Ultrasound and Wood Science at the Forestry Research Centre in Nancy, France, for a period of twenty years, between 1979 and 1999

Topical Issue - Development of European Acoustics in 20th Century

Open Access

Table 7.

Velocity of surface wave and off diagonal term of stiffness matrix [27].

Characteristics	Parameters
Wood density	.ρ
Free surface orientation	Plane 12
Propagation direction	Axis 1
Polarisation direction	Axis 3
Off-diagonal term to calculate	C₁₃
Velocity of surface waves measured	V
Equation
$M \cdot C_{ij}^{4} + N \cdot C_{ij}^{2} + P = 0$ $\displaystyle M\cdot C_{ij}^{4} +N\cdot C_{ij}^{2} +P=0$
Coefficients of the equation for the calculation of C₁₃	M, N, P
$\begin{matrix} M = \frac{(ρ V^{2} - C_{55})}{C_{33} \cdot C_{55}} \\ [10 p t] N = 2 [\frac{{(ρ V^{2})}^{2}}{C_{55}} - ρ V^{2} (1 + \frac{C_{11}}{C_{55}}) + C_{11}] \\ [10 p t] P = {(ρ V^{2})}^{3} \frac{(C_{33} - C_{55})}{C_{55}} + {(ρ V^{2})}^{2} (C_{33} - C_{11} + \frac{2 C_{11} \cdot C_{33}}{C_{55}}) + ρ V^{2} \frac{2 C_{11} C_{55}}{C_{11} C_{33} - C_{11} \cdot C_{55}} \end{matrix}$ $\begin{array}{l} \displaystyle M= \frac{\left( \rho V^{2}- C_{55} \right)}{C_{33}\cdot C_{55}}\\ \displaystyle N=2 \left[ \frac{\left( \rho V^{2} \right)^{2}}{C_{55}} - \rho V^{2}\left( 1+ \frac{C_{11}}{C_{55}} \right)+C_{11}\right]\\ \displaystyle P= \left( \rho V^{2} \right)^{3}\frac{\left( C_{33}-C_{55}\right)}{C_{55}}+ \left( \rho V^{2} \right)^{2} \left( C_{33}- C_{11}+ \frac{2C_{11}\cdot C_{33}}{C_{55}} \right)+\rho V^{2}\frac{2C_{11} C_{55}}{C_{11}C_{33}- C_{11}\cdot C_{55}}\end{array}$

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