Speed ​​of sound

Sound quantities

Sound deflection ${\displaystyle \xi }$ Sound pressure ${\displaystyle p}$ Sound pressure level ${\displaystyle L_{p}}$ Sound energy density ${\displaystyle E}$ Sound energy ${\displaystyle W}$ Sound flow ${\displaystyle q}$ Speed ​​of sound ${\displaystyle c_{\text{S}}}$ Acoustic impedance ${\displaystyle Z}$ Sound intensity ${\displaystyle I}$ Sound power ${\displaystyle P_{\text{ak}}}$ Speed ​​of sound ${\displaystyle v}$ Fast sound amplitude ${\displaystyle v}$ Sound radiation pressure

The speed of sound is the speed at which sound waves propagate in a medium. Its SI unit is meters per second  (m / s). ${\displaystyle c_{\text{S}}}$

It is not to be confused with the speed of sound , i.e. H. the instantaneous speed at which the individual particles of the medium move in order to build up and break down the deformation associated with the sound wave. ${\displaystyle v}$

The speed of sound is generally dependent on the medium (in particular elasticity and density ), and its temperature in fluids in addition the pressure and in solids largely on the wave type ( longitudinal wave , shear wave , Rayleigh wave , Lamb wave , etc.) and of the frequency. In anisotropic media it is also directional. In gases or gas mixtures such as air under conditions around 1 bar and 20 ° C, only the temperature dependence plays a significant role.

The speed of sound in dry air at 20 ° C is 343.2 m / s (1236 km / h).

For the relationship between the speed of sound and the frequency of a monochromatic sound wave of wavelength , as for all such waves, the following applies: ${\displaystyle c}$ ${\displaystyle f}$ ${\displaystyle \lambda }$

${\displaystyle c_{\text{S}}=\lambda \,f}$

Speed ​​of sound in liquids and gases

In liquids and gases , only pressure or density waves can propagate in which the individual particles move back and forth in the direction of wave propagation ( longitudinal wave ). The speed of sound is a function of the density and the ( adiabatic ) compression modulus and is calculated as follows: ${\displaystyle \rho }$ ${\displaystyle K}$

${\displaystyle c_{\,{\text{Flüssigkeit, Gas}}}={\sqrt {\frac {K}{\rho }}}}$

Speed ​​of sound in solids

Sound waves in solids can propagate both as a longitudinal wave (the direction of oscillation of the particles is parallel to the direction of propagation) or as a transverse wave (direction of oscillation perpendicular to the direction of propagation).

For longitudinal waves, the speed of sound in solids generally depends on the density , Poisson's number and the modulus of elasticity of the solid. The following applies ${\displaystyle \rho }$ ${\displaystyle \nu }$ ${\displaystyle E}$

${\displaystyle c_{\text{Festkörper, longitudinal}}={\sqrt {\frac {E\,(1-\nu )}{\rho \,(1-\nu -2\nu ^{2})}}}}$

${\displaystyle c_{\text{Festkörper, transversal}}={\sqrt {\frac {E}{2\,\rho \,(1+\nu )}}}={\sqrt {\frac {G}{\rho }}}}$

with the shear module . ${\displaystyle G={\frac {E}{2\,(1+\nu )}}}$

For a surface wave on an extended solid (Rayleigh wave) the following applies:

${\displaystyle c_{\text{Festkörper, Oberfläche}}\approx 0{,}922\cdot c_{\text{Festkörper, transversal}}}$

The term is also known as the longitudinal module , so for the longitudinal wave too ${\displaystyle M={\frac {E\,(1-\nu )}{(1-\nu -2\nu ^{2})}}}$

${\displaystyle c_{\text{Festkörper, longitudinal}}={\sqrt {\frac {M}{\rho }}}}$

can be written.

