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 sea water page. Below further details can be found relating to each of these equations as well as information on the use of pressure and depth in sound speed equations and details on the uncertainties.

### Simple Equations

#### Mackenzie

c(D,S,T) =
1448.96 + 4.591T - 5.304 x 10-2T2 + 2.374 x 10-4T3 + 1.340 (S-35) + 1.630 x 10-2D + 1.675 x 10-7D2 - 1.025 x 10-2T(S - 35) - 7.139 x 10-13TD3

T = temperature in degrees Celsius
S = salinity in parts per thousand
D = depth in metres

Range of validity: temperature 2 to 30 °C, salinity 25 to 40 parts per thousand, depth 0 to 8000 m

The above equation for the speed of sound in sea-water as a function of temperature, salinity and depth is given by Mackenzie equation (1981).

#### Coppens

c(D,S,t) =
c(0,S,t) + (16.23 + 0.253t)D + (0.213-0.1t)D2 + [0.016 + 0.0002(S-35)](S - 35)tD

c(0,S,t) =
1449.05 + 45.7t - 5.21t2 + 0.23t3 + (1.333 - 0.126t + 0.009t2)(S - 35)

t = T/10 where T = temperature in degrees Celsius
S = salinity in parts per thousand
D = depth in kilometres

Range of validity: temperature 0 to 35 °C, salinity 0 to 45 parts per thousand, depth 0 to 4000 m

The above equation for the speed of sound in sea-water as a function of temperature, salinity and depth is given by Coppens equation (1981).

### Refined Equations

#### The UNESCO equation: Chen and Millero

The international standard algorithm, often known as the UNESCO algorithm, is due to Chen and Millero (1977), and has a more complicated form than the simple equations above, but uses pressure as a variable rather than depth. For the original UNESCO paper see Fofonoff and Millard (1983). Wong and Zhu (1995) recalculated the coefficients in this algorithm following the adoption of the International Temperature Scale of 1990 and their form of the UNESCO equation is:

c(S,T,P) =
Cw(T,P) + A(T,P)S + B(T,P)S3/2 + D(T,P)S2

Cw(T,P) =
(C00 + C01T + C02T2 + C03T3 + C04T4 + C05T5) +
(C10 + C11T + C12T2 + C13T3 + C14T4)P +
(C20 +C21T +C22T2 + C23T3 + C24T4)P2 +
(C30 + C31T + C32T2)P3

A(T,P) =
(A00 + A01T +A02T2 + A03T3 + A04T4) +
(A10 + A11T + A12T2 + A13T3 + A14T4)P +
(A20 + A21T + A22T2 + A23T3)P2 +
(A30 + A31T + A32T2)P3

B(T,P) =
B00 + B01T + (B10 + B11T)P

D(T,P) =
D00 + D10P

T = temperature in degrees Celsius
S = salinity in Practical Salinity Units (parts per thousand)
P = pressure in bar

Range of validity: temperature 0 to 40 °C, salinity 0 to 40 parts per thousand, pressure 0 to 1000 bar (Wong and Zhu, 1995).

Please note that for consistency, within the interactive version, the pressure must be input in kPa.

Table of Coefficients
Coefficients Numerical values Coefficients Numerical values
C00 1402.388 A02 7.166E-5
C01 5.03830 A03 2.008E-6
C02 -5.81090E-2 A04 -3.21E-8
C03 3.3432E-4 A10 9.4742E-5
C04 -1.47797E-6 A11 -1.2583E-5
C05 3.1419E-9 A12 -6.4928E-8
C10 0.153563 A13 1.0515E-8
C11 6.8999E-4 A14 -2.0142E-10
C12 -8.1829E-6 A20 -3.9064E-7
C13 1.3632E-7 A21 9.1061E-9
C14 -6.1260E-10 A22 -1.6009E-10
C20 3.1260E-5 A23 7.994E-12
C21 -1.7111E-6 A30 1.100E-10
C22 2.5986E-8 A31 6.651E-12
C23 -2.5353E-10 A32 -3.391E-13
C24 1.0415E-12 B00 -1.922E-2
C30 -9.7729E-9 B01 -4.42E-5
C31 3.8513E-10 B10 7.3637E-5
C32 -2.3654E-12 B11 1.7950E-7
A00 1.389 D00 1.727E-3
A01 -1.262E-2 D10 -7.9836E-6

