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2.4 Acoustics

2.4.1 The speed and attenuation of sound

Gases and vapours

The attenuation of plane sound waves in neper/unit length is α = (l/2d)ln(I0/Id), where the initial intensity I0, has decreased to Id after traversing distance d. When expressed in decibels per unit length, convenient for practical application, the value is 8.686α. In normal monatomic gases α varies classically as f 2, f being the frequency, but in polyatomic gases relaxation phenomena may dominate over classical absorption. (See Herzfeld and Litovitz, 1959.) For gases at moderate pressure the speed of sound c = (γp/ρ) or alternatively c = /(γRT/M), p being the ambient gas pressure, γ the ratio of the specific heat at constant pressure to that at constant volume, R the gas constant, T the absolute temperature and M the molecular weight. Thus, to first order, c is proportional to T and is independent of gas pressure; at higher pressures, however, the value is changed due to contributions from the second and higher virial coefficients. The speed of sound waves bounded by walls or tubes is less than the free-space value. Dispersion is observed for sound propagation in many polyatomic gases due to molecular relaxation: above the relaxation frequency the rotational or vibrational degrees of freedom are not excited, the specific heats are modified and the speed of sound is increased. The table below gives experimentally determined values of sound speed in gases and vapours, selected from the literature. Where known, both the low-frequency and high-frequency limiting values are given, the latter being in brackets.




Speed of sound in gases and vapours

Gas

t/ºC

  c/(m s−1 )

Gas

t/ºC  

  c/(m s−1)

 

 

 

 

 

 

Air     .      .      .      .      .      .

(see "Speed of Sound in Air" table)

  Freon 22 (CHCl F2)       .      .

17  

  179 (193)

Acetaldehyde   .      .      .      .

0    

  278

  Freon 113 (CCl2FCCl F2)     .

53  

  124 (139)

Acetylene         .      .      .      .

0    

  329

  Helium     .    .    .    .    .    .    .

0  

  972.5

Ammonia         .      .      .      .

30    

  440

  Hydrogen bromide  .    .    .    .

0  

  200

Argon       .      .      .      .      .

0    

  307.85

  Hydrogen chloride        .    .    .

0  

  296

Benzene           .      .      .      .

90    

  200

  Hydrogen iodide     .    .    .    .

0  

  157

Bromine           .      .      .      .

58    

  149

  Hydrogen sulfide     .    .    .    .

24  

  309

Carbon dioxide     .      .      .

51    

  280 (293)

  Krypton         .    .    .    .    .    .

30  

  224

Carbon disulfide      .      .      .

35    

  206

  Methane       .    .    .    .    .    .

41  

  466

Carbon tetrachloride

22    

  133 (146)

  Neon       .    .    .    .    .    .    .

30  

  461

Chloroform     .      .      .      .

22    

  154 (166)

  Nitric oxide        .    .    .    .    .

16  

  334

Cyclohexane          .      .      .

30    

  181 (200)

  Nitrogen       .    .    .    .    .    .

29  

  354.4

Deuterium       .      .      .      .

0    

  888 (969)

  Nitrous oxide    .    .    .     .    .

25  

  268 (281)

Diethyl ether          .      .      .

40    

  187

  Oxygen         .    .    .    .    .    .

30  

  332.2

Ethane           .      .      .      .

31    

  316 (335)

  Sulfur hexafluoride        .    .    .

11  

  133 (147)

Ethylene        .      .      .      .

20    

  327

  Water      .    .    .    .    .    .    .

100  

  477.5

Fluorine        .      .      .      .

102    

  332 (339)

  Water (6 MPa)  .    .    .    .    .

350  

  571

Freon 11 (CCl3F)      .      .

18    

  143 (154)

  Water (heavy)    .    .    .    .    .

100  

  450

Freon 12 (CCl2F2)      .      .

17    

  140 (152)

 

 

 

 

 

 

 

 

 




Attenuation of sound in air

The attenuation of sound in air due to viscous, thermal and rotational loss mechanisms is simply proportional to f 2. However, losses due to vibrational relaxation of oxygen molecules are generally much greater than those due to the classical processes, and the attenuation of sound varies significantly with temperature, water-vapour content and frequency. A method for calculating the absorption at a given temperature, humidity, and pressure can be found in ISO 9613-1 (1993). The table gives values of attenuation in dB km−1 for a temperature of 20°C and a pressure of 101.325 kPa. The uncertainty is estimated to be ± 10%.



Attenuation of sound in air (dB km−1)


Frequency
(kHz)

Relative Humidity %

10   

20   

30   

40   

50   

60   

70   

80   

90   

1

14

  6.5

5

  4.7

  4.7

  4.8

    5

   5.1

   5.3

     1.25

21

  9.4

  6.7

  5.9

  5.7

  5.7

     5.9

   6.1

   6.3

   1.6

32

14  

  9.8

  8.1

  7.5

  7.2

     7.2

   7.4

   7.5

2

45

22  

14  

11  

  9.9

  9.3

  9

9

  9.1

   2.5

63

32  

21  

16  

14  

12  

12

11  

11  

     3.15

85

49  

32  

24  

20  

17  

16

15  

15  

4

110 

75  

49  

36  

30  

26  

23

21  

20  

5

130  

110   

74  

55  

44  

38  

33

31  

28  

  6.3

160  

160   

110   

84  

68  

57  

50

45  

42  

180  

220  

170   

130   

110   

89  

78

69  

63  

10  

190  

280  

240  

190   

160   

130   

120 

100   

95  

 12.5

210  

360  

340  

280  

240  

200  

180 

160   

140   

16  

230  

430  

470  

420  

360  

320  

280

250  

230  

20  

260  

510  

600  

580  

520  

470  

420

380  

350  

25  

300  

580  

740  

770  

730  

680  

620

570  

520  

 31.5

360  

670  

890  

990  

1000    

960  

900

840  

790  

40  

460  

780  

1100   

1200   

1300    

1300    

1300  

1200    

1200   

50  

600  

940  

1300  

1500   

1700    

1700   

1700  

1700    

1700   

63  

840  

1200    

1500  

1800   

2100   

2200  

2300  

2300   

2300   

80  

1200   

1600    

2000  

2300  

2600   

2800  

3000  

3100   

3100   

100    

1800   

2200    

2500  

2900  

3300   

3600  

3800  

4000  

4100  




Speed of sound in air

The speed of free progressive sound waves in standard dry air containing 0.03% CO2 by volume is 331.46 ± 0.10 m s−1 at a temperature of 0°C and a pressure of 101.325 kPa (see Cramer, 1993). The speed of sound in air changes with temperature, water vapour content, and CO2 content. The table gives values of the speed in m s−1 for a range of temperatures and humidities at 0.03% CO2 by volume. The uncertainty in the values in the table is estimated to be 0.1 m s−1.



Speed of sound in air (m s−1)

  Temperature
        °C

Relative Humidity %

10

20

30

40

50

60

70

80

90

 

 

 

 

 

 

 

 

 

 

  0

331.5

331.5

331.5

331.6

331.6

331.6

331.7

331.7

331.7

  5

334.5

334.6

334.6

334.7

334.7

334.7

334.8

334.8

334.9

10

337.5

337.6

337.7

337.7

337.8

337.9

337.9

338.0

338.0

15

340.5

340.6

340.7

340.8

340.9

341.0

341.1

341.2

341.2

20

343.5

343.6

343.7

343.9

344.0

344.1

344.2

344.4

344.5

25

346.4

346.6

346.8

347.0

347.1

347.3

347.5

347.6

347.8

30

349.4

349.6

349.9

350.1

350.3

350.5

350.8

351.0

351.2

 

 

 

 

 

 

 

 

 

 




Speed, attenuation and non-linearity of sound in liquids

The speed of dilatational waves in unbounded fluids is c = aρ)−1/2, βa being the adiabatic compressibility and ρ the density. In practice the propagation velocity of each part of a sound wave depends on the local amplitude and the importance of this effect is determined by the ratio B/A = (ρ/c2)∂2p/∂ρ2 (where p is the pressure and all quantities are at ambient pressures and constant entropy). For plane waves the sound speed is then c + βu, where u is the particle velocity and β = 1 + B/2A (see Hamilton and Blackstock, 1988). In normal fluids the classical attenuation is proportional to f 2, f being the frequency, but relaxation phenomena may dominate; the attenuation generally decreases with increasing temperature. Values for various liquids are given below, selected from numerous sources; f is in Hz and α is in neper m−1. Where known, the frequency at which the attenuation was measured is given in brackets, in MHz. The temperature for the B/A values is given in brackets, in °C.



Properties of sound in liquids

Liquid

t

°C

c

m s−1

dc/dt

m s−1 K−1

(α/f 2)

10−15 neper m−1 Hz−2

B

A

t

°C

 

 

 

 

 

 

Acetic acid

    19.6

1173

Acetone

  25

1170

− 4.5

~ 35

9.2 (20)     

Aniline

  25

1640

− 3.6

~ 50

Argon

− 243   

  840

− 6.5

Benzene

  25

1300

− 4.7

870 (< 70)

9.0 (20)     

Bismuth

280

1651

  − 0.13

~8 (20)

  7.1 (318)     

n-Butanol

  25

1242

− 3.4

 85 (25)

10.7 (20)       

Caesium

  40

  980

  − 0.31

112 (30) 

Carbon disulfide

  25

1141

− 3.2

~ 5600 (3)

Carbon tetrachloride

25

  921

− 3.0

535

9.0 (25)       

Chloroform

25

  984

− 3.5

370

9 (25)         

Chlorobenzene

25

1270

− 3.9

140 (< 200)

9.3 (30)      

Cyclohexane

20

1280

− 5.4

~ 180

10.1 (30)        

Cyclohexanol

25

1465

− 3.7

− 500 (< 45)

Ethyl alcohol

25

1145

− 3.3

51

10.5 (20)         

Ethylene glycol

25

1660

− 2.1

120  

  9.7 (30)        

    (ethane-1,2-diol)

 

 

 

 

 

Freon (C51–12)

25

  524

− 3.4

Glycerol

25

1920

− 1.9

~ 3000 (4–12)

  9.0 (30)         

*Helium (4He)

− 269    

  180

− 50    

n-Hexanol

25

1303

− 3.4

Hydrogen

− 255    

1246

− 26     

Indium

160 

2313

  −0.29

  4.6 (160)         

Lead

340 

1766

  −0.28

~9 (20)

Mercury

25

1449

  −0.46

5.6 (100)

7.8 (30)         

Methane

− 170    

1420

− 9.7

Methyl alcohol

25

1103

− 3.3

32

9.6 (20)         

Naphthalene

100 

1248

− 2.5

Neon

− 243   

  540

 −16.9   

Nitrogen

− 202   

  912

− 9.8

    6.6 (−196)        

Oil (castor)

  25

1490

− 3.1

~5300 (3)

Oil (lubricating)

  25

1461

− 3.4

(at 2 MHz)

Oil (sperm)

  32

1411

Oxygen

− 202   

1056

− 7.8

n-Pentanol

  25

1277

− 3.3

Potassium

 80

1869

  −0.49

34 (30)

2.9 (100)         

n-Propanol

 25

1207

− 3.3

67

10.7 (20)             

Pyridine

 25

1417

− 4.2

Rubidium

 50

1247

  −0.38

78 (30)

Sodium

110 

2520

  −0.52

12 (30)

2.7 (110)         

Tin

240 

2471

  −0.25

~6 (20)

4.4 (240)         

Toluene

25

1306

− 4.3

84 (15–200)

Water (distilled)

 

 

see below

 

 

Water (heavy)

20

   1383.6

+ 3.2

32 (50)

Water (sea)

 

 

see below

 

 

Zinc

450  

2780

− 4.3

~4 (20)

 

 

 

 

 

 

    * The acoustic properties of helium are highly complex and dependent on temperature; see Rudnick (1980).




