in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness given by where is the cross-sectional area, and is the Young's modulus of the material. The wave equation becomes is equal to the volume of the bar, and therefore where is the density of the material. The wave equation reduces to The speed of a stress wave in a bar is therefore . General solution Algebraic approach The one-dimensional wave equation is unusual for a partial differential equation in that a relatively simple general solution may be found. Defining new variables changes the wave equation into which leads to the general solution or equivalently, In other words, solutions of the 1D wave equation are sums of a right-traveling function and a left-traveling function . "Traveling" means that the shape of these individual arbitrary functions with respect to stays constant, however, the functions are translated left and right with time at the speed . This was derived by Jean le Rond d'Alembert. Another way to arrive at this result is to factor the wave equation into two one-way wave equations: i.e. As a result, if we define then From this, must have the form , and from this the correct form of the full solution can be deduced. The usual second-order wave equation is sometimes called the "two-way wave equation" (superposition of two waves) to distinguish it from the first-order one-way wave equation describing the wave propagation of a single wave in a pre-defined direction. For an initial-value problem, the arbitrary functions and can be determined to satisfy initial conditions: The result is d'Alembert's formula: In the classical sense, if , and , then . However, the waveforms and may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left. The basic wave equation is a linear differential equation, and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal