Speed of sound

For other uses, see Speed of sound (disambiguation).

Sound measurements
Sound pressure p
Particle velocity v
Particle velocity level (SVL)
(Sound velocity level)
Particle displacement ξ
Sound intensity I
Sound intensity level (SIL)
Sound power P_ac
Sound power level (SWL)
Sound energy density E
Sound energy flux q
Surface S
Acoustic impedance Z
Speed of sound c v • d • e

Sound is a vibration that travels through an elastic medium as a wave. Sound pressure is the local Pressure deviation from the ambient (average or equilibrium pressure caused by a Sound Wave. Particle velocity is the velocity v of a particle (real or imagined in a medium as it transmits a Wave. The particle velocity level or the sound velocity level tells the ratio of a sound incidence in comparison to a reference level of 0 dB in a medium mostly air Particle displacement or particle amplitude (represented in Mathematics by the lower-case Greek letter &xi) is a Measurement The sound intensity, I, (acoustic intensity is defined as the Sound power Pac per unit area A. Sound intensity level or acoustic intensity level is a Logarithmic measure of the Sound intensity in comparison to the reference level of 0 dB ( Decibels Sound power or acoustic power P ac is a measure of sonic Energy E per Time t unit Sound power level or acoustic power level is a logarithmic measure of the Sound power in comparison to a specified reference level The sound energy density or sound density (symbol E or w) is an adequate measure to describe the sound field at a given point as a sound energy value The sound energy q results from the integral Particle velocity v of the surface A, whereby only the portions perpendicularly to the surface In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The acoustic impedance Z (or sound impedance) is a frequency f dependent parameter and is very useful for example for describing the behaviour of musical Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies A material is said to be elastic if it deforms under stress (e A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. The speed of sound describes how much distance such a wave travels in a certain amount of time. In dry air at 20 °C (68 °F), the speed of sound is 343 m/s (1235 km/h, or 770 mph, or 1129 ft/s). The Celsius Temperature scale was previously known as the centigrade scale. Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit (1686–1736 a German Physicist who proposed it in 1724 (For the South African airport with IATA code "KMH" see Johan Pienaar Airport. The foot per second (plural feet per second) is a unit of both Speed (scalar and Velocity (vector quantity which includes direction Although it is commonly used to refer specifically to air, the speed of sound can be measured in virtually any substance. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five Sound travels faster in liquids and non-porous solids than it does in air.

Basic concept

The transmission of sound can be illustrated by using a toy model consisting of an array of balls interconnected by springs. In Physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full non-simplified For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). Effects like dispersion and reflection can also be understood using this model.

In a real material, the stiffness of the springs is called the elastic modulus, and the mass corresponds to the density. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones. For instance, sound will travel faster in iron than uranium, and faster in hydrogen than nitrogen, due to the lower density of the first material of each set. At the same time, sound will travel faster in iron than hydrogen, because the internal bonds in a solid like iron are much stronger than the gaseous bonds between hydrogen molecules. In general, solids will have a higher speed of sound than liquids, and liquids will have a higher speed of sound than gases.

Some textbooks mistakenly state that the speed of sound increases with increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel. With only these three examples it indeed appears that speed is correlated to density, yet including only a few more examples would show this assumption to be incorrect.

General formulae

In general, the speed of sound c is given by

$c = \sqrt{\frac{C}{\rho}}$

where

C is a coefficient of stiffness

ρ

is the density

Thus the speed of sound increases with the stiffness of the material, and decreases with the density. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by

$c^2=\frac{\partial p}{\partial\rho}$

where differentiation is taken with respect to adiabatic change.

If relativistic effects are important, the speed of sound may be calculated from the relativistic Euler equations. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Fluid mechanics and Astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of Special

In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds air is a non-dispersive medium. But air does contain a small amount of CO₂ which is a dispersive medium, and it introduces dispersion to air at ultrasonic frequencies (> 28 kHz). Not to be confused with Supersonic. Ultrasound is cyclic Sound pressure with a Frequency greater than the upper The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. ^[1]

In a dispersive medium sound speed is a function of sound frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity, while the energy of the disturbance propagates at the group velocity. The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope The same phenomenon occurs with light waves -- see optical dispersion for a description. In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency

Dependence on the properties of the medium

The speed of sound is variable and depends mainly on the temperature and the properties of the substance through of which the wave is traveling. For example, in low molecular weight gases, such as helium, sound propagates faster compared to heavier gases, such as xenon. The molecular mass (abbreviated m of a substance, more commonly referred to as molecular weight and abbreviated as MW, is the Mass of one Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical Xenon (ˈzɛnɒn or) is a Chemical element represented by the symbol Xe. In a given ideal gas the sound speed depends only on its temperature. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature At a constant temperature, the ideal gas pressure has no effect on the speed of sound, because pressure and density (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In non-ideal gases, such as a van der Waals gas, the proportionality is not exact, and there is a slight dependence on the gas pressure, even at a constant temperature. The van der Waals equation is an Equation of state for a Fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle Force Humidity also has a small, but measurable effect on sound speed (increase of about 0. 1%-0. 6%), because some oxygen and nitrogen molecules of the air are replaced by the lighter molecules of water. Oxygen (from the Greek roots ὀξύς (oxys (acid literally "sharp" from the taste of acids and -γενής (-genēs (producer literally begetteris the Nitrogen (ˈnaɪtɹəʤɪn is a Chemical element that has the symbol N and Atomic number 7 and Atomic weight 14 Water is a common Chemical substance that is essential for the survival of all known forms of Life.

Implications for atmospheric acoustics

In the Earth's atmosphere, the most important factor affecting the speed of sound is the temperature (see Details below). Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Sound is a vibration that travels through an elastic medium as a Wave. Since temperature and thus the speed of sound normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. Refraction is the change in direction of a Wave due to a change in its Speed. An acoustic shadow is an area through which sound waves fail to propagate due to topographical obstructions or disruption of the waves via phenomena such as wind currents ^[2] The decrease of the sound speed with height is referred to as a negative sound speed gradient. In Acoustics, the sound speed gradient is the rate of change of the Speed of sound with distance for example with depth in the Ocean,or height in the However, in the stratosphere, the speed of sound increases with height due to heating within the ozone layer, producing a positive sound speed gradient. The stratosphere is the second major layer of Earth's atmosphere, just above the Troposphere, and below the Mesosphere. The photochemical mechanisms that give rise to the ozone layer were worked out by the British physicist Sidney Chapman in 1930

Practical formula for dry air

The approximate speed of sound in dry (0% humidity) air, in metres per second (m·s^-1), at temperatures near 0 °C, can be calculated from:

$c_{\mathrm{air}} = 331{.}3 + (0{.}606 \cdot \vartheta) \ \mathrm{m \cdot s^{-1}}\,$

where $\vartheta,$ is the temperature in degrees Celsius (°C). The Celsius Temperature scale was previously known as the centigrade scale.

This equation is derived from the first two terms of the Taylor expansion of the following much more accurate equation:

$c_{\mathrm{air}} = 331.3 \sqrt{1+\frac{\vartheta}{273.15}}\ \mathrm{m \cdot s^{-1}}$

The value of 331. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives 3 m/s, which represents the 0 °C speed, is based on theoretical (and some measured) values of the heat capacity ratio, $γ$ , as well as on the fact that at 1 atm real air is very well described by the ideal gas approximation. Ideal gas relations For an ideal gas the heat capacity is constant with temperature The Standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of Pressure (symbol atm Commonly found values for the speed of sound at 0 °C may vary from 331. 2 to 331. 6 due to the assumptions made when it is calculated. If ideal gas $γ$ is assumed to be 7/5 = 1. 4 exactly, the 0 °C speed is calculated (see section below) to be 331. 3 m/s, the coefficient used above.

This equation is correct to a much wider temperature range, but still depends on the approximation of heat capacity ratio being independent of temperature, and will fail, particularly at higher temperatures. It gives good predictions in relatively dry, cold, low pressure conditions, such as the Earth's stratosphere. The stratosphere is the second major layer of Earth's atmosphere, just above the Troposphere, and below the Mesosphere. A derivation of these equations will be given in a later section.