Acoustic properties of distilled water

Air-free distilled water is non-dispersive but the temperature dependence of sound speed is anomalous, mainly because of the temperature dependence of the adiabatic compressibility of the water molecule itself; the maximum speed occurs at a temperature of approximately 74.16 °C. The pressure coefficient is approximately + 0.156 m s−1 atm−1at 20 °C (see Del Grosso and Mader, 1972, Chen and Millero, 1976, Bilaniuk and Wong, 1993). The estimated uncertainty in the tabulated values is less than ± 0.02 m s−1. The attenuation of sound in distilled water is proportional to f 2, at least for the range 3–70 MHz, but is considerably in excess of that expected from the shear viscosity alone. The excess is attributed to structural relaxation between an open type of molecular structure, where each water molecule has four nearest neighbours tetrahedrally arranged about it, and a closer type of packing in which there are twelve nearest neighbours (see Litovitz and Davis, 1965).



Properties of sound in distilled water

t/°C

0

10

20

30

40

50

60

70

80

90

100

 

           

 

 

 

 

 

c/(m s−1)

(α/f 2)

(10−15 neper m−1 Hz−2)

B/A

1402.39

1447.28

1482.36

1509.14

1528.88

1542.56

1551.00

1554.80

1554.49

1550.48

1543.09

           

 

 

 

 

 

57 36 25

18

14

12

10

    8.7

    7.9

    7.3

    7.0

 

 

 

 

 

 

 

 

 

 

 

    4.0

    4.5

    4.9

     5.2

     5.4

     5.6

     5.7

    5.8

    5.9

    6.0

    6.1

 

           

 

 

 

 

 




Speed and attenuation in sea water

The speed of sound in sea water is a function of the temperature, the excess pressure, and the salinity, their relative importance being in that order (see Lovett, 1978, and Spiesberger, 1993); values in the table relate to surface depth and a salinity of 3.5%. The attenuation of sound in sea water is much higher than in distilled water owing to complex relaxation phenomena involving MgSO4, MgCO3 and B(OH)3 (see Mellen et al., 1980); values in the table, relating to surface depth, a pH of 8.0 and a salinity of 3.5%, are based on Schulkin and Marsh (1962, 1978).



Velocity of sound in sea water

t/°C

0

5

10

15

20

25

30

c/(m s−1)

1449.0

1470.6

1489.8

1506.7

1521.5

1534.4

1545.5




Attenuation of sound in sea water (dB km−1)

t/°C

Frequency/kHz

0.5

1

2

5

10

20

 

 

 

 

 

 

 

  0

0.03

0.07

0.14

0.41

1.3  

4.6

10

0.02

0.07

0.14

0.33

0.92

3.2

20

0.02

0.06

0.13

0.30

0.70

2.2

30

0.01

0.05

0.13

0.29

0.58

1.6

 

 

 

 

 

 

 




Speed and attenuation in solids

In isotropic solids, both shear (transverse) and longitudinal waves can be propagated. The velocity of shear waves in an extensive medium is cS = √() = √{E/2ρ(1 + σ)}, E being Young’s modulus, G the rigidity modulus and σ Poisson’s ratio, and this is also the velocity of torsional waves in thin cylindrical bars. The speed of longitudinal or irrotational waves in an extensive medium is cL = √{(K + G)/ρ} = √{E(1 − σ)/ρ(1 − 2σ)(1 + σ)}, K being the bulk modulus. In straight uniform bars and in tubes thin compared with a wavelength, the speed of longitudinal waves is cE = √(E/ρ). Surface waves propagating along the surface of an extensive solid are generally known as Rayleigh waves and propagate with speed cSR = acS, where a is the least positive root of the equation



   a6

 + a2 =

 

1

8(1 − a2)

1 − σ



In plates, the propagating modes are dispersive and for thin isotropic plates these Lamb waves reduce to two types: flexural (tending to zero speed) and antisymmetric (P0) waves (Achenbach, 1984).



