膨胀波波速
一种材料属性：膨胀波的波速。（注：Meyer（见下）也将其称为压缩波波速或纵波波速。）
膨胀波波速与其他材料属性的关系为 —
\begin{align} \label{eq:12001a} c_d &= c_{tw} \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2}\\[0.7em]%eqn_interline_spacing &= \left[ \frac{E}{\rho} \right]^{1/2} \, \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing &= \left[ \frac{E^\prime}{\rho} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing \end{align}
式中 —
| \( c_d \) | = 膨胀波波速 |
| \( c_{tw} \) | = 细杆波速 |
| \( \nu \) | = 泊松比 |
| \( E \) | = 弹性模量（杨氏模量） |
| \( \rho \) | = 密度 |
| \( E^\prime \) | = 无限介质的有效模量 |
\begin{align} \label{eq:12002a} ~~~~~~&= E \, \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing \end{align}
由于介质无限大，该波不会因泊松耦合而发生横向膨胀或收缩。由于介质由此受到约束，有效模量 \( E^\prime \) 大于细杆模量 \( E \)，因此所得波速 \( c_d \) 高于细杆波速 \( c_{tw} \)。
对于功率超声而言，膨胀波波速主要具有理论意义。
参考文献：Meyer (1)，公式 1.58，第 18 页
Dilatational wave speed
A material property: the wave speed of a dilatational wave. (Note: Meyer (below) also refers to this as compressional or longitudinal wave speed.)
The dilatational wave speed is related to other material properties by —
\begin{align} \label{eq:12001a} c_d &= c_{tw} \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2}\\[0.7em]%eqn_interline_spacing &= \left[ \frac{E}{\rho} \right]^{1/2} \, \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing &= \left[ \frac{E^\prime}{\rho} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing \end{align}
where —
| \( c_d \) | = dilatational wave speed |
| \( c_{tw} \) | = thin-wire wave speed |
| \( \nu \) | = Poisson's ratio |
| \( E \) | = modulus of elasticity (Young's modulus) |
| \( \rho \) | = density |
| \( E^\prime \) | = effective modulus for an infinite medium |
\begin{align} \label{eq:12002a} ~~~~~~&= E \, \left[ \frac{1 - \nu}{(1 + \nu) \, (1 - 2 \nu)} \right]^{1/2} \nonumber \\[0.7em]%eqn_interline_spacing \end{align}
Because the medium is infinite, the wave cannot expand or contract laterally due to Poisson coupling. Because the medium is thereby constrained, the effective modulus \( E^\prime \) is greater than the thin-wire modulus \( E \) and the resulting wave speed \( c_d \) is higher than the thin-wire wave speed \( c_{tw} \).
For power ultrasonics, the dilatational wave speed is mainly of theoretical interest.
Reference: Meyer (1), equation 1.58, p. 18
Also see —
Thin-plate wave speed
Shear wave speed