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Encyclopedia > Speed of sound
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 Pac
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 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.[1] The decrease of the sound speed with height is referred to as a negative sound speed gradient. However, in the stratosphere, the speed of sound increases with height due to heating within the ozone layer, producing a positive sound speed gradient. For the property of metals, see refraction (metallurgy). ... 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. ... In acoustics, the sound speed gradient is the rate of change of the speed of sound with depth in the ocean,[1] or height in the Earths atmosphere. ... Atmosphere diagram showing stratosphere. ... The ozone layer is a layer in Earths atmosphere which contains relatively high concentrations of ozone (O3). ...

## Practical formulae for dry air GA_googleFillSlot("encyclopedia_square");

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). For other uses, see Celsius (disambiguation). ...

This equation is derived from the first two terms of the Taylor expansion of the following much more accurate equation: As the degree of the taylor series rises, it approaches the correct function. ...

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

The value of 331.3 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. 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. The heat capacity ratio is simply the ratio of the heat capacity at constant pressure to that at constant volume It should be noted that chemical engineers and many others commonly refer to the heat capacity ratio as rather than . ... Standard atmosphere (symbol: atm) is a unit of pressure. ...

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. A derivation of these equations will be given in a later section. Atmosphere diagram showing stratosphere. ...

## Basic concept

The transmission of sound can be explained using a toy model consisting of an array of balls interconnected by springs. For a 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 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 theory. ...

In a real material, the stiffness of the springs is called the elastic modulus, and the mass corresponds to the density. 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. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substances tendency to be deformed when a force is applied to it. ... For other uses, see Density (disambiguation). ...

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.

## Details

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. For general equations of state, if classical mechanics is used, the speed of sound c is given by An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substances tendency to be deformed when a force is applied to it. ... For other uses, see Density (disambiguation). ...

$c^2=frac{partial p}{partialrho}$

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. For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity. ...

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).[2] For other uses, see Ultrasound (disambiguation). ... A kilohertz (kHz) is a unit of frequency equal to 1,000 hertz (1,000 cycles per second). ...

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 same phenomenon occurs with light waves -- see optical dispersion for a description. The phase velocity 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 waves amplitude (known as the modulation or envelope of the wave) propagate through space. ... Dispersion of a light beam in a prism. ...

### 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.

In a solid rod (with thickness much smaller than the wavelength) the speed of sound is given by:

$c_{mathrm{solids}} = sqrt{frac{E}{rho}}$

where

E is Young's modulus
ρ (rho) is density

Thus in steel the speed of sound is approximately 5,100 m·s-1. In beryllium, a substance with the greatest known ratio of stiffness to density at room temperature and pressure, the speed of sound reaches 12,870 m·s-1, which is the highest known for solids in standard conditions.[3] In solid mechanics, Youngs modulus (E) is a measure of the stiffness of a given material. ... For other uses, see Density (disambiguation). ... For other uses, see Steel (disambiguation). ... General Name, symbol, number beryllium, Be, 4 Chemical series alkaline earth metals Group, period, block 2, 2, s Appearance white-gray metallic Standard atomic weight 9. ...

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: In solid mechanics, Youngs modulus (E) is a measure of the stiffness of a given material. ... In solid mechanics, Youngs modulus (E) is a measure of the stiffness of a given material. ... Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ...

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

### Speed in liquids

In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).

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

The bulk modulus (K) of a substance essentially measures the substances resistance to uniform compression. ...

#### 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. Examples are sonar, acoustic communication and acoustical oceanography. 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. 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. ... This article is about underwater sound propagation. ... 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. ...

#### 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.

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.[4] Other factors affecting sound speed are minor. For more information see Dushaw et al.[5] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The World Ocean Atlas (WOA) is a data product of the Ocean Climate Laboratory of the National Oceanographic Data Center (USA). ... SOFAR channel stands for SONAR Fixing and Ranging channel. It is a layer of water deep in the ocean (near Bermuda its around 1000 metres deep) where the speed of sound is at a minimum. ... Annual mean sea surface salinity for the World Ocean. ...

A simple empirical equation for the speed of sound in sea water with reasonable accuracy for the world's oceans is due to Mackenzie:[6]

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. The three-letter acronym PPT may stand for: Microsoft Powerpoint, a Microsoft program intedended to construct presentations. ...

