In the case that only the propagation of a single wave in a predefined direction is of interest, a first-order partial differential equation one-way wave equation can be considered. Wave equation in one space dimension The wave equation in one space dimension can be written as follows: This equation is typically described as having only one space dimension , because the only other independent variable is the time . Nevertheless, the dependent variable may represent a second space dimension, if, for example, the displacement takes place in direction, as in the case of a string that is located in the plane. Derivation of the wave equation The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. Another physical setting for derivation of the wave equation in one space dimension uses Hooke's law. In the theory of elasticity, Hooke's law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). From Hooke's law The wave equation in the one-dimensional case can be derived from Hooke's law in the following way: imagine an array of little weights of mass interconnected with massless springs of length . The springs have a spring constant of : Here the dependent variable measures the distance from the equilibrium of the mass situated at , so that essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass at the location are: The equation of motion for the weight at the location is given by equating these two forces: If the array of weights consists of weights spaced evenly over the length of total mass , and the total spring constant of the array , we can write the above equation as Taking the limit and assuming smoothness, one gets which is from the definition of a second derivative. is the square of the propagation speed in this particular case. Stress pulse in a bar In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs