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Mathematical physics

Mathematical physics

Fundamental constants set upper limit for the speed of sound

09 Oct 2020
Speed of sound
Upper limit: a US Navy F/A-18 travelling near the speed of sound in air. The white halo comprises water droplets that have condensed from the air because of the sudden drop in pressure behind the shock cone around the aircraft . (Courtesy: John Gay/US Navy)

The upper limit on the speed of sound in solids and liquids depends on just two dimensionless quantities – the fine structure constant and the proton-to-electron mass ratio. That is the surprising conclusion of physicists in the UK and Russia, who calculate that the speed limit is twice that of the highest speed of sound measured to date.

Sound propagates as a series of compressions and rarefactions in an elastic medium, with its speed varying significantly from one material to another. Typically, sound is slowest in gases, higher in liquids and higher still in solids. In air at ambient conditions sound travels at about 340 m/s, while in water it reaches about 1500 m/s and in iron more than 5000 m/s.

These differences are due to the way that a passing wave disturbs atoms and molecules. Thought of as hard spheres linked to one another with springs, the particles are knocked forward by their neighbours in the direction of sound propagation and in turn go on to nudge other neighbouring particles ahead of them. But this transmission is delayed by inertia, meaning that waves move faster when the particles are less massive.

Stiffer links mean less delay

However, stiffer links also mean less delay – each particle having to move less before it triggers the movement of its neighbour. This is why sound travels faster through iron than it does through water, for example.

Expressed mathematically, sound’s longitudinal speed in a material is equal to the square root of that material’s elastic modulus – which quantifies its resistance to compression – divided by its density.

In the latest research, Kostya Trachenko of Queen Mary University of London and colleagues at the University of Cambridge and the Russian Academy of Sciences’ Institute for High Pressure Physics set out to recast this formula in terms of fundamental constants. Their first step was to link a material’s bulk modulus with the energy that binds its atoms together, given that greater stiffness implies a higher binding energy. They then assumed that the latter term could be equated to the Rydberg energy, which is the characteristic binding energy in condensed matter.

Eye-catching formula

The resulting formula proved eye-catching, particularly when expressed in terms of the fine-structure constant – which sets the strength of fundamental electromagnetic interactions  – and then written specifically as an upper limit on the speed of sound. The upper limit comes about when the mass of the atoms in question is the lowest mass possible – that of the hydrogen atom. In that case, the velocity of sound can simply be expressed in terms of the proton-to-electron mass ratio (inverted, halved and square-rooted), the fine-structure constant and the speed of light in a vacuum.

Inserting the relevant numbers into their formula, the researchers worked out that the highest possible speed of sound in solids and liquids (exposed to moderate pressures) comes in at a little over 36,000 m/s. That is still nearly 10,000 times slower than light’s upper limit but about twice as high as the fastest ever recorded sound wave – which stands at around 18,000 m/s and obtained in (very stiff) diamond.

To establish whether their equation is broadly in line with measured velocities, the researchers compared its predictions against the experimentally obtained speed of sound in 36 different elemental solids. To do so they used a log-log plot to show how these speeds vary with the solids’ atomic mass and on the same graph drew the straight sloping line generated by their equation – which terminates at the high end with hydrogen. They conclude that the experimental data points more or less follow their sloping line, even if a coefficient they dropped to simplify their equation means that not all are that close.

Metallic hydrogen

As an additional check on their work, Trachenko’s team used density functional theory to calculate the speed of sound through metallic hydrogen from first principles. When exposed to very high pressures, hydrogen becomes a molecular solid, and at pressures above about 400 GPa it is predicted then to become an atomic metal. It is this metallic state that should hold the speed record. Modelling hydrogen in these conditions, they found that sound should propagate at up to 35,000 m/s – faster than in any other material, but still below their upper limit.

This work follows similar research that Trachenko and fellow group member Vadim Brazhkin published earlier this year, which revealed a universal lower limit on viscosity expressed in terms of the proton-to-electron mass ratio and Planck’s constant. By incorporating the fine-structure constant, they ended up with an expression containing two fundamental constants whose fine tuning yields a stable proton and also allows stars to ignite and produce heavier elements, enabling, as they put it, “the existence of solids and liquids where sound can propagate”.

Kamran Behnia of PSL University in Paris describes the work as “simultaneously simple and deep”. He says he had not expected to be able to work out the rough speed of sound in a particular material “by hand-waving arguments” – particularly, he adds, when the resulting formula involves only fundamental constants. As he points out, quantum mechanics is not needed to explain the propagation of sound – even if quantum mechanics is what makes the solid state possible. “This is why the main message of this paper comes as a surprise,” he says.

The research is described in Science Advances.

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