Details

Speed in ideal gases and in air

For a gas, K (the bulk modulus in equations above, equivalent to C, the coefficient of stiffness in solids) is approximately given by

$K = \gamma \cdot p$ thus $c = \sqrt{\gamma \cdot {p \over \rho}}$

Where:

γ

is the adiabatic index also known as the isentropic expansion factor. Ideal gas relations For an ideal gas the heat capacity is constant with temperature It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume(

C p / C v

), and arises because a classical sound wave induces an adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse, and thus contributes to the pressure induced by the compression.

p is the pressure. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

$ρ$ is the density

Using the ideal gas law to replace $p$ with nRT/V, and replacing ρ with nM/V, the equation for an ideal gas becomes:

$c_{\mathrm{ideal}} = \sqrt{\gamma \cdot {p \over \rho}} = \sqrt{\gamma \cdot R \cdot T \over M}= \sqrt{\gamma \cdot k \cdot T \over m}$

where

$c ideal$ is the speed of sound in an ideal gas. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion
$R$ (approximately 8. 3145 J·mol^-1·K^-1) is the molar gas constant. [1]
$k$ is the Boltzmann constant
$γ$ (gamma) is the adiabatic index (sometimes assumed 7/5 = 1. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics Ideal gas relations For an ideal gas the heat capacity is constant with temperature 400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1. 3991 to 1. 403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1. 6667 for monoatomic molecules such as noble gases). History Noble gas is translated from the German noun de ''Edelgas'' first used in 1898 by Hugo Erdmann to indicate their extremely low level of reactivity
$T$ is the absolute temperature in kelvins. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic
$M$ is the molar mass in kilograms per mole. The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this The mean molar mass for dry air is about 0. 0289645 kg/mol.
$m$ is the mass of a single molecule in kilograms.

This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for $c air$ have been found to vary slightly from experimentally determined values. ^[3]

Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " An isothermal process is a Thermodynamic process in which the Temperature of the System stays Constant: &Delta T = 0 His result was missing the factor of $γ$ but was otherwise correct.

Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of $\ \gamma\, = 1.4000$ requires that the gas exist in a temperature range high enough that rotational heat capacity is fully excited (i. e. , molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i. e. , insigificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in heat capacity for a more complete discussion of this phenomenon. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity

If temperatures in degrees Celsius(°C) are to be used to calculate air speed in the region near 273 kelvins, then Celsius temperature $\vartheta = T - 273.15$ may be used. The Celsius Temperature scale was previously known as the centigrade scale. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic

$c_{\mathrm{ideal}} = \sqrt{\gamma \cdot R \cdot T} = \sqrt{\gamma \cdot R \cdot (\vartheta + 273.15)}$

$c_{\mathrm{ideal}} = \sqrt{\gamma \cdot R \cdot 273.15} \cdot \sqrt{1+\frac{\vartheta}{273.15}}$

For dry air, where $\vartheta\,$ (theta) is the temperature in degrees Celsius(°C). The Celsius Temperature scale was previously known as the centigrade scale.

Making the following numerical substitutions: $\ R = R_*/M_{\mathrm{air}}$ , where $\ R_* = 8.315410 \cdot J \cdot mol^{-1} \cdot K^{-1}$ is the molar gas constant, $\ M_{\mathrm{air}} = 0.0289645 \cdot kg \cdot mol^{-1}$ , and using the ideal diatomic gas value of $\ \gamma\, = 1.4000$

Then:

$c_{\mathrm{air}} = 331.3 \ \mathrm{m \cdot s^{-1}} \sqrt{1+\frac{\vartheta}{273.15}}$

Using the first two terms of the Taylor expansion:

$c_{\mathrm{air}} = 331.3 \ \mathrm{m \cdot s^{-1}} (1 + \frac{\vartheta}{2 \cdot 273.15}) \,$

$c_{\mathrm{air}} = 331{.}3 + (0{.}606 \cdot \vartheta) \ \mathrm{m \cdot s^{-1}}\,$

The derivation includes the two approximate equations which were given in the introduction. For Celsius temperatures which are negative, the second term of the equation right hand side, is negative.

Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. Refraction is the change in direction of a Wave due to a change in its Speed. An acoustic shadow is an area through which sound waves fail to propagate due to topographical obstructions or disruption of the waves via phenomena such as wind currents ^[4] Wind shear of 4 m/s/km can produce refraction equal to a typical temperature lapse rate of 7. The lapse rate is defined as the negative of the rate of change in an atmospheric variable usually Temperature, with height in an atmosphere 5 °C/km. ^[5] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,^[6] eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the sound is not being carried along by the wind. ^[7]

For sound propagation, the exponential variation of wind speed with height can be defined as follows:^[8]

$\ U(h) = U(0) h ^ \zeta$

$\ \frac {dU} {dH} = \zeta \frac {U(h)} {h}$

where:

$\ U(h)$ = speed of the wind at height $\ h$ , and $\ U(0)$ is a constant

$\ \zeta$ = exponential coefficient based on ground surface roughness, typically between 0. 08 and 0. 52

$\ \frac {dU} {dH}$ = expected wind gradient at height

h

In the 1862 American Civil War Battle of Iuka, an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,^[9] because they could not hear the sounds of battle only six miles downwind. Causes of the war See also Origins of the American Civil War, Timeline of events leading to the American Civil War The coexistence of a slave-owning South Background As Confederate General Braxton Bragg moved north from Tennessee into Kentucky in September 1862 Union Maj ^[10]

Tables

In the standard atmosphere:

T₀ is 273. In Physical sciences standard conditions for temperature and pressure are Standard sets of conditions for experimental measurements to allow comparisons to be made 15 K (= 0 °C = 32 °F), giving a theoretical value of 331. 3 m·s^-1 (= 1086. 9 ft/s = 1193 km·h^-1 = 741. 1 mph = 644. 0 knots). Values ranging from 331. 3-331. 6 may be found in reference literature, however.
T₂₀ is 293. 15 K (= 20 °C = 68 °F), giving a value of 343. 2 m·s^-1 (= 1126. 0 ft/s = 1236 km·h^-1 = 767. 8 mph = 667. 2 knots).
T₂₅ is 298. 15 K (= 25 °C = 77 °F), giving a value of 346. 1 m·s^-1 (= 1135. 6 ft/s = 1246 km·h^-1 = 774. 3 mph = 672. 8 knots).

In fact, assuming an ideal gas, the speed of sound c depends on temperature only, not on the pressure or density (since these change in lockstep for a given temperature and cancel out). These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Air is almost an ideal gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere - actual conditions may vary.

Effect of temperature
$\vartheta$ in °C	c in m·s^-1	ρ in kg·m^-3	Z in N·s·m^-3
−10	325. The Celsius Temperature scale was previously known as the centigrade scale. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical 2	1. 342	436. 1
−5	328. 3	1. 317	432. 0
0	331. 3	1. 292	428. 4
+5	334. 3	1. 269	424. 3
+10	337. 3	1. 247	420. 6
+15	340. 3	1. 225	416. 8
+20	343. 2	1. 204	413. 2
+25	346. 1	1. 184	409. 8
+30	349. 0	1. 165	406. 3

$\vartheta$ is the temperature in °C

c is the speed of sound in m·s^-1

ρ is the density in kg·m^-3

Z is the characteristic acoustic impedance in N·s·m^-3 (Z=ρ·c)

Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:

Altitude	Temperature	m·s^-1	km·h^-1	mph	knots
Sea level	15 °C (59 °F)	340	1225	761	661
11 000 m−20 000 m (Cruising altitude of commercial jets, and first supersonic flight)	−57 °C (−70 °F)	295	1062	660	573
29 000 m (Flight of X-43A)	−48 °C (−53 °F)	301	1083	673	585

Effect of frequency and gas composition

The medium in which a sound wave is travelling does not always respond adiabatically, and as a result the speed of sound can vary with frequency. The acoustic impedance Z (or sound impedance) is a frequency f dependent parameter and is very useful for example for describing the behaviour of musical WikipediaWikiProject Aircraft. Please see WikipediaWikiProject Aircraft/page content for recommended layout The X-43 is an unmanned experimental Hypersonic Aircraft design with multiple planned scale variations meant to test different ^[11]

The limitations of the concept of speed of sound due to extreme attenuation are also of concern. The attenuation which exists at sea level for high frequencies applies to successively lower frequencies as atmospheric pressure decreases, or as the mean free path increases. In Physics the mean free path of a particle is the average distance covered by a particle ( Photon, Atom or Molecule) between subsequent impacts For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes. :^[3]

