A brief history of sound of speed published: 13^th November 2024

“We say that we hear a sound, which means that somewhere or other an air quiver has been started and has reached our ears. As the life and processes of the world go on the actions which take place are accompanied by these tremors, and we live in this world of sound.”
- Sir William Bragg

Velocity, the unit used to measure the speed of sound through a medium comes from Latin celeritas (root celer meaning quick or swift). You see c as universal symbol for speed of light but V was the most common symbol used in the 19th century started by Maxwell in 1865. Actual history is a bit ambiguous on why is c the symbol of light so it stands for either "constant" or "celeritas".

Sound being a vibrational phenomenon was known as early as 6^th BC, the Greek mathematician Pythagoras was aware that sound is vibration. We also have written records from the Greek philosopher Aristotle (350 BC) with rudimentary knowledge of sound propagation. He also believed that high frequencies traveled faster through air than low frequencies.

“All sounds are produced because air is set in motion by expansion or compression or when it is clashed together by an impact from the breath or from the strings of musical instruments”

The first explicit mention of sound comes from Francis Bacon where he discussed the possibility of comparing the speed of sound to that of the light by calculating the time taken by a church bell sound to travel a mile. He compared this value with a simultaneous light signal over the same distance using his own pulse as the timing mechanism. But the first quantitative value came from Marin Mersènne a French mathematician, natural philosopher, and catholic priest (a highly influential figure in the development of acoustic science) who in 1640 recorded the speed of sound as it travels through air. He used musical instruments and gun sounds and estimated the distance traveled in one second to be 230 French Toises which is 448m/s which has an error of around 10%. He used a pendulum to measure the time between the flash of exploding gunpowder and the arrival of the sound. Although it might seem like a large error it is still quite impressive given the technology during his time. He also, incorrectly asserted that the same speed is observed at day and night, either with wind or against it. Pierre Gassandi a French philosopher, Catholic priest, astronomer, and mathematician used a mechanical timepiece to measure the time between the flash of exploding gunpowder and the arrival of the sound. He discovered that all pitches travel at the same speed but he incorrectly concluded as that of Mersènne that wind has no effect on the velocity. This was when importance of air as a medium for sound was still not realised but the Italian physicist and mathematician Evangelista Torricelli's invention of the braometer helped others conduct their experiments in air pumps. In the 17th century Sir Issac Newton showed that in an elastic fluid the sound speed propagation is proportional to the square root of elasticity divided by density. The reasoning for which is unclear and mathematicians Lagrange actually claimed the derivation to be illogical. Newton then used Boyle's law which is true only for constant temperature to derive the speed as c = √p/e which comes to 295m/s. He concluded that the experimental data then available shows velocity lies between 290-330 m/s.

Several experiments for the speed of sound were conducted in this time. Reverend William Derham in 1708, fired guns from various church towers, covering a distance of 20.1 km and used a pendulum beating every half second for timing the arrival of the report of the sound and arrived at a mean value of 348m/s. With his series of tests conducted he was able to show that favourable winds accelerated sound propagation while opposing winds retarded it. Unfortunately, he used a fifteen point scale and translating that to true wind speed has not been done. These experiments were been conducted while thermometry was still new. Celsius scale was first proposed in 1742. While Derham did not measure the temperature he incorrectly concluded that wind speed was same in summer and winter both.

In 1738, the Académie des Sciences organised experiments to measure the speed of sound more accurately. These experiments involved prominent members including Jacques Cassini, Pierre Louis Maupertuis, Claude Louis d'Épinay, and others. They were the first to state that with a prevailing wind speed denoted as u. In the direction of the wind sound propagates with speed (c+u) but against the wind with speed (c-u). With their experiments they came to the value of 337m/s for the speed of sound. While they concluded that the temperature does not affect the speed of sound but it was Italian scientist G.L. Bianconi who in 1740 demonstrated by comparing velocities in winter and summer. He correctly concluded with rise in temperature the speed of sound increases and vice versa.

The rest of the 18th century mostly about consolidating various studies. Joseph-Louis Lagrange Italian mathematician tackled Newton's reasoning on the elasticity of the medium and generalised his reasoning to cover sound waves with complex harmonics. However, the calculated values and the experimental data had large discrepancies. John Dalton an English chemist and physicist made observations on the behaviour of gases particularly around how sudden compression affects their temperature. Dalton found an important distinction between isothermal and adiabatic processes. An isothermal process is in which the temperature of medium remains constant since it can exchange heat with its surroundings. If a gas is compressed slowly it has time to release heat to the environment preventing temperature change. The change of volume with change in pressure in isothermal conditions is directly proportional to the static pressure of the gas. An adiabatic process is in which there is no heat transfer from the medium to its surroundings (i.e. no heat is gained or lost). This happens when the compression is rapid as there is no time for heat exchange to happen. In such cases the gas heats up significantly. Dalton noted that assuming isothermal conditions for sound waves might be incorrect and for rapid compressions like sound waves adiabatic conditions should be applied. This led to Pierre-Simon Laplace a French scholar to revise Newton's equation to c = √γp/e where γ is called the adiabatic index. γ = C_p/C_v where C_p is the heat capacity needed to raise the temperature of gas by 1 degree Celsius (or 1 Kelvin) at a constant pressure. C_v is the heat capacity required to raise the temper by 1 degree Celsius (or 1 Kelvin) at a constant volume. Although inclusion of γ brought the calculated and observed values into agreement it was still not obvious that sound propagation was completely adiabatic process. Sir George Stokes an Irish mathematician and physicist showed that sound propagation has to be either substantially isothermic or substantially adiabatic else large damping factors and high attenuation would be observed and helped clear lingering doubts. Karl Ferdinand Herzfeld an Austrian-American physicist and Francis Owen Rice an English chemist showed that for an unbounded wave (wave that propagates through an infinite or large medium without significant constraints or reflections from boundaries) the time required for temperature to achieve equilibrium is proportional to the square of the wave length, whereas the actual time available for this to occur is directly proportional to the wavelength. This confirms that sound is an adiabatic process.

With time other techniques for measuring the speed were put forward. However, experimenting in open air hindered in getting an accurate speed of sound. A wind speed of 8km/hr corresponds to a possible error of 2.2m/s. Also, a change of temperature of 1 degree Celsius produces a speed change of about 0.6m/s.

Is light a wave? Light travels in a straight line, you can hear sounds around a corner but certainly can't see. Interference is a fundamental property of waves where two or more waves superpose to form a combined wave. The result depends on the phases of the signals interacting leading to constructive or destructive interference. Constructive interference happens when the waves meet in phase i.e. their peaks and troughs align. The amplitude of the resulting wave is the sum of the amplitude of the individual waves. Destructive interference occurs when the waves meet out of phase i.e. one wave's peak aligns with the other's trough. The amplitude of the resulting wave is the difference of the individual amplitudes. Interference is a property only of waves. Detection of object at point locations only happens with particles. The result of the double slit experiment by Thomas Young is that on the photographic plate you get an interference pattern, but the pattern is composed of a smattering of point-wise detections. If it were just a particle, you would not get the interference pattern. If if were just a wave, you would not get the point-wise detections. No wave like interference effects are very evident for light. Sound is compressions in air, if light is a wave, what is it waving? It's not air since light comes from the sun which is so far where there is no air. Despite, around 1800 it was established that light is a wave. The wavelength is just really short. Light wavelength is about one fifty thousands of an inch. Compared to the wavelength detected by human ears is about half an inch.

So, if light is a wave just what is it waving?
Sound needs a medium to travel, it can travel through solids, liquids, and gases. It was natural to assume that light too travels through a medium and this mysterious medium was called aether. This medium must fill all space out to the stars because we can see them so aether has to fill it to let light travel.

Detecting ether: the Michelson-Morley experiment
Michelson and Morley used an interferometer an instrument that could split a light beam into two perpendicular paths using a half silvered mirror. The initial light beam was split into two beams traveling at right angles to each other. The beams traveled a fixed distance were reflected back and recombined to form an interference pattern. The expectation was that if aether fills space the beam traveling along the direction of earth's motion would experience the aether wind and would slow down compared to the beam traveling perpendicular to earth's motion. he experiment's outcome was surprising: no shift in the interference pattern was detected, regardless of the direction of the light beams or Earth's movement.
Michelson performed series of experiments but without any luck. The only possible conclusion from this series of experiments was that the whole concept of an all pervading aether was wrong from the start. Albert Einstein addressed this with his Theory of Special Relativity in 1905. Einstein proposed that the speed of light is a fundamental constant in nature, unaffected by the motion of the observer or source, eliminating the need for the aether concept altogether.

So what does this have to do with the speed of sound? Thomas Carlyle Hebb while working at Ryerson Physical Laboratory in the USA made direct measurement in free air of the wavelength of the sound corresponding to a signal of known frequency using the method devised by Michelson for measuring light wavelength.
Two paraboloidal reflectors (focal length approximately 0.38 m, diameter 1.5 m) were arranged coaxially, one of them being moveable on a parallel track. A pure- tone air-whistle source whose frequency f = 2376.5 Hz was found by comparison with a tuning fork, was placed at the focus of one of the mirrors whilst a carbon microphone was placed at the focus of each of the mirrors. Using a split-primary transformer the outputs of the two microphones were combined in such a way that when the output from the secondary was monitored using headphones the sound heard was proportional to the vector sum of the two microphone outputs. This method therefore provided a direct determination of wavelength and, as the frequency was known, this yielded the sound velocity. The experiment was carried out in a hall 36m long, thus no wind was encountered and the temperature was found to remain very constant. The separation between mirrors could be increased to as much as 100 λ and the positions of the minima could be located to within about 1 cm, so that an accuracy of the order of 0.1% was achieved. In this way Hebb reported a value of c_o = (331.29 ± 0.04) m/s but subsequently he revealed that this result was slightly in error due to the method he had used to correct the velocity measured in air containing moisture to obtain the dry air value. His revised estimate for dry air was remarkable c_o = 331.41 m/s. Cyril M. Harris an acoustic engineer made careful measurements of relative sound velocity and showed that at room temperature it reaches a minimum value at relative humidity of about 14% at which point it is approximately 0.2 m/s lower than the dry-air value: at 100% Relative Humidity (RH) the velocity is about 1.1 m/s higher than the dry air value. It is this value of 331m/s which is commonly referenced to in text books today.

The speed (velocity = speed with direction) of sound in dry air is generally given as: V_{sound in air} ≈ 331.6 + 0.6 T_c (m/s)