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[7] and the Chen-Millero-Li Equation.[8] [5]

### 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. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume(Cp / Cv), 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.
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: The adiabatic index of a gas, is the ratio of its specific heat capacity at constant pressure (CP) to its specific heat capacity at constant volume (CV). ... This article is about pressure in the physical sciences. ... For other uses, see Density (disambiguation). ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

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

where

• cideal is the speed of sound in an ideal gas.
• 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 = 1.400 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).
• T is the absolute temperature in kelvins.
• M is the molar mass in kilograms per mole. 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 cair have been found to vary slightly from experimentally determined values.[9] The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... The adiabatic index of a gas, is the ratio of its specific heat capacity at constant pressure (CP) to its specific heat capacity at constant volume (CV). ... This article is about the chemical series. ... For other uses, see Kelvin (disambiguation). ... Kg redirects here. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ...

Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of γ but was otherwise correct. Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. ... This article covers adiabatic processes in thermodynamics. ...

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. To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. For other uses, see Celsius (disambiguation). ... For other uses, see Kelvin (disambiguation). ...

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

For dry air, where $vartheta,$ (theta) is the temperature in degrees Celsius(°C). For other uses, see Celsius (disambiguation). ...

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.[10] Wind shear of 4 m/s/km can produce refraction equal to a typical temperature lapse rate of 7.5 °C/km.[11] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,[12] 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.[13] For the property of metals, see refraction (metallurgy). ... 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. ... The lapse rate is defined as the negative of the rate of change in an atmospheric variable, usually temperature, with height observed while moving upwards through an atmosphere. ...

For sound propagation, the exponential variation of wind speed with height can be defined as follows:[14]

$U(h) = U(0) h ^ zeta$
$frac {dU} {dH} = zeta frac {U(h)} {h}$

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,[15] because they could not hear the sounds of battle only six miles downwind.[16] Combatants United States of America (Union) Confederate States of America (Confederacy) Commanders Abraham Lincoln, Ulysses S. Grant Jefferson Davis, Robert E. Lee Strength 2,200,000 1,064,000 Casualties 110,000 killed in action, 360,000 total dead, 275,200 wounded 93,000 killed in action, 258,000 total... The Battle of Iuka was a United States Civil War battle fought from October 3 - September 19, 1862 in Iuka, Mississippi. ...

### Tables

In the standard atmosphere: Temperature and air pressure can vary from one place to another on the Earth, and can also vary in the same place with time. ...

T0 is 273.15 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). A knot is a non SI unit of speed equal to one nautical mile per hour. ... A knot is a non SI unit of speed equal to one nautical mile per hour. ... A knot is a non SI unit of speed equal to one nautical mile per hour. ...

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). 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. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

Effect of temperature
$vartheta$ in °C c in m·s-1 ρ in kg·m-3 Z in N·s·m-3
−10 325.2 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: For other uses, see Celsius (disambiguation). ... Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. ... Kg redirects here. ... For other uses, see Newton (disambiguation). ... 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 wind instruments. ...

 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

The Bell X-1, originally designated XS-1, was a joint NACA-U.S. Army Air Forces/US Air Force supersonic research project and the first aircraft to exceed the speed of sound in controlled, level flight. ... NASA technicians working on the X-43A at the tip of a Pegasus rocket attached to a Boeing B-52B prior to launch (March 27, 2004) The X-43 is an unmanned experimental hypersonic aircraft design with multiple planned scale variations meant to test different aspects of highly supersonic flight. ...

## 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.[17]

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. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes.:[9] For sound waves in an enclosure, the mean free path is the average distance the wave travels between reflections off of the enclosures walls. ...

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) due to the fact that they have a higher γ (5/3 = 1.67) 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

$c_{mathrm{gas-monatomic/diatomic}} = sqrt{{{1.67 over 1.40}}}$ = 1.09

This gives the 9 % difference, and would be a typical ratio for sound speeds at room temperature in helium vs. deuterium, each with a molecular weight of 4. 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). General Name, symbol, number helium, He, 2 Chemical series noble gases Group, period, block 18, 1, s Appearance colorless Standard atomic weight 4. ... Deuterium, also called heavy hydrogen, is a stable isotope of hydrogen with a natural abundance in the oceans of Earth of approximately one atom in 6500 of hydrogen (~154 PPM). ...

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). 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. To meet Wikipedias quality standards, this article or section may require cleanup. ...

