many other physical quantities besides temperatures. For another example, we can describe all possible sounds echoing within a container of gas by a function that gives the pressure at a point and time within that container. If the gas was initially at uniform temperature and composition, the evolution of is constrained by the formula Here is some extra compression force that is being applied to the gas near by some external process, such as a loudspeaker or piston right next to . This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is , the second derivative of with respect to time, rather than the first derivative . Yet this small change makes a huge difference on the set of solutions . This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves. Wave in elastic medium Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling in the direction in space. For example, let the positive direction be to the right, and the negative direction be to the left. with constant amplitude with constant velocity , where is independent of wavelength (no dispersion) independent of amplitude (linear media, not nonlinear). with constant waveform, or shape This wave can then be described by the two-dimensional functions (waveform traveling to the right) (waveform traveling to the left) or, more generally, by d'Alembert's formula: representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation General solutions are based upon Duhamel's principle. Beside the second order wave equations that are describing a standing wave field, the one-way wave equation describes the propagation of single wave in a defined direction. Wave forms The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x-direction at velocity v (and G will