Variation of a with Poisson’s ratio


σ

       0.20

       0.25

       0.30

       0.35

       0.40

a

       0.9110

       0.9194

       0.9274

       0.9350

       0.9422



In anisotropic solids, which may have as many as 21 independent elastic constants, there may exist, for a given direction of the wave normal, three distinct displacement vectors each associated with a distinct plane wave velocity. Of the three waves, one is analogous to the longitudinal and the others to transverse waves in the isotropic case. The directions of the respective displacement vectors are mutually orthogonal and are in general oblique to the wave normal. A generalized Rayleigh-type wave may be propagated, in a limited number of directions, along the surface of an extensive anisotropic medium. See Achenbach (1984).

Velocities and attenuation constants for various solids in the region of 20 °C are given below. In the case of metals, factors such as texture, cold work, stress, hardening, tempering and aging can cause significant departures from the values given in the table. Properties of plastics vary considerably with molecular weight, with additives and with temperature. Rocks or building materials can be equally variable. In view of this, many values are approximate and relate to materials of variable composition; the velocities given should therefore only be regarded as typical. The bracketed figures following the attenuation are frequencies in MHz; for plastics and many other solids, the variation of attenuation with frequency is often approximately linear. Composition of materials is given as percentage by weight.




Speed and attenuation of waves in solids

Material

Speed/(m s−1)

 

α longit. waves

 
 

neper m−1

 

cL
longitudinal
bulk waves

cE
irrot. rod
waves

cS
shear
waves

cSR
Rayleigh
waves

           

Aluminium

6 374

5102

3111

2906

0.40 (10)

ADP crystal, X-cut

6 250

10.8   (10)  

            ,,      Y-cut

6 250

            ,,      Z-cut

4 300

3500

9.69 (10)

Barium titanate ceramic

4 000

Beryllium

12 890  

8880

Bone, human tibia

4 000

1970

460 (2.9)    

Brass

4 372

3451

2100

1964

Brick

3650

Butyl rubber/carbon (100/40)

1 600

133 (0.35)  

Cadmium

2 780

2400

Cellulose acetate butyrate

2 080

103 (2.5)    

Chromium

6608

6229

4005

3655

Concrete

3900–4700

7–13 (0.2)

Constantan

5177

4276

2625

2445

Copper

4759

3813

2325

2171

Cork

 500

Duralumin

6398

5120

3122

2917

1.23 (10)

Ebonite

2500

Glass (crown)

5660

5342

3420

3127

2 (10)

    ,,    (heavy flint)

5260

4717

2960

2731

    ,,    (pyrex)

5640

5170

3280

Gold (hard-drawn)

3240

2030

1200

Ice (maximum density, polar ice

 

 

 

 

 

   sheets, firn temperature − 20 °C

3840

Invar (36 Ni, 63.8 Fe, 0.2 C)

4657

4216

2658

2447

Iron (soft)

5957

5189

3224

2986

   ,,   (cast)

4994

4477

2809

2590

Lead

2160

1188

  700

Magnesium

5823

5082

3163

2930

Manganese

4600

3830

Marble

3810

Molybdenum

6475

5636

3505

3248

Monel metal

5350

4400

2720

Neoprene

1510

230 (2.5)

Neoprene/carbon (100/60)

1690

Nickel (unmag. soft)

5608

4787

2929

2722

    ,,    (unmag. hard)

5814

4974

3078

2857

Niobium

5068

3497

2092

1970

Ni-Span-C

4831

2799

Nylon

2680

13.0 (5)    

Perspex

2700

2177

1330

1242

57 (2.5)    

Platinum

3260

2800

1730

Polycarbonate

2220

  910

240 (5)    

Polyethylene

2100–2400

850-920

100–300 (5)  

Polyethylene terephthalate (Mylar)

2400

1000

2000 (50)  