The molecular composition of the gas contributes both as the mass (M) of the molecules, and their heat capacities, and so both have an influence on speed of sound. In general, at the same molecular mass, monatomic gases have slightly higher sound speeds (over 9% higher) because they have a higher $γ$ (5/3 = 1. 66. . . ) than diatomics do (7/5 = 1. 4). Thus, at the same molecular mass, the sound speed of a monatomic gas goes up by a factor of

${ c_{\mathrm{gas: monatomic}} \over c_{\mathrm{gas: diatomic}} } = \sqrt{{{{5 / 3} \over {7 / 5}}}} = \sqrt{25 \over 21}$ = 1. 091. . .

This gives the 9 % difference, and would be a typical ratio for sound speeds at room temperature in helium vs. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical deuterium, each with a molecular weight of 4. Deuterium, also called heavy hydrogen, is a Stable isotope of Hydrogen with a Natural abundance in the Oceans of Earth Sound travels faster in helium than deuterium because adiabatic compression heats helium more, since the helium molecules can store heat energy from compression only in translation, but not rotation. Thus helium molecules (monatomic molecules) travel faster in a sound wave and transmit sound faster. (Sound generally travels at about 70% of the mean molecular velocity in gases).

Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see heat capacity). Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas gives the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between sound speed in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.

Mach number

Main article: Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of an object's speed to the speed of sound in the medium through which it is passing (again, usually air). Mach number (\mathrm{Ma} or M (generally ˈmɑːk sometimes /ˈmɑːx/ or /ˈmæk/ is the speed of an object moving through air or any Fluid Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of At altitude, for reasons explained, Mach number is a function of temperature.

Aircraft flight instruments, however, operate using pressure differential to compute Mach number; not temperature. Most aircraft are equipped with a standard set of flight instruments which give the pilot information about the aircraft's attitude airspeed and altitude The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the impact pressure sensed by a Pitot tube is dependent on altitude as well as speed. A Pitot (ˈpiːtoʊ tube is a Pressure measurement instrument used to measure Fluid flow Velocity.

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation for M<1:^[12]

${M}=\sqrt{5\left[\left(\frac{q_c}{P}+1\right)^\frac{2}{7}-1\right]}$

where

M

is Mach number

q c

is impact pressure and

P

is static pressure. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion In Fluid dynamics, Bernoulli's principle states that for an Inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation:

${M}=0.88128485\sqrt{\left[\left(\frac{q_c}{P}+1\right)\left(1-\frac{1}{[7M^2]}\right)^{2.5}\right]}$

where

M

is Mach number

q c

is impact pressure measured behind a normal shock

P

is static pressure. In Fluid mechanics, the Rayleigh number for a fluid is a Dimensionless number associated with buoyancy driven flow (also known as Free convection or natural

As can be seen, M appears on both sides of the equation. The easiest method to solve the supersonic M calculation is to enter both the subsonic and supersonic equations into a computer spread sheet such as Microsoft Excel, OpenOffice.org Calc, or some equivalent program. In Computing, Microsoft Excel (full name Microsoft Office Excel) consists of a proprietary Spreadsheet -application written and distributed OpenOfficeorg Calc is the Spreadsheet component of the OpenOffice First determine if M is indeed greater than 1. 0 by calculating M from the subsonic equation. If M is greater than 1. 0 at that point, then use the value of M from the subsonic equation as the initial condition in the supersonic equation. Then perform a simple iteration of the supersonic equation, each time using the last computed value of M, until M converges to a value--usually in just a few iterations. ^[12]

Experimental methods

A range of different methods exist for the measurement of sound in air.

Single-shot timing methods

The simplest concept is the measurement made using two microphones and a fast recording device such as a digital storage scope. A digital system uses discrete (discontinuous values usually but not always Symbolized Numerically (hence called "digital" to represent information for This method uses the following idea.

If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:

1. The distance between the microphones (x), called microphone basis. 2. The time of arrival between the signals (delay) reaching the different microphones (t)

Then v = x / t

An older method is to create a sound at one end of a field with an object that can be seen to move when it creates the sound. When the observer sees the sound-creating device act they start a stopwatch and when the observer hears the sound they stop their stopwatch. Again using v = x / t you can calculate the speed of sound. A separation of at least 200 m between the two experimental parties is required for good results with this method.

Other methods

In these methods the time measurement has been replaced by a measurement of the inverse of time (frequency). For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Frequency is a measure of the number of occurrences of a repeating event per unit Time.

Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume. Kundt's tube is an experimental setup of August Kundt for the measurement of the Speed of sound in both a Gas and a Solid rod It has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes and antinodes visible to the human eye. A node is a point along a Standing wave where the wave has minimal Amplitude. A node is a point along a Standing wave where the wave has minimal Amplitude. This is an example of a compact experimental setup.

A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water. A tuning fork is an acoustic Resonator in the form of a two-pronged Fork with the tines formed from a U-shaped bar of elastic A pipe is a tube or hollow cylinder used to convey materials or as a structural component Water is a common Chemical substance that is essential for the survival of all known forms of Life. In this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ({1+2n}λ/4) where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these. A node is a point along a Standing wave where the wave has minimal Amplitude.

Here it is the case that v = fλ

Non-gaseous media

Speed in solids

In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode. A sound wave generating volumetric deformations is called longitudinal and a transversal wave generates shear deformations. The velocities of these two different sound waves can be calculated in isotropic solids by:

$c_{\mathrm{long}} = \sqrt{\frac{E}{\rho}}$

$c_{\mathrm{trans}} = \sqrt{\frac{G}{\rho}}$

where

E is Young's modulus

G is Shear modulus

ρ (rho) is density

Thus in steel the speed of sound is approximately 5,100 m·s^-1. Isotropy is uniformity in all directions Precise definitions depend on the subject area In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 In beryllium, a substance with relatively high stiffness and low density the speed of sound is 12,870 m·s^-1. Beryllium (bəˈrɪliəm is a Chemical element with the symbol Be and Atomic number 4 ^[13]

In a solid rod (with thickness much smaller than the wavelength) only longitudinal waves occur.

In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found by replacing Young's modulus E in the above formula by the plane wave modulus M, which can be expressed in terms of the Young's modulus and Poisson's ratio as:

$M = E \frac{1-\nu}{1-\nu-2\nu^2}$

Speed in liquids

In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces). In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material In Linear elasticity, the P-wave modulus M also known as the longitudinal modulus, is one of the elastic moduli available to describe isotropic homogeneous In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to

Hence the speed of sound in a fluid is given by

$c_{\mathrm{fluid}} = \sqrt {\frac{K}{\rho}}$

where

K is the bulk modulus of the fluid

Water

The speed of sound in water is of interest to anyone using underwater sound as a tool, whether in a laboratory, a lake or the ocean. Underwater acoustics is the study of the propagation of Sound in Water and the interaction of the mechanical waves that constitute sound with the water and its boundaries Examples are sonar, acoustic communication and acoustical oceanography. Sonar (which started as an Acronym for sound navigation and ranging) is a technique that uses Sound propagation (usually underwater to navigate Underwater acoustics is the study of the propagation of Sound in Water and the interaction of the mechanical waves that constitute sound with the water and its boundaries Acoustical oceanography is the use of Underwater sound to study the Sea, its boundaries and its contents See Discovery of Sound in the Sea for other examples of the uses of sound in the ocean (by both man and other animals). In fresh water, sound travels at about 1497 m/s at 25 °C. See Technical Guides - Speed of Sound in Pure Water for an online calculator.

Seawater

Sound speed as a function of depth at a position north of Hawaii in the Pacific Ocean derived from the 2005 World Ocean Atlas. The SOFAR channel is centered on the minimum in sound speed at ca. 750-m depth.

Sound speed as a function of depth at a position north of Hawaii in the Pacific Ocean derived from the 2005 World Ocean Atlas. The Pacific Ocean is the largest of the Earth 's Oceanic divisions The World Ocean Atlas ( WOA) is a Data product of the Ocean Climate Laboratory of the National Oceanographic Data Center ( U The SOFAR channel is centered on the minimum in sound speed at ca. The SOFAR channel (Sound Fixing And Ranging channel or deep sound channel (DSCis a horizontal layer of water in the ocean centered around the depth at which the Speed 750-m depth.

