thin disk. The results at intermediate diameters are approximated by a method called "apparent elasticity".
The Rayleigh equation is a commonly used (and better known) alternative to the Mori equation. However, the Rayleigh equation is only are exact at \( D \) = 0. At larger diameters the Rayleigh correction is not sufficient (i.e., the predicted half wavelength is too long).
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The effective axial wave speed \( c_{eff} \) can be calculated by substituting equation \eqref{eq:14502a} into —
\begin{align} \label{eq:14503a} c_{eff} = 2 \, \Gamma_D \, f \end{align}
where —
| \( c_{eff} \) | = effective axial wave speed of a cylinder whose diameter is \( D \) |
| \( f \) | = frequency |
Also see —
Inverse Mori equation
Rayleigh equation
References:
Mori (1)
Derks (1), pp. 41 - 45