#### Del Grosso’s Equation

An alternative equation to the UNESCO algorithm, which has a more restricted range of validity, but which is preferred by some authors, is the Del Grosso equation (1974). Wong and Zhu (1995) also reformulated this equation for the new 1990 International Temperature Scale and their version is:

c(S,T,P) =
C000 + ΔCT + ΔCS + ΔCP + ΔCSTP

ΔCT(T) = CT1T + CT2T2 + CT3T3
ΔCS(S) = CS1S + CS2S2
ΔCP(P) = CP1P + CP2P2 + CP3P3
ΔCSTP(S,T,P) = CTPTP + CT3PT3P + CTP2TP2 + CT2P2T2P2 + CTP3TP3 + CSTST + CST2ST2 + CSTPSTP + CS2TPS2TP + CS2P2S2P2

T = temperature in degrees Celsius
S = salinity in Practical Salinity Units
P = pressure in kg/cm2

Range of validity: temperature 0 to 30 °C, salinity 30 to 40 parts per thousand, pressure 0 to 1000 kg/cm2, where 100 kPa=1.019716 kg/cm2. (Wong and Zhu, 1995).

Please note that for consistency, within the interactive version, the pressure must be input in kPa

Table of Coefficients
Coefficients Numerical values
C000 1402.392
CT1 0.5012285E1
CT2 -0.551184E-1
CT3 0.221649E-3
CS1 0.1329530E1
CS2 0.1288598E-3
CP1 0.1560592
CP2 0.2449993E-4
CP3 -0.8833959E-8
CST -0.1275936E-1
CTP 0.6353509E-2
CT2P2 0.2656174E-7
CTP2 -0.1593895E-5
CTP3 0.5222483E-9
CT3P -0.4383615E-6
CS2P2 -0.1616745E-8
CST2 0.9688441E-4
CS2TP 0.4857614E-5
CSTP -0.3406824E-3

#### NPL Equation

The most recent equation for the speed of sound in seawater is the NPL Equation as formulated by Leroy, Robinson and Goldsmith (2008). This equation was created out of a need for an equation that was suitable for use in all ‘neptunian’ waters excluding the abnormal ‘hot spots’ of abnormally high temperature and salinity, in order to enable a person to accurately calculate the speed of sound in various scenarios using only a single equation.

c =
1402.5 + 5T - 5.44 x 10-2T2 + 2.1 x 10-4T3
+ 1.33S - 1.23 x 10-2ST + 8.7 x 10-5ST2
+1.56 x 10-2Z + 2.55 x 10-7Z2 - 7.3 x 10-12Z3
+ 1.2 x 10-6Z(Φ - 45) - 9.5 x 10-13TZ3
+ 3 x 10-7T2Z + 1.43 x 10-5SZ

This equation is valid for use in any ocean or sea with a salinity that does not exceed 42‰

### Pressure of Depth

Both the UNESCO equation and Del Grosso’s equation use pressure as a variable instead of depth because they are based on measurements made in a small laboratory’s pressurised chamber. Useful guidance and suitable equation for converting pressure into depth and depth into pressure can be found in Leroy and Parthiot (1998). The key equations here are:

#### Conversion of pressure into depth

ZS(P, Φ) =
9.72659 x 102P - 2.2512 x 10-1P2 + 2.279 x 10-4P3 - 1.82 x 10-7P4
g(Φ) + 1.092 x 10-4P

Where g(Φ), the international forumla for gravity, is given by:

g(Φ) =
9.780318 (1 + 5.2788 x 10-3 sin2 + 2.36 x 10-5 sin4 Φ
Z = depth in metres
P = pressure in MPa (relative to atmospheric pressure)
Φ = latitude

The above equation is true for the oceanographers’ standard ocean, defined as an ideal medium with a temperature of 0°C and salinity of 35 parts per thousand.

Leroy and Parthiot (1998) give a table of corrections which are needed when the standard formula is applied to specific oceans and seas. The above equation and interactive version do not apply any corrections.

Please note that for consistency, within the interactive version, the pressure must be input in kPa.

#### Converstion of depth to pressure

P(Z,Φ) =
h(Z,Φ) - h0Z
h(ZmΦ) =
h(Z,45) x k(Z,Φ)
h(Z,45) =
1.00818 x 10-2Z + 2.465 x 10-8Z2 - 1.25 x 10-13Z3 + 2.8 x 10-19Z4
k(Z,Φ) =
(g(Φ) - 2 x 10-5) / (9.80612 - 10-5Z
g(Φ) =
9.7803 (1 + 5.3 x 10-3 sin2Φ)
h0Z =
1.0 x 10-2Z / (Z+100) + 6.2 x 10-6Z
Z = depth in metres
h = pressure in MPa (relative to atmospheric pressure)
Φ = latitude

In the above equation, P (=h(Z,Φ)) would apply to the oceanographers’ standard ocean, defined as an ideal medium with a temperature of 0 °C and salinity of 35 parts per thousand.