## 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). At altitude, for reasons explained, Mach number is a function of temperature. An F/A-18 Hornet breaking the sound barrier. ... This article does not cite any references or sources. ...

Aircraft flight instruments, however, operate using pressure differential to compute Mach number; not temperature. 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. Six basic instruments in a light twin-engine airplane arranged in the basic-T. From top left: airspeed indicator, attitude indicator, altimeter, turn coordinator, heading indicator, and vertical speed indicator Most aircraft are equipped with a standard set of flight instruments which give the pilot information about the aircrafts... A Pitot tube is a measuring instrument used to measure fluid flow. ...

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:[18] An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... In fluid dynamics, Bernoullis equation, derived by Daniel Bernoulli, describes the behavior of a fluid moving along a streamline. ...

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

where

M is Mach number
qc is impact pressure and
P is static pressure.

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation: In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with the heat transfer within the fluid. ...

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

where

M is Mach number
qc is impact pressure measured behind a normal shock
P is static pressure.

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. 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.[18] Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation and graphing tools which, along with aggressive marketing, have made Excel one of the most popular microcomputer applications to date. ... OpenOffice. ...

## 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. This method uses the following idea. Microphones redirects here. ... For other uses, see Digital (disambiguation). ...

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). This article is about the concept of time. ... For other uses, see Frequency (disambiguation). ...

Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume. 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. This is an example of a compact experimental setup. Kundts tube is an experimental setup of August Kundt for the measurement of the speed of sound in both a gas and a solid rod. ... A standing wave. ... A standing wave. ...

A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water. 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 tuning fork is a simple metal two-pronged fork with the tines formed from a U-shaped bar of elastic material (usually steel). ... Pipe is a tube or hollow cylinder for the conveyance of fluid, gas and sometimes other materials. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... A standing wave. ...

Here it is the case that v =

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. 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 result is that sound gets confined in the layer, much the way light can be confined in a sheet of glass or optical fiber. SOFAR channel stands for SONAR Fixing and Ranging channel. It is a layer of water deep in the ocean (near Bermuda its around 1000 metres deep) where the speed of sound is at a minimum. ... 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. ... For the property of metals, see refraction (metallurgy). ... Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide 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, whose primary purpose was long-distance detection of sound waves generated by Soviet atomic bomb tests and ballistic missiles. ...

## References

1. ^ Everest, F. (2001). The Master Handbook of Acoustics. New York: McGraw-Hill, pp. 262-263. ISBN 0071360972.
2. ^ Dean, E. A. (August 1979). Atmospheric Effects on the Speed of Sound, Technical report of Defense Technical Information Center
3. ^ http://www.sizes.com/natural/sound.htm Accessed Jan 5, 2007
4. ^ APL-UW TR 9407 High-Frequency Ocean Environmental Acoustic Models Handbook, pp. I1-I2.
5. ^ 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.
6. ^ 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.
7. ^ 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.
8. ^ 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.
9. ^ a b U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.
10. ^ Everest, F. (2001). The Master Handbook of Acoustics. New York: McGraw-Hill, pp. 262-263. ISBN 0071360972.
11. ^ Uman, Martin (1984). Lightning. New York: Dover Publications. ISBN 0486645754.
12. ^ Volland, Hans (1995). Handbook of Atmospheric Electrodynamics. Boca Raton: CRC Press, p. 22. ISBN 0849386470.
13. ^ 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.”
14. ^ 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.”
15. ^ Cornwall, Sir (1996). Grant as Military Commander. Barnes & Noble Inc. ISBN 1566199131 pages = p. 92.
16. ^ 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.
17. ^ A B Wood, A Textbook of Sound (Bell, London, 1946)
18. ^ a b Olson, Wayne M. (2002). "AFFTC-TIH-99-02, Aircraft Performance Flight Testing." (PDF). Air Force Flight Test Center, Edwards AFB, CA, United States Air Force.
• Applied Physics Laboratory - University of Washington, 1994

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Dr Albert Beaumont Wood OBE DSc (1890 - 19 July 1964) was a British physicist, known for his pioneering work in the field of underwater acoustics and sonar. ...

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. ... U.S. Navy F/A-18 at transonic speed. ... SOFAR channel stands for SONAR Fixing and Ranging channel. It is a layer of water deep in the ocean (near Bermuda its around 1000 metres deep) where the speed of sound is at a minimum. ... 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. ...

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