In the special case of a long rod, the diameter of which is significantly smaller than the wavelength of the sound wave, the influence of the transverse contraction can be neglected (i.e. ), and one obtains: ${\displaystyle \nu =0}$

${\displaystyle c_{\text{langer Stab, longitudinal}}={\sqrt {\frac {E}{\rho }}}}$

${\displaystyle c_{\text{langer Stab, transversal}}={\sqrt {\frac {E}{2\rho }}}}$

Speed ​​of sound in the ideal gas

Classic ideal gas

Since the compression modulus of a classical, ideal gas only depends on the adiabatic exponent (" kappa ") of the gas and the prevailing pressure , the following results for the speed of sound: ${\displaystyle K=\kappa \,p}$ ${\displaystyle \kappa }$ ${\displaystyle p}$

${\displaystyle c_{\text{Ideales Gas}}={\sqrt {\kappa \,{\frac {p}{\rho }}}}={\sqrt {\kappa \,{\frac {RT}{M}}}}}$

Here is the universal gas constant , the molar mass (mass / amount of substance) of the gas, and the absolute temperature . For fixed values and , therefore, for a given ideal gas, the speed of sound depends only on the temperature. In particular, it is not dependent on the pressure or the density of the gas. The adiabatic exponent is calculated approximately , where the number of degrees of freedom of movement of a particle (atom or molecule) is. The following applies to a mass point: for a rigid dumbbell with two mass points (molecule with two atoms) , for a rigid body with more than two mass points (strongly angled molecule) , for non-rigid bodies with more than two mass points (molecule with a missing rigid connection ) . For complex molecules the degree of freedom increases by every missing rigid connection . Without taking into account the vibration of all polyatomic molecules in the higher temperature range, the adiabatic exponent can only assume the following values: ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle T}$ ${\displaystyle M}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ={\tfrac {f+2}{f}}}$ ${\displaystyle f}$ ${\displaystyle f=3}$ ${\displaystyle f=5}$ ${\displaystyle f=6}$ ${\displaystyle f=7}$ ${\displaystyle f>7}$

• ${\displaystyle \kappa ={\tfrac {5}{3}}\approx 1{,}67}$ for monatomic gases (e.g. all noble gases)
• ${\displaystyle \kappa ={\tfrac {7}{5}}=1{,}40}$for diatomic gases (e.g. nitrogen N ₂ , hydrogen H ₂ , oxygen O ₂ , carbon monoxide CO)
• ${\displaystyle \kappa ={\tfrac {8}{6}}\approx 1{,}33}$for rigid molecules with more than two atoms (e.g. water vapor H ₂ O, hydrogen sulfide H ₂ S, methane CH ₄ )
• ${\displaystyle \kappa ={\tfrac {9}{7}}\approx 1{,}29}$for molecules with a missing rigid connection (e.g. nitrogen oxides NO ₂ and N ₂ O, carbon dioxide CO ₂ , sulfur dioxide SO ₂ , ammonia NH ₃ )
• ${\displaystyle \kappa ={\tfrac {10}{8}}=1{,}25}$for larger molecules with missing rigid connections, (e.g. ethane C ₂ H ₆ , ethene C ₂ H ₄ , methanol CH ₃ OH)

For dry air (mean molar mass , normal temperature , ) one obtains ${\displaystyle M=0{,}02896\,\mathrm {kg/mol} }$ ${\displaystyle T=293{,}15\,\mathrm {K} }$ ${\displaystyle \kappa =1{,}402}$

${\displaystyle c_{\text{Luft}}\,\approx \,{\sqrt {1{,}402\,{\frac {8{,}3145\,\mathrm {\frac {J}{mol\,K}} 293{,}15\,\mathrm {K} }{0{,}02896\,\mathrm {\frac {kg}{mol}} }}}}=343{,}5\,\mathrm {\frac {m}{s}} }$, in good agreement with the value measured in dry air.

The speed of sound is somewhat lower than the mean translation speed of the particles moving in the gas. This is in line with the clear interpretation of sound propagation in the kinetic gas theory : a small local deviation of the pressure and density from their average values ​​is carried into the environment by the particles flying around one another. ${\displaystyle c_{\text{Ideales Gas}}={\sqrt {\kappa \,{\tfrac {RT}{M}}}}}$ ${\displaystyle {\sqrt {\overline {v^{2}}}}={\sqrt {3\,{\tfrac {RT}{M}}}}}$

The factor comes from the adiabatic equation of state , which describes processes in which the temperature does not remain constant although no heat is exchanged. Sound waves consist of periodic fluctuations in density and pressure, which occur so quickly that heat can neither flow in nor out during them. Because of the associated temperature fluctuations, the above formula only applies in the limiting case of small amplitudes, whereby the average temperature should be used. In fact, at large amplitudes, e.g. B. After a detonation, non-linear effects are noticeable in that the wave crests - wave fronts with maximum density - run faster than the wave troughs, which leads to steeper waveforms and the formation of shock waves . ${\displaystyle \kappa }$ ${\displaystyle c_{\text{Luft}}}$ ${\displaystyle T}$

Quantum effects

Since the speed of sound was relatively easy to measure precisely with Kundt's tube on the one hand and is directly linked to an atomic physical quantity, the number of degrees of freedom, on the other hand, it led to the early discovery of important effects that could only be explained with quantum mechanics .

