# 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