الجمعة، 10 أغسطس 2012

Speed of Sound in Almost Incompressible Liquid

Even liquid normally is assumed to be incompressible in reality has a small and important compressible aspect. The ratio of the change in the fractional volume to pressure or compression is referred to as the bulk modulus of the material. For example, the average bulk modulus for water is $ 2.2 \times 10^9$ $ N/m^2$ . At a depth of about 4,000 meters, the pressure is about $ 4 \times 10^7$ $ N/m^2$ . The fractional volume change is only about 1.8% even under this pressure nevertheless it is a change.
The compressibility of the substance is the reciprocal of the bulk modulus. The amount of compression of almost all liquids is seen to be very small as given in Table (3.5). The mathematical definition of bulk modulus as following

In physical terms can be written as

For example for water
$\displaystyle c = \sqrt{2.2 \times 10^9 N /m^2 \over 1000 kg /m^3} = 1493 m/s %\label{sound:eq:waterSpeed} $    

This agrees well with the measured speed of sound in water, 1482 m/s at $ 20^{\circ}C$ . Many researchers have looked at this velocity, and for purposes of comparison it is given in Table (3.5)

Table 3.1: Water speed of sound from different sources
Remark reference Value [m/sec]
Fresh Water ( $ 20^{\circ}C$ ) Cutnell, John D. & Kenneth W. Johnson. Physics. New York: Wiley, 1997: 468. 1492
Distilled Water at ( $ 25^{\circ}C$ ) The World Book Encyclopedia. Chicago: World Book, 1999. 601 1496
Water distilled Handbook of Chemistry and Physics. Ohio: Chemical Rubber Co., 1967-1968:E37 1494

The effect of impurity and temperature is relatively large, as can be observed from the equation (3.37). For example, with an increase of 34 degrees from $ 0 ^{\circ}C$ there is an increase in the velocity from about 1430 m/sec to about 1546 [m/sec]. According to Wilson3.5, the speed of sound in sea water depends on temperature, salinity, and hydrostatic pressure.
Wilson's empirical formula appears as follows:

where $ c_0 = 1449.14[m/sec]$ is about clean/pure water, $ c_{T}$ is a function temperature, and $ c_S$ is a function salinity, $ c_P$ is a function pressure, and $ c_{STP}$ is a correction factor between coupling of the different parameters.

Table 3.2: Liquids speed of sound, after Aldred, John, Manual of Sound Recording, London: Fountain Press, 1972
material reference Value [m/sec]
Glycerol   1904
Sea water $ 25^{\circ}C$ 1533
Mercury   1450
Kerosene   1324
Methyl alcohol   1143
Carbon tetrachloride   926

In summary, the speed of sound in liquids is about 3 to 5 relative to the speed of sound in gases
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