Atoms as mass points

The first chemical methods as monatomic identified gas -  mercury vapor at high temperature - was in 1875 for the first time the value , that is . According to the kinetic gas theory, this value is reserved for a gas consisting of ideal mass points. From 1895, the same findings were made on the newly discovered noble gases argon , neon, etc. On the one hand, this supported the atomic hypothesis of the time , according to which all matter is made up of tiny spheres, but, on the other hand, raised the question of why these spheres do not, like any rigid body, have three further degrees of freedom for rotational movements. The quantum mechanical explanation found at the end of the 1920s states that for rotational movements excited energy levels must be occupied whose energy is so high that the kinetic energy of the colliding gas particles is far from sufficient. This also applies to the rotation of a diatomic molecule around the connecting line of the atoms and thus explains why there are not three, but only two degrees of freedom for the rotation. ${\displaystyle \kappa =1{,}667}$ ${\displaystyle f=3}$

Freezing the rotation

A marked temperature dependence of the adiabatic coefficient was measured at 1912 hydrogen discovered: During cooling from 300 K to 100 K increases monotonically from to , d. H. from the value for a dumbbell to the value for a mass point. It is said that the rotation “freezes”, at 100 K the whole molecule behaves like a mass point. The quantum mechanical justification follows on from the above explanation for single atoms: At 100 K the collision energy of the gas molecules is practically never enough to excite an energy level with a higher angular momentum, at 300 K it is practically always. The effect is so clearly not observable with other gases because they are already liquefied in the respective temperature range. However, this explains why the measured adiabatic coefficients of real gases usually deviate somewhat from the simple formula . ${\displaystyle \kappa }$ ${\displaystyle 1{,}400}$ ${\displaystyle 1{,}667}$ ${\displaystyle \kappa ={\tfrac {f+2}{f}}}$

Speed ​​of sound in real gas / phenomena in the air atmosphere

The ideas and formulas developed for the ideal gas also apply to a very good approximation for most real gases. In particular, their adiabatic exponent varies over wide ranges neither with temperature nor with pressure. The linear approximation formula is often used for the temperature dependence of the speed of sound in air in the range of normal ambient temperatures ${\displaystyle \kappa =c_{\mathrm {p} }/c_{\mathrm {V} }}$

${\displaystyle c_{\mathrm {Luft} }\approx (331{,}5+0{,}6\,\vartheta /{}^{\circ }\mathrm {C} ){\frac {\mathrm {m} }{\mathrm {s} }}}$

used. This approximation applies in the temperature range −20 ° C <<+40 ° C ${\displaystyle \vartheta }$ with an accuracy of more than 99.8%. The absolute temperature was converted into ° C here . ${\displaystyle \vartheta /{}^{\circ }\mathrm {C} =T/\mathrm {K} -273{,}15}$

In addition to the temperature dependence of the speed of sound in air, the influence of air humidity must be taken into account. This causes the speed of sound to increase slightly, because the mean molar mass of moist air decreases more strongly than the mean adiabatic coefficient due to the addition of lighter water molecules . For example, at 20 ° C, the speed of sound at 100% humidity is 0.375% higher than at 0% humidity. The same increase in the speed of sound compared to dry air would result from a temperature increase to a good 22 ° C. ${\displaystyle M}$ ${\displaystyle \kappa }$

In the normal atmosphere, the speed of sound therefore decreases with altitude. It reaches a minimum of about 295 m / s (1062 km / h) in the tropopause (about 11 km altitude). On the other hand, the speed of sound increases with altitude in the case of an inversion weather situation , since a warmer layer of air then lies over a colder one. Often this happens in the evening after a warm sunny day, because the ground cools faster than the higher layers of air. Then the waves advance faster upwards than downwards, so that a wave front that strives diagonally upwards from a sound source close to the ground is directed downwards again (see sound propagation ). It is said that the sound beams are curved towards the ground. On summer evenings this can often be seen in the greater range of sound propagation.

