A common lecture demonstration is as follows: hold or clamp a onemeter long thin aluminium bar at the center, strike one end longitudinally (i.e. parallel to the axis of the bar) with a hammer, and the result is a sound wave of frequency $2500 \mathrm{~Hz}$.

From this experiment, calculate the speed of sound in aluminium.
The point where the bar is struck is an antinode and the point where it is held a node. With the bar held at the center and its one end struck, the wavelength $\lambda$ is related to its length $L$ by $\lambda=2 L$. Hence the speed of sound propagation in the aluminium bar is

$$
v_{\mathrm{Al}}=\nu \lambda=2 \nu L=2 \times 2500 \times 1=5000 \mathrm{~m} / \mathrm{s} \text {. }
$$

The speed of sound in a solid is

$$
v=\sqrt{\frac{Y}{\rho}},
$$

where $Y$ is the Young's modulus of its material and $\rho$ its density. The speed of sound in a fluid is

$$
v=\sqrt{\frac{M}{\rho}}
$$

where $M$ is its bulk modulus and $\rho$ its density. For adiabatic compression of a gas, $M=\gamma p$, where $p$ is its pressure and $\gamma$ the ratio of its principal specific heats; $\gamma=1.4$ for air, a diatomic gas. Hence

$$
\frac{v_{\text {air }}}{v_{\mathrm{Al}}}=\sqrt{\frac{1.4 p \rho_{\mathrm{Al}}}{Y \rho_{\mathrm{air}}}} .
$$

With

$$
\begin{aligned}
p &=1.013 \times 10^{6} \mathrm{dyn} / \mathrm{cm}^{2} \quad(\text { standard atmosphere) }\\
Y &=7.05 \times 10^{11} \mathrm{dyn} / \mathrm{cm}^{2} \\
\rho_{\mathrm{Al}} &=2.7 \mathrm{~g} / \mathrm{cm}^{3} \\
\rho_{\text {air }} &=1.165 \times 10^{-3} \mathrm{~g} / \mathrm{cm}^{3} \quad\left(\text { at } 30^{\circ} \mathrm{C}\right) \\
v_{\text {air }} &=6.83 \times 10^{-2} \times 5000=341 \mathrm{~m} / \mathrm{s} .
\end{aligned}
$$$v_{\mathrm{Al}}=5000 \mathrm{~m} / \mathrm{s}$.