Polypropylene

2600

1200

Polystyrene

2350

1840

1120

1047

23 (2.5)    

Polysulfone

2260

  920

Polyvinyl chloride

2330

1070

3.5 (0.35)    

Polyvinyl chloride acetate

2250

1270 (10)  

Polyvinyl formal

2680

115 (2.5)  

Polyvinylidene chloride

2400

207 (2.5)  

Polyvinylidene fluoride

2560

1040

1100 (10)  

Quartz (crystal) X-cut

5720

5440

0.0127 (10)  

     ,,     (fused)

5970

5759

3765

3410

Rock (metamorphic/igneous)

4600–6200

    ,,    (limestones)

3100–6200

Rubber (natural)

1600

15 (0.35)

Rubber/carbon (100/40)

1680

36.6 (0.35)

Rubber (RTV silicone)

900–1050

Silica (fused)

5968

5760

3764

Silver

3704

2806

1698

1592

— 

Steel (mild)

5960

5196

3235

2996

    ,,    (tool) hardened

5874

5116

3179

2945

4.94 (10)

    ,,    (stainless)

5980

5282

3297

3049

Tantalum

4159

3337

2036

1902

Teflon

1400

440

430 (5)

Tin

3380

2626

1594

1491

Titanium

6130

5164

3182

2958

Tourmaline (crystal) Z-cut

7250

7170

Tungsten (annealed)

5221

4619

2887

2668

       ,,       carbide

6655

6223

3984

3643

       ,,       (drawn)

5410

4320

2640

Uranium

3370

1940

Vanadium

6023

4584

2774

2600

Wood (Ash) with grain

4670

     ,,    across grain

1390

     ,,    (Oak) with grain

4100

     ,,    (Pine) with grain

3600

Zinc (rolled)

4187

3826

2421

2225

Zirconium

4650

2250

           



References

J. D. Achenbach (1984) Wave propagation in elastic solids, North-Holland, Amsterdam.
N. Bilaniuk and G. S. K. Wong (1993) J. Acoust. Soc. Am., 93, 1609–12.
C.-T. Chen and F. J. Millero (1976) J. Acoust. Soc. Am., 60, 1270–3.
O. Cramer (1993) J. Acoust. Soc. Am., 93, 2510–2513.
V. A. Del Grosso and C. W. Mader (1972) J. Acoust. Soc. Am., 52, 1442–6.
C. M. Davis and J. Jarzynski (1972) Liquid water–acoustic properties: Absorption and relaxation, in Water–a comprehensive treatise (ed. F. Franks) vol. 1, Plenum, New York, 443–61.
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M. F. Hamilton and D. T. Blackstock (1988) J. Acoust. Soc. Am., 83, 74–7.
K. F. Herzfeld and T. A. Litovitz (1959) Absorption and Dispersion of Ultrasonic Waves, Academic Press.
T. A. Litovitz and C. M. Davis (1965) Structural and shear relaxation in liquids, in Physical Acoustics (ed. W. P. Mason), vol. II Part A: Properties of Gases, Liquids and Solutions, Academic Press, 281–349.
J. R. Lovett (1978) J. Acoust. Soc. Am., 63, 1713–18.
R. H. Mellen et al. (1980) J. Acoust. Soc. Am., 68, 248–57.
J. M. M. Pinkerton (1947) Proc. Phys. Soc., 62B, 129–41.
I. Rudnick (1980) J. Acoust. Soc. Am. 68, 36–45.
M. Schulkin and H. W. Marsh (1962) J. Acoust. Soc. Am., 34, 864–5 (errata in (1963) J. Acoust. Soc. Am., 35, 739).
M. Schulkin and H. W. Marsh (1978) J. Acoust. Soc. Am., 63, 43–8.
J. L. Spiesberger (1993) J. Acoust. Soc. Am., 93, 2235–7.
V. Uhlendorf et al. (1985) J. Phys. E: Sci. Instrum., 18, 151–7.

D.R.Bacon, D.R.Jarvis

 

 

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