# 11.3.3: Speed of Sound in Almost Incompressible Liquid

Every liquid 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 the Book "Fundamentals of Compressible Flow." The mathematical definition of bulk modulus as following

\[

B_T = \rho \,{\dfrac{\partial P}{\partial \rho}}

\label{gd:sd:eq:bulkModulus} \tag{16}

\]

In physical terms can be written as

Liquid/Solid Sound Speed

\[

\label{gd:sd:eq:sondLiquid}

c = \sqrt{\dfrac{elastic\;\; property }{ inertial\;\; property} }

= \sqrt{\dfrac{B_T }{ \rho}} \tag{17}

\]

For example for water

\[

\nonumber

c = \sqrt{\dfrac{2.2 \times 10^9 N /m^2 }{ 1000 kg /m^3}} = 1493 m/s \tag{18}

\]

This value agrees well with the measured speed of sound in water, 1482 m/s at \(20^{\circ}C\). A list with various typical velocities for different liquids can be found in "Fundamentals of Compressible Flow'' by by this author. The interesting topic of sound in variable compressible liquid also discussed in the above book. It can be shown that velocity in solid and and slightly compressible liquid is expressed by In summary, the speed of sound in liquids is about 3 to 5 relative to the speed of sound in gases.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.