Speed of sound

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United States Navy F18 flying at transonic speed

The speed of sound is the phase speed of sound waves in a medium, that is, it is the speed at which a wave front propagates in said medium. In the Earth's atmosphere it is 343.2 m/s (1235.52 km/h at a temperature of 20 °C, with 50% humidity and at sea level). The speed of sound varies depending on the medium in which it is transmitted. Since the speed of sound varies depending on the medium, the number Mach 1 is used to indicate it. Thus a body moving through the air at Mach 2 advances at twice the speed of sound, regardless of air pressure or temperature.

The propagation speed of the sound wave depends on the characteristics of the medium in which said propagation takes place and not on the characteristics of the wave or the force that generates it. Its propagation in a medium can be used to study some properties of said transmission medium.

History

Isaac Newton's 1687 Principia includes a calculation of the speed of sound in air as 298 m/s. This is about 15% too low. The discrepancy is mainly due to ignoring the (then unknown) effect of rapidly fluctuating temperature on a sound wave (in modern terms, the compression and expansion of the sound wave of the air is an adiabatic process, not an isothermal process). This error was later corrected by Laplace.

During the 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630 (1,380 Parisian feet per second), Pierre Gassendi in 1635 (1,473 Parisian feet per second), and Robert Boyle (1,125 Parisian feet per second.) In 1709, the Reverend William Derham, Rector of Upminster, published a more precise measurement of the speed of sound, at 1,072 Parisian feet Parisian feet per second. (The Parisian foot was 325 mm. This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8mm, making the speed of sound at 20°C 1055ft. Parisians per second).

Derham used a telescope in the tower of St. Laurence's Church, Upminster to watch the flash of a distant shotgun being fired, then measured the time until he heard the shot with a half-second pendulum. Shot measurements were taken of several local landmarks, including North Ockendon Church. The distance was known by triangulation, and therefore the speed at which sound had traveled was calculated.

Means of propagation

3D animation of a plane breaking the sound barrier, exceeding 1234.8 km/h

Main article: Propagation of sound

The speed of sound varies depending on the medium through which the sound waves travel.

The thermodynamic definition of the speed of sound, for any medium, is:

${displaystyle c={sqrt {left({frac {partial P}{partial rho }}right)_{S}}}}$

That is, the partial derivative of pressure with respect to density at constant entropy.

The speed of sound also varies with changes in the temperature of the medium. This is because an increase in temperature translates into an increase in the frequency with which the interactions between the particles that carry the vibration occur, and this increase in activity increases the speed.

For example, on a snowy surface, sound is capable of traveling great distances. This is possible thanks to the refractions produced under the snow, which is not a uniform medium. Each layer of snow has a different temperature. The deepest ones, where the sun does not reach, are colder than the superficial ones. In these colder layers close to the ground, sound propagates with less speed.

In general, the speed of sound is greater in solids than in liquids, and in liquids it is greater than in gases. This is due to the greater degree of cohesion that atomic or molecular bonds have as the matter becomes more solid.

The sound speed in the air (at a temperature of 20 °C) is 343,2 m/s. If we want to obtain the equivalence in kilometers per hour we can determine it by following physical conversion: Sound speed in the air in km/h = (343.2 m/1 s) · (3600 s/1 h) · (1 km/1000 m) = 1235.5 km/h.

In the air, at 0 °C, the sound travels at a speed of 331 m/s (per every degree Celsius that increases the temperature, the speed of sound increases by 0.6 m/s).
In the water (at 25 °C) is 1593 m/s.
In the tissues it is 1540 m/s.
In the wood is 3700 m/s.
In the concrete is 4000 m/s.
In steel it is 6100 m/s.
In aluminum is 6400 m/s.
In the cadmium it is 12400 m/s.

Speed of sound in gases

In gases, the equation of the speed of sound is the following:

${displaystyle v={sqrt {frac {gamma RT}{M}}}}$
Symbol	Name	Unit
$v$	Sound speed	m/s
$T$	Temperature	K
$R$	Universal gas strength	J/(kg K)
$M$	Gas grinding mass
$gamma$	Adiabatic expansion ratio

Typical values for the standard atmosphere at sea level are as follows:

gamma

= 1.4 for air

R

= 8,314 J/(mol·K) = 8,314 kg·m2/(mol·K·s)²)

T

= 293.15 K (20 °C)

M

= 0.029 kg/mould for air

Applying the ideal gas equation:

${displaystyle PV={frac {m}{M}}{R}{T}}$
Symbol	Name	Unit
$P$	Gas pressure	Pa
$V$	Volume	m³
$T$	Temperature	K
$R$	Universal gas strength	J/(kg K)
$m$	Masa	kg
$M$	Gas grinding mass

It can also be written as:

${displaystyle v={sqrt {frac {gamma P}{rho }}}}$
Symbol	Name	Unit
$v$	Sound speed	m/s
$P$	Gas pressure	Pa
$rho$	Density of the medium	kg/m³
$gamma$	Adiabatic expansion ratio

Speed of sound in solids

In solids the speed of sound is given by:

${displaystyle v_{s}={sqrt {frac {E}{rho }}}}$
Symbol	Name	Unit
$v$	Sound speed in solids	m/s
$E$	Young Module	Pa
$rho$	Density	kg/m³

This way you can calculate the speed of sound for steel, which is about 5148 m/s.

Speed of sound in liquids

The speed of sound in water is of interest for mapping the ocean floor. In salt water, sound travels at approximately 1,500 m/s and in fresh water at 1,435 m/s. These speeds vary mainly according to pressure, temperature and salinity.

${displaystyle v={sqrt {frac {K}{rho }}}}$
Symbol	Name	Unit
$v$	Sound speed in liquids	m/s
$K$	Understandability Module	Pa
$rho$	Density	kg/m³

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