All that noise, and all that sound

The speed of sound

• decreases with density

• increases with inter-particle forces.

Diamond has a moderate density (much less than gold), but is held together by strong covalent bonds. This makes for a very high speed of sound.

$\begin{array}{lr} \text{air} & \SI{340}{m/s} \\ \text{helium} & \SI{1\,000}{m/s} \\ \text{water} & \SI{1\,500}{m/s} \\ \text{gold} & \SI{3\,240}{m/s} \\ \text{marble} & \SI{3\,800}{m/s} \\ \text{diamond} & \SI{12\,000}{m/s} \end{array}$

In the following I will briefly sketch the derivation for the speed of sound: In a sound wave, the particles of the medium are displaced from their original position, so that they move from the position $x$ to the location $x + u (x, t)$ with the displacement $u ( x, t)$ . Due to the gradient in the displacement, there is a local volume change of the medium: $\frac{\Delta V}{V} = \frac{\partial u}{\partial x}$ The expansion or compression of the medium leads to an additional pressure $\Delta p$ in the medium $\Delta p = - \frac{1}{\kappa} \frac{\Delta V}{V}$ where $\kappa = - \dfrac{1}{V} \dfrac{dV}{dp}$ is the compressibility. A small volume $V$ of the medium is accelerated by the gradient in the pressure field $p = p_0 + \Delta p$ . The volume $V$ has the mass $m = \rho V$ with the mass density $\rho$ .Therefore, Newton's law results $\begin{aligned} m \frac{d^2 x}{dt^2} = \rho \frac{d^2 u}{dt^2} V &= - \frac{\partial p}{\partial x} V = + \frac{1}{\kappa} \frac{d^2 u}{dx^2} V \\ \Rightarrow \quad \frac{d^2 u}{dt^2} &= c^2 \frac{d^2 u}{dx^2}, \quad c = \frac{1}{\sqrt{\rho \kappa}} \end{aligned}$ The result corresponds to a wave equation with the speed of sound $c$ .

The speed of sound is greatest when the density $\rho$ and compressibility $\kappa$ are as small as possible. For an ideal gas the compressibility yields $\kappa \propto \frac{1}{p}$ , so that speed of sound $c \propto \sqrt{\frac{T}{M}}$ depends only on the temperature $T$ and the molar mass $M$ of the molecules regardless of the pressure. In particular, the speed of sound of helium is greater than of normal air, since the helium atoms have a lower molar mass $M$ than nitrogen and oxygen molecules. During the phase transition from gaseous to liquid and from liquid to solid, the mechanical properties of the medium change, so that the medium becomes stiffer and the compressibility decreases. Therefore, the highest values for the speed of sound are achieved in solids. As an example, the sound velocities of water vapor, water and ice are given here: $\begin{aligned} \text{water vapor:} & & c_\text{gas} &\approx 450 \,\frac{\text{m}}{\text{s}} \\ \text{water:} & & c_\text{liquid} &\approx 1500 \,\frac{\text{m}}{\text{s}} \\ \text{ice:} & & c_\text{solid} &\approx 4000 \,\frac{\text{m}}{\text{s}} \end{aligned}$ Among the options mentioned the solid with the highest stiffness and the lowest density is diamond, because

• Diamond is the hardest material and therefore has the smallest compressibility
• Diamond consists of carbon atoms, which have only a small mass in comparison with calcium and gold, so that the density of diamond is relatively low.

Therefore, diamond has the largest speed of sound with $c \approx 18,000 \,\frac{\text{m}}{\text{s}}$ .