The bulk modulus ( or ) of a substance is a measure of the resistance of a substance to compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear stress, and Young's modulus describes the response to normal (lengthwise stretching) stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law. The reciprocal of the bulk modulus at fixed temperature is called the isothermal compressibility. Definition The bulk modulus (which is usually positive) can be formally defined by the equation where is pressure, is the initial volume of the substance, and denotes the derivative of pressure with respect to volume. Since the volume is inversely proportional to the density, it follows that where is the initial density and denotes the derivative of pressure with respect to density. The inverse of the bulk modulus gives a substance's compressibility. Generally the bulk modulus is defined at constant temperature as the isothermal bulk modulus, but can also be defined at constant entropy as the adiabatic bulk modulus. 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 ), constant-entropy (isentropic ), and other variations are possible. Such distinctions are especially relevant for gases. For an ideal gas, an isentropic process has: where is the heat capacity ratio. Therefore, the isentropic bulk modulus is given by Similarly, an isothermal process of an ideal gas has: Therefore, the isothermal bulk modulus is given by . When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus and the density determine the speed of sound (pressure waves), according to the Newton-Laplace formula In solids, and 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. Measurement It is possible to measure the bulk modulus using powder diffraction under applied pressure. It is a property of