In salt water that is free of air bubbles or suspended sediment, sound travels at about 1500 m/s. The speed of sound in seawater depends on pressure (hence depth), temperature (a change of 1 °C ~ 4 m/s), and salinity (a change of 1‰ ~ 1 m/s), and empirical equations have been derived to accurately calculate sound speed from these variables. Salinity is the Saltiness or dissolved salt content of a body of Water. ^[14] Other factors affecting sound speed are minor. For more information see Dushaw et al. ^[15]

A simple empirical equation for the speed of sound in sea water with reasonable accuracy for the world's oceans is due to Mackenzie:^[16]

c(T, S, z) = a₁ + a₂T + a₃T² + a₄T³ + a₅(S - 35) + a₆z + a₇z² + a₈T(S - 35) + a₉Tz³

where T, S, and z are temperature in degrees Celsius, salinity in parts per thousand and depth in metres, respectively. The constants a₁, a₂, . . . , a₉ are:

a₁ = 1448. 96, a₂ = 4. 591, a₃ = -5. 304×10^-2, a₄ = 2. 374×10^-4, a₅ = 1. 340, a₆ = 1. 630×10^-2, a₇ = 1. 675×10^-7, a₈ = -1. 025×10^-2, a₉ = -7. 139×10^-13

with check value 1550. 744 m/s for T=25 °C, S=35‰, z=1000 m. This equation has a standard error of 0. 070 m/s for salinities between 25 and 40 ppt. See Technical Guides - Speed of Sound in Sea-Water for an online calculator.

Other equations for sound speed in sea water are accurate over a wide range of conditions, but are far more complicated, e. g. , that by V. A. Del Grosso^[17] and the Chen-Millero-Li Equation. ^[18] ^[15]

Speed in plasma

The speed of sound in a plasma for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see here)

$c_s = (\gamma ZkT_e/m_i)^{1/2} = 9.79\times10^5\,(\gamma ZT_e/\mu)^{1/2}\,\mbox{cm/s}$

In contrast to a gas, the pressure and the density are provided by separate species, the pressure by the electrons and the density by the ions. In Physics and Chemistry, plasma is an Ionized Gas, in which a certain proportion of Electrons are free rather than being bound Plasma parameters define various characteristics of a plasma, an electrically conductive collection of Charged particles that responds collectively The two are coupled through a fluctuating electric field.

Gradients

When sound spreads out evenly in all directions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean there is a layer called the 'deep sound channel' or SOFAR channel which can confine sound waves at a particular depth, allowing them to travel much further. The SOFAR channel (Sound Fixing And Ranging channel or deep sound channel (DSCis a horizontal layer of water in the ocean centered around the depth at which the Speed In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher index, sound waves will refract towards a region where their speed is reduced. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium Refraction is the change in direction of a Wave due to a change in its Speed. The result is that sound gets confined in the layer, much the way light can be confined in a sheet of glass or optical fiber. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length

A similar effect occurs in the atmosphere. Project Mogul successfully used this effect to detect a nuclear explosion at a considerable distance. Project Mogul (sometimes referred to as Operation Mogul) was a top secret project by the US Army Air Forces involving High altitude balloons

References

^ Dean, E. A. (August 1979). Atmospheric Effects on the Speed of Sound, Technical report of Defense Technical Information Center
^ Everest, F. (2001). The Master Handbook of Acoustics. New York: McGraw-Hill, pp. 262-263. ISBN 0071360972.
^ ^a ^b U. S. Standard Atmosphere, 1976, U. S. Government Printing Office, Washington, D. C. , 1976.
^ Everest, F. (2001). The Master Handbook of Acoustics. New York: McGraw-Hill, pp. 262-263. ISBN 0071360972.
^ Uman, Martin (1984). Lightning. New York: Dover Publications. ISBN 0486645754.
^ Volland, Hans (1995). Handbook of Atmospheric Electrodynamics. Boca Raton: CRC Press, p. 22. ISBN 0849386470.
^ Singal, S. (2005). Noise Pollution and Control Strategy. Alpha Science International, Ltd, p. 7. ISBN 1842652370. “It may be seen that refraction effects occur only because there is a wind gradient and it is not due to the result of sound being convected along by the wind. ”
^ Bies, David (2003). Engineering Noise Control; Theory and Practice. London: Spon Press, p. 235. ISBN 0415267137. “As wind speed generally increases with altitude, wind blowing towards the listener from the source will refract sound waves downwards, resulting in increased noise levels. ”
^ Cornwall, Sir (1996). Grant as Military Commander. Barnes & Noble Inc. ISBN 1566199131 pages = p. 92.
^ Cozzens, Peter (2006). The Darkest Days of the War: the Battles of Iuka and Corinth. Chapel Hill: The University of North Carolina Press. ISBN 0807857831.
^ A B Wood, A Textbook of Sound (Bell, London, 1946)
^ ^a ^b Olson, Wayne M. Dr Albert Beaumont Wood OBE DSc ( 1890 - 19 July 1964) was a British physicist, known for his pioneering work in the field of (2002). "AFFTC-TIH-99-02, Aircraft Performance Flight Testing. " (PDF). Air Force Flight Test Center, Edwards AFB, CA, United States Air Force.
^ http://www.sizes.com/natural/sound.htm Accessed Jan 5, 2007
^ APL-UW TR 9407 High-Frequency Ocean Environmental Acoustic Models Handbook, pp. I1-I2.
^ ^a ^b Dushaw, Brian D. ; Worcester, P. F. ; Cornuelle, B. D. ; and Howe, B. M. (1993). "On equations for the speed of sound in seawater". Journal of the Acoustical Society of America 93 (1): 255-275. doi:10.1121/1.405660. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Mackenzie, Kenneth V. (1981). "Discussion of sea-water sound-speed determinations". Journal of the Acoustical Society of America 70 (3): 801-806. doi:10.1121/1.386919. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Del Grosso, V. A. (1974). "New equation for speed of sound in natural waters (with comparisons to other equations)". Journal of the Acoustical Society of America 56 (4): 1084-1091. doi:10.1121/1.1903388. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Meinen, Christopher S. ; Watts, D. Randolph (1997). "Further evidence that the sound-speed algorithm of Del Grosso is more accurate than that of Chen and Millero". Journal of the Acoustical Society of America 102 (4): 2058-2062. doi:10.1121/1.419655. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

Applied Physics Laboratory - University of Washington, 1994

External links

Calculation: Speed of sound in air and the temperature
Speed of sound - temperature matters, not air pressure
Properties Of The U.S. Standard Atmosphere 1976
The Speed of Sound at MathPages
How to measure the speed of sound in a laboratory
Speed of sound in water and water steam as function of pressure & temperature
Teaching resource for 14-16yrs on sound including speed of sound
Technical Guides - Speed of Sound in Pure Water
Technical Guides - Speed of Sound in Sea-Water
NewByte standard atmosphere calculator and speed converter
Did sound once travel at light speed? - If the speed of sound was greater just after the big bang, it could solve a longstanding mystery over the universe's background temperature (New Scientist, 9 April 2008)

Second sound is a Quantum mechanical phenomenon in which Heat transfer occurs by wave -like motion rather than by the more usual mechanism of Diffusion WikipediaWikiProject Aircraft. Please see WikipediaWikiProject Aircraft/page content for recommended layout The SOFAR channel (Sound Fixing And Ranging channel or deep sound channel (DSCis a horizontal layer of water in the ocean centered around the depth at which the Speed Underwater acoustics is the study of the propagation of Sound in Water and the interaction of the mechanical waves that constitute sound with the water and its boundaries

Dictionary

-noun

(physics) the speed at which sound is propagated through some medium under specified conditions

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Contents

Basic concept

General formulae

Dependence on the properties of the medium

Implications for atmospheric acoustics

Practical formula for dry air

Details

Speed in ideal gases and in air

Effects due to wind shear

Tables

Effect of frequency and gas composition

Mach number

Experimental methods

Single-shot timing methods

Other methods

Non-gaseous media

Speed in solids

Speed in liquids

Water

Seawater

Speed in plasma

Gradients

References

See also

External links

Dictionary

speed of sound

-noun