This page lists the underlying physics on the equations that are available to be automatically calculated in the technical guide for the speed of sound in pure water page. Below further details can be found relating to each of these equations as well as all the references for all of the information regarding the speed of sound in pure water on these pages.

### Simple Equations

#### Bilaniuk and Wong

112 point equation

c =
1.40238742 x 103 + 5.03821344 T - 5.80539349 x 10-2 T2 + 3.32000870 x 10-4 T3 - 1.44537900 x 10-6 T4 + 2.99402365 x 10-9 T5

36 point equation

c =
1.40238677 x 103 + 5.03798765 T - 5.80980033 x 10-2 T2 + 3.34296650 x 10-4 T3 - 1.47936902 x 10-6 T4 + 3.14893508 x 10-9 T5

148 point equation

c =
1.40238744 x 103 + 5.03836171 T - 5.81172916 x 10-2 T2 + 3.34638117 x 10-4 T3 - 1.48259672 x 10-6 T4 + 3.16585020 x 10-9 T5
T = temperature in degrees Celsius

Range of validity: 0-100 °C at atmospheric pressure

Bilaniuk and Wong (1993,1996) converted Del Grosso and Mader’s 1972 data to the 1990 International Temperature Scale and then produced three sets of coefficients depending on the number of temperature points which were converted and considered in their data fitting routines.

#### Marczak

c =
1.402385 x 103 + 5.038813 T - 5.799136 x 10-2 T2 + 3.287156 x 10-4 T3 - 1.398845 x 10-6 T4 + 2.787860 x 10-9 T5
T = temperature in degrees Celsius

Range of validity: 0-95 °C at atmospheric pressure.

Marczak (1997) combined three sets of experimental measurements, Del Grosso and Mader (1972), Kroebel and Mahrt (1976) and Fujii and Masui (1993) and produced a fifth order polynomial based on the 1990 International Temperature Scale.

#### Lubbers and Graaff’s simplified equations

A simple equation for use in the temperature interval 15-35°C

c =
1404.3 + 4.7T - 0.04 T2

Range of validity: 15-35°C at atmospheric pressure. Claimed accuracy - maximum error 0.18 ms-1

c =
1405.03 + 4.624 T - 3.83 x 10-2 T2

Range of validity: 10-40 °C at atmospheric pressure

Lubbers and Graaff (1998) produced these simple equations with a restricted temperature range for medical ultrasound applications, including tissue mimicking materials and test objects. Within the quoted temperature ranges they claim that the maximum error is approximately 0.18 ms-1 in comparisons with experimental data and more detailed equations such as Bilaniuk and Wong (1993,1996).

### Refined Eqautions

#### Belogol’skii, Sekoyan et al: speed of sound as a function of temperature and pressure

c(T,P) =
c(T,0) + M1(T)(P - 0.101325) + M2(T) (P- 0.101325)2 + M3(T)(P - 0.101325)3
c(T,0) =
a00 + a10T + a20T2 + a30T3 + a40T4 + a50T5
M1(T) =
a01 + a11T + a21T2 + a31T3
M2(T) =
a02 + a12T + a22T2 + a32T3
M3(T) =
a03 + a13T + a23T2 + a33T3
T = temperature in degrees Celsius
P = pressure in megapascals

Range of validity: 0-40& deg;C, 0.1 - 60 MPa

This version uses the 1990 International Temperature Scale

Belogol’skii, Sekoyan et al (1999) made their own measurements of sound speed as a function of pressure and temperature and also used the equation of Bilaniuk and Wong (1996) for sound speed at atmospheric pressure.

Table of Coefficients
Coefficient Numerical value
a00 1402.38744
a10 5.03836171
a20 -5.81172916 x 10-2
a30 3.34638117 x 10-4
a40 -1.48259672 x 10-6
a50 3.16585020 x 10-9
a01 1.49043589
a11 1.077850609 x 10-2
a21 -2.232794656 x 10-4
a31 2.718246452 x 10-6
a02 4.31532833 x 10-3
a12 -2.938590293 x 10-4
a22 6.822485943 x 10-6
a32 -6.674551162 x 10-8
a03 -1.852993525 x 10-5
a13 1.481844713 x 10-6
a23 -3.940994021 x 10-8
a33 3.939902307 x 10-10

Any comments or suggestions about further speed of sound equations?

### References

1. V.A. Belogol'skii, S.S. Sekoyan, L.M. Samorukova, S.R. Stefanov and V.I. Levtsov (1999), Pressure dependence of the sound velocity in distilled water, Measurement Techniques, Vol 42, No 4, pp 406-413.
2. N. Bilaniuk and G. S. K. Wong (1993), Speed of sound in pure water as a function of temperature, J. Acoust. Soc. Am. 93(3) pp 1609-1612, as amended by N. Bilaniuk and G. S. K. Wong (1996), Erratum: Speed of sound in pure water as a function of temperature [J. Acoust. Soc. Am. 93, 1609-1612 (1993)], J. Acoust. Soc. Am. 99(5), p 3257.
3. C-T Chen and F.J. Millero (1977), The use and misuse of pure water PVT properties for lake waters, Nature Vol 266, 21 April 1977, pp 707-708.
4. V.A. Del Grosso and C.W. Mader (1972), Speed of sound in pure water, J. Acoust. Soc. Am. 52, pp 1442-1446.
5. K. Fujii and R. Masui (1993), Accurate measurements of the sound velocity in pure water by combining a coherent phase-detection technique and a variable path-length interferometer, J. Acoust. Soc. Am. 93, pp 276-282.
6. International Association for the Properties of Water and Steam (1995), Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water substance for General and Scientific Use, Executive Secretary R.B. Dooley, Electric Power Research Inst., Palo Alto.
7. G. S. Kell (1977), Thermodynamic and transport properties of fluid water, in Water: a comprehensive treatise, edited by F. Franks, Plenum, New York, (1972) 1st edition, 4th printing (1983), Vol 1, Ch 10, pp 376-378.
8. W. Kroebel and K-H. Mahrt (1976), Recent results of absolute sound velocity measurements in pure water and sea water at atmospheric pressure, Acustica 35, pp 154-164.
9. V.I.Levtsov, G.N. Ovodov, L.M. Samorukova, S.S. Sekoyan and S.R. Stefanov (1997), Measuring the acoustic velocity of distilled water in the hydrostatic temperature and pressure range, Measurement Techniques, Vol. 40, No 10, pp 990-994.
10. J. Lubbers and R. Graaff (1998), A simple and accurate formula for the sound velocity in water, Ultrasound Med. Biol. Vol 24, No 7, pp 1065-1068.
11. W. Marczak (1997), Water as a standard in the measurements of speed of sound in liquids J. Acoust. Soc. Am. 102(5) pp 2776-2779.
12. M. O'Donnell, E.T. Jaynes and J.G. Miller (1981), Kramers-Kronig relationship between ultrasonic attenuation and phase velocity (1981), J. Acoust. Soc. Am. Vol 69, No 3, pp 696-701.
13. A. Saul and W. Wagner (1989), A fundamental equation for water covering the range from the melting line to 1273K at pressures up to 25,000 MPa , J. Phys. Chem. Ref. Data, Vol 18. No 4 pp 1537-1564.
14. S. Wiryana, L.J. Slutsky and J.M. Brown (1998), The equation of state of water to 200°C and 3.5 GPa: model potentials and the experimental pressure scale, Earth and Planetary Science Letters 163, pp 123-130.

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