The reasoning for hearing better with the wind than against the wind is similar. Although the movement of the medium air should not have any influence on the propagation of sound as such, since the wind speed is always small compared to the speed of sound, the range of the sound improves. The wind almost always has a speed profile with an increasing speed, which, as described above, leads to a deflection of the sound propagation, namely a deflection upwards with headwind and downwards with headwind.

Examples of the speed of sound in different media

The following tables list some examples of sound velocities in different media at a temperature of 20 ° C. The speed of sound for the pressure wave (longitudinal wave) is given for all materials; shear waves (transverse waves) also propagate in solids.

Speed ​​of sound in selected gases at 20 ° C

gas	longitudinal in m / s
air	343
helium	981
hydrogen	1280
Oxygen (at 0 ° C)	316
Carbon dioxide	266
argon	319
krypton	221
methane	466
Water vapor (at 100 ° C)	477
Sulfur hexafluoride (at 0 ° C)	129

Speed ​​of sound in selected liquids at 20 ° C

medium	longitudinal in m / s
water	1484
Water (at 0 ° C)	1407
Sea water	≈1500
2.5 mol sodium chloride solution (at 25 ° C)	1540
Oil (SAE 20/30)	1340
benzene	1326
Ethyl alcohol	1168
mercury	1450

Speed ​​of sound in selected solids at 20 ° C

medium	longitudinal in m / s	transversal in m / s
Ice (at −4 ° C)	3250	1990
rubber	1500	150
Silicone rubber (RTV)	≈ 1000
Plexiglass	2670	1120
PVC -P (soft)	80
PVC-U (hard)	2250	1060
POM	2470	1200
Concrete (C20 / 25)	3655	2240
Concrete (C30 / 37)	3845	2355
Beech wood	3300
marble	6150
aluminum	6250-6350	3100
beryllium	12,800, 12,900	8710, 8880
lead	2160	700
gold	3240	1200
copper	4660	2260
magnesium	5790	3100
Magnesium / Zk60	4400	810
nickel	4900
zinc	4170	2410
steel	5850, 5920	3230
titanium	6100	3120
Brass	3500
tungsten	5180	2870
iron	5170
silver	3600	1590
boron	16,200
diamond	18,000
Graph	20,000

Speed ​​of sound under extreme conditions

medium	longitudinal in m / s
Dense molecular cloud	1,000
Earth's core ( seismic P waves )	8,000 ... 11,000
Interplanetary medium at the level of the earth's orbit	60,000
Interstellar medium (depends strongly on the temperature)	10,000 ... 100,000
Nuclear matter	60,000,000

Temperature dependence

Speed ​​of sound as a function of the air temperature

Temperature in ° C ${\displaystyle \vartheta }$	Speed ​​of sound in m / s ${\displaystyle c_{\text{S}}}$	Speed ​​of sound in km / h ${\displaystyle c_{\text{S}}}$
+50	360.57	1298.0
+40	354.94	1277.8
+30	349.29	1257.2
+20	343.46	1236.5
+10	337.54	1215.1
0	331.50	1193.4
−10	325.35	1171.3
−20	319.09	1148.7
−30	312.77	1126.0
−40	306.27	1102.6
−50	299.63	1078.7

Frequency dependence

In a dispersive medium, the speed of sound depends on the frequency . The spatial and temporal distribution of a reproductive disorder is constantly changing. Each frequency component propagates with its own phase velocity , while the energy of the disturbance propagates with the group velocity . Rubber is an example of a dispersive medium: at a higher frequency it is stiffer, i.e. it has a higher speed of sound.

In a non-dispersive medium, the speed of sound is independent of the frequency. Hence the speeds of energy transport and sound propagation are the same. Water and dry air are non-dispersive media in the frequency range that can be heard by humans. At high humidity and in the near ultrasonic range (100 kHz), air is dispersive.

Speed ​​of sound and thermodynamics

The speed of sound plays a special role in thermodynamics , especially in pressure relief devices, where it defines the maximum attainable speed. Because it can be measured with extreme accuracy, it plays a major role in the establishment of highly precise equations of state and in the indirect measurement of the heat capacity of an ideal gas . The general equation for calculating the speed of sound is

${\displaystyle c^{2}=-v^{2}\left({\frac {\partial p}{\partial v}}\right)_{\!\!s}}$

with as the spec. Volume, the reciprocal of density (v = 1 / ρ). The index s in the differential quotient means "with constant specific entropy " ( isentropic ). For the ideal gas, this results as stated above ${\displaystyle v}$