Leroy and Parthiot (1998) give a table of corrections which are needed when the standard formula is applied to specific oceans and seas. The correction h0Z is the correction applicable to common oceans. These are defined as open oceans between the latitudes of 60°N and 40°S, and excluding closed ocean basins and seas. A full range of corrections may be found in Leroy and Parthiot (1998).

For consistency the interactive version returns a pressure in kPa.

### Uncertainties

Although the UNESCO algorithm is the International Standard algorithm, there is much debate in the scientific literature about the accuracy and range of applicability of this equation and of Del Grosso’s equation. Some researchers prefer Del Grosso’s equation, especially for calculations within its own domain of validity. It is important to recognise that the equations presented here are derived from fitting to experimental data from several different experiments and each has an associated uncertainty in its prediction of sound speed. The choice of equation may depend on the accuracy and precision which is acceptable for the particular application in which it is being employed. For further discussion on this topic, please refer to Dushaw et al (1993), Meinen and Watts (1997), Millero and Xu Li (1994), Speisberger and Metzger (1991a, 1991b) and Speisberger (1993).

The Hydrographic Society (Pike and Beiboer, 1993) has a good summary of the main algorithms for sound speed in the ocean. This sets out more detailed advice and information than can be provided on this web-site, including information of the domains of validity of the main equations and on depth to pressure conversions.

### References

1. C-T. Chen and F.J. Millero, Speed of sound in seawater at high pressures (1977) J. Acoust. Soc. Am. 62(5) pp 1129-1135
2. A.B. Coppens, Simple equations for the speed of sound in Neptunian waters (1981) J. Acoust. Soc. Am. 69(3), pp 862-863
3. V.A. Del Grosso, New equation for the speed of sound in natural waters (with comparisons to other equations) (1974) J. Acoust. Soc. Am 56(4) pp 1084-1091
4. B.D. Dushaw, P.F. Worcester, B.D. Cornuelle and B.M. Howe, On equations for the speed of sound in sea water (1993) J. Acoust. Soc. Am. 93(1) pp 255-275
5. N.P. Fofonoff and R.C. Millard Jr. Algorithms for computation of fundamental properties of seawater (1983), UNESCO technical papers in marine science. No. 44, Division of Marine Sciences. UNESCO, Place de Fontenoy, 75700 Paris.
6. C. C. Leroy and F Parthiot, Depth-pressure relationship in the oceans and seas (1998) J. Acoust. Soc. Am. 103(3) pp 1346-1352
7. K.V. Mackenzie, Nine-term equation for the sound speed in the oceans (1981) J. Acoust. Soc. Am. 70(3), pp 807-812
8. C.S. Meinen and D.R. Watts, Further evidence that the sound-speed algorithm of Del Grosso is more accurate than that of Chen and Millero (1997) J. Acoust. Soc. Am. 102(4) pp 2058-2062
9. F.J. Millero and Xu Li, Comments on "On equations for the speed of sound in seawater" (1994), J. Acoust. Soc. Am. 95(5), pp 2757-2759
10. J.M. Pike and F.L. Beiboer, A comparison between algorithms for the speed of sound in seawater (1993) The Hydrographic Society, Special Publication no. 34
11. J.L. Speisberger and K. Metzger, New estimates of sound speed in water (1991a) J. Acoust. Soc. Am. 89(4) pp 1697-1700
12. J.L. Speisberger and K. Metzger, A new algorithm for sound speed in seawater (1991b) J. Acoust. Soc. Am. 89(6) pp 2677-2687
13. J.L. Speisberger, Is Del Grosso's sound-speed algorithm correct ? (1993) J. Acoust. Soc. Am. 93(4) pp 2235-2237
14. G.S.K. Wong and S Zhu, Speed of sound in seawater as a function of salinity, temperature and pressure (1995) J. Acoust. Soc. Am. 97(3) pp 1732-1736
15. Leroy, C. C., Robinson, S. P. and Goldsmith, M. J. “A new simple equation using depth and latitude for the accurate calculation of sound speed in all ocean acoustics applications”, J. Acoust. Soc .Am., vol 124 (5), p2774-2782, 2008.
16. Leroy, C. C., Robinson, S. P. and Goldsmith, M. J. Erratum: “A new equation for the accurate calculation of sound speed in all oceans” [J. Acoust. Soc. Am. 124(5), 2774-2783 (2008)] J. Acoust. Soc. Am. 126 (4), p2117, October 2009.