Editing Bulk modulus (section)

==Thermodynamic relation==
Strictly speaking, the bulk modulus is a [[thermodynamic]] quantity, and in order to specify a bulk modulus it is necessary to specify how the pressure varies during compression: constant-[[temperature]] (isothermal <math>K_T</math>), constant-[[entropy]] ([[isentropic process|isentropic]] <math>K_S</math>), and other variations are possible. Such distinctions are especially relevant for [[gas]]es.

For an [[Ideal gas#Speed of sound|ideal gas]], an isentropic process has:

:<math>PV^\gamma=\text{constant} \Rightarrow P\propto \left(\frac{1}{V}\right)^\gamma\propto \rho ^\gamma,
</math>

where <math>\gamma </math> is the [[heat capacity ratio]]. Therefore, the isentropic bulk modulus <math>K_S</math> is given by

:<math>K_S=\gamma P.</math>

Similarly, an isothermal process of an ideal gas has:

:<math>PV=\text{constant} \Rightarrow P\propto \frac{1}{V} \propto \rho,
</math>

Therefore, the isothermal bulk modulus <math>K_T</math> is given by

:<math>K_T = P </math> .

When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus <math>K</math> and the [[density]] <math>\rho</math> determine the [[speed of sound]] <math>c</math> ([[P-wave|pressure waves]]), according to the Newton-Laplace formula

:<math>c=\sqrt{\frac{K_S}{\rho}}.</math>

In solids, <math>K_S</math> and <math>K_T</math> have very similar values. Solids can also sustain [[transverse waves]]: for these materials one additional [[elastic modulus]], for example the shear modulus, is needed to determine wave speeds.