${\displaystyle c_{id}={\sqrt {\kappa RT}}}$

With

${\displaystyle \kappa ={\frac {c_{p}^{id}}{c_{v}^{id}}}}$

than the ratio of the isobaric and the isochoric spec. Heat capacities and R as the special gas constant (related to mass). The common thermal equations of state have the form . It follows after some transformations ${\displaystyle p=f(T,v)}$

${\displaystyle c^{2}=v^{2}\left[{\frac {T}{c_{v}}}\left({\frac {\partial p}{\partial T}}\right)_{v}^{2}-\left({\frac {\partial p}{\partial v}}\right)_{T}\right]}$

with the real spec. isochoric heat capacity

${\displaystyle c_{v}=c_{v}^{id}+T\int _{\infty }^{v}\left({\frac {\partial ^{2}p}{\partial T^{2}}}\right)_{v}dv}$

Fig. 1: Velocity of sound of ethylene at 100 ° C as a function of pressure

Fig. 2: Mass flow density of a gas flow as a function of the outlet pressure

Image 3: Shape of a Laval nozzle

Fig. 4: Laval nozzles on the engine model of the Saturn V rocket in Cape Canaveral

With these relationships, one can take into account the influence of pressure on the speed of sound if a thermal equation of state is known. Figure 1 shows the dependence of the speed of sound on the pressure in ethylene for a temperature of 100 ° C.

The speed of sound has become particularly important because of its easy experimental accessibility. The specific heat capacity of ideal gases, which can hardly be measured directly, is linked to the speed of sound of the ideal gas:

${\displaystyle c_{p}^{id}={\frac {Rc^{2}}{c^{2}-RT}}}$

The gas constant can also be determined very precisely with sound velocity measurements. For monatomic noble gases (He, Ne, Ar) is independent of the temperature. Then follows ${\displaystyle c_{p}^{id}=2{,}5R}$

${\displaystyle R={\frac {3}{5}}{\frac {c^{2}}{T}}}$

Since and can be measured very precisely, this is an extremely precise method of determining the gas constant. The speed of sound is decisive for the pressure relief of gases via a valve or a diaphragm. Depending on the condition in the container to be relieved, there is a maximum mass flow density in the narrowest cross-section of the valve, which cannot be exceeded, even if the pressure beyond the valve is further reduced (Fig. 2). The speed of sound of the gas then occurs in the narrowest cross-section. In the case of ideal gases, this is approximately the case when the outlet pressure is less than half the container pressure. The max. Mass flow density also applies when a gas flows through a pipe with a constant cross-section. The speed of sound cannot then be exceeded, which is also of considerable safety significance for the design of pressure relief devices. In order to accelerate a gas beyond the speed of sound, specially shaped flow channels are required, which expand in a defined manner according to a narrowest cross section, so-called Laval nozzles (Fig. 3). An example of this are the outlet nozzles of rocket engines (Fig. 4). ${\displaystyle c}$ ${\displaystyle T}$

Others

In aviation, the speed of an aircraft is also measured relative to the speed of sound. The unit Mach (named after Ernst Mach ) is used, where Mach 1 is equal to the respective speed of sound. In contrast to other units of measurement, when measuring speed in Mach, the unit is placed in front of the number.

The distance of a lightning bolt, and thus a thunderstorm, can be estimated by counting the seconds between the lightning bolt flashing and the thundering . Sound travels a kilometer in the air in around three seconds, while light flashes in a negligibly short three microseconds . If you divide the number of seconds counted by three, the result is roughly the distance to the lightning in kilometers.

See also

• Sound propagation
• Acoustic impedance
• Supersonic speed
• Supersonic flight
• sound

Web links

Wiktionary: Speed ​​of sound  - explanations of meanings, word origins, synonyms, translations

• Experiment A6: Speed ​​of sound in gases and solids. (PDF; 169 kB).
• The speed of sound, the temperature ... and not the air pressure. (PDF; 32 kB).
• Calculate the speed of sound in air and the effective temperature.

Individual evidence

1. ↑ Douglas C. Giancoli: Physics . Pearson Deutschland GmbH, 2010, p. 561 ( limited preview in Google Book search).
2. ↑ The surface wave velocity depends on the Poisson's number . For is a factor of 0.8741 (e.g. cork ) instead of the specified 0.92, for is 0.9194 (e.g. iron ) and for is 0.9554 (e.g. rubber ). See Arnold Schoch: Sound reflection, sound refraction and sound diffraction . In: Results of the exact natural sciences . tape ${\displaystyle \nu }$ ${\displaystyle \nu =0}$ ${\displaystyle \nu =0{,}25}$ ${\displaystyle \nu =0{,}5}$ 23 , 1950, pp. 127-234 .
3. ↑ ^{a b} Jörn Bleck-Neuhaus: Elementary Particles. Modern physics from the atoms to the standard model . Springer-Verlag (Heidelberg), 2010, ISBN 978-3-540-85299-5 , doi : 10.1007 / 978-3-540-85300-8 .
4. ^ Owen Cramer: The variation of the specific heat ratio and the speed of sound in air with temperature, pressure, humidity, and CO ₂ concentration. In: The Journal of the Acoustical Society of America. Vol. 93 (5), p. 2510, 1993.
5. ^ Dennis A. Bohn: Environmental Effects on the Speed ​​of Sound. In: Journal of the Audio Engineering Society . 36 (4), April 1988. PDF version.
6. ↑ ^{a b c d} A. J. Zuckerwar: Handbook of the Speed ​​of Sound in Real Gases. Academic Press 2002.
7. ↑ ^{a b c d} David R. Lide (Ed.): CRC Handbook of Chemistry and Physics . 57th edition. (Internet version:), CRC Press / Taylor and Francis, Boca Raton, FL, , p E-47.
8. ↑ Sonja Kerstin Leicht: Investigation of mechanical parameters of the degenerative changes in cartilage and subchondral bone in primary osteoarthritis of the knee using high-frequency ultrasound techniques. Dissertation, University of Halle 2007, Section 3.
9. ↑ Compression module EÖl (K). ( Memento from January 5, 2013 in the web archive archive.today ). Fluid technology from A to Z. At: vfmz.com.
10. ↑ ^{a b c d e f g h i j k l m n o p q r s t u v w} Joseph L. Rose: Ultrasonic Waves in Solid Media . Cambridge University Press, 2004, ISBN 978-0-521-54889-2 ( limited preview in Google Book Search).
11. ^ Y. Yamashita, Y. Hosono, K. Itsumi: Low-Attenuation Acoustic Silicone Lens for Medical Ultrasonic Array Probes. Pp. 169 and 175. In: Ahmad Safari, E. Koray Akdogan (Ed.): Piezoelectric and Acoustic Materials for Transducer Applications. Springer-Verlag , 2008, ISBN 0-387-76540-9 , pp. 161-178.
12. ↑ Vadim Adamyan, Vladimir Zavalniuk: phonon in graphene with point defects. In: J. Phys. Condens. Matter 23 (1), 2011, p. 15402.
13. ^ IS Glass, Glass I. S. Handbook of Infrared Astronomy . Cambridge University Press, Cambridge 1999, ISBN 978-0-521-63385-7 , pp. 98 ( books.google.de ).
14. ↑ Imke de Pater, Jack J. Lissauer: Planetary Sciences . Cambridge University Press, Cambridge 2015, ISBN 978-1-316-19569-7 , pp. 286 ( books.google.de ).
15. ↑ JEDyson, DA Williams: The Physics of the Interstellar Medium , Tylor & Francis, New York (1997), 2nd ed., P. 123.
16. ^ John Hussey: Bang to Eternity and Betwixt: Cosmos . John Hussey, 2014 ( books.google.de ).
17. ^ Walter Greiner, Horst Stöcker, André Gallmann: Hot and Dense Nuclear Matter, Proceedings of a NATO Advanced Study , ISBN 0-306-44885-8 , 1994 Plenum Press, New York p. 182.
18. ↑ Source unknown, s. also David R. Lide (Ed.): CRC Handbook of Chemistry and Physics . 57th edition. (Internet version:), CRC Press / Taylor and Francis, Boca Raton, FL, , p e-54th
19. ↑ Dispersion relation for air via Kramers-Kronig analysis . In: The Journal of the Acoustical Society of America . tape 124 , no. 2 , July 18, 2008, ISSN  0001-4966 , p. EL57-EL61 , doi : 10.1121 / 1.2947631 .
20. ↑ ^{a b c d} Jürgen Gmehling, Bärbel Kolbe, Michael Kleiber, Jürgen Rarey: Chemical Thermodynamics for Process Simulation . Wiley-VCH, Weinheim 2012, ISBN 978-3-527-31277-1 .