# Basic formula

In general, the speed of sound c is given by the Newton-Laplace equation: where K is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gases), is the density Thus the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density. For ideal gases the bulk modulus P is simply the gas pressure multiplied by the adiabatic index. For general equations of state, if classical mechanics is used, the speed of sound is given by where is the pressure and is the density and the derivative is taken adiabatically, that is, at constant entropy per particle (s). If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations. In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds, a mixture of oxygen and nitrogen constitutes a non-dispersive medium. But air does contain a small amount of CO2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic frequencies (> 28 kHz). In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity, while the energy of the disturbance propagates at the group velocity. The same phenomenon occurs with light waves; see optical dispersion for a description.The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). It is sometimes also known as the isentropic expansion factor and is denoted by (gamma) or (kappa). The latter symbol kappa is primarily used by chemical engineers. Mechanical engineers use the Roman letter . where, is the heat capacity and the specific heat capacity (heat capacity per unit mass) of a gas. Suffix and refer to constant pressure nd constant volume conditions respectively. To understand this relation, consider the following experiment: A closed cylinder with a locked piston contains air. The pressure inside is equal to the outside air pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant, while temperature and pressure rise. When the target temperature is reached, the heating is stopped. The piston is now freed and moves outwards, expanding without exchange of heat (adiabatic expansion). Doing this work cools the air inside the cylinder to below the target temperature. To return to the target temperature (still with a free piston), the air must be heated. This extra heat amounts to about 40% more than the previous amount added. In this example, the amount of heat added with a locked piston is proportional to , whereas the total amount of heat added is proportional to . Therefore, the heat capacity ratio in this example is 1.4. Another way of understanding the difference between and is that applies if work is done to the system which causes a change in volume (e.g. by moving a piston so as to compress the contents of a cylinder), or if work is done by the system which changes its temperature (e.g. heating the gas in a cylinder to cause a piston to move). applies only if - that is, the work done - is zero. Consider the difference between adding heat to the gas with a locked piston, and adding heat with a piston free to move, so that pressure remains constant. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. In the first, constant-volume case (locked piston) there is no external motion, and thus no mechanical work is done on the atmosphere; is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant pressure case.