is approximated with 100 discrete mass points, one gets the 100 coupled second-order differential equations (), () and () or, equivalently, 200 coupled first-order differential equations. Propagating these up to the times using an 8th-order multistep method, the 6 states displayed in figure are found: The red curve is the initial state at time zero, at which the string is "let free" in a predefined shape with all . The blue curve is the state at time i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity would need for one fourth of the length of the string. The wave travels towards right with the speed without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form . The constraint on the right extreme starts to interfere with the motion, preventing the wave to raise the end of the string. The direction of motion is reversed. The red, green and blue curves are the states at the times while the 3 black curves correspond to the states at with the wave starting to move back towards left. The wave now travels towards left, and the constraints at the end points are not active any more. When finally the other extreme of the string is reached, the direction will again be reversed in a way similar to what is displayed in the figure above for Several space dimensions The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain in -dimensional space, with boundary . Then the wave equation is to be satisfied if is in , and . On the boundary of , the solution shall satisfy where is the unit outward normal to , and is a non-negative function defined on . The case where vanishes on is a limiting case for approaching infinity. The initial conditions are where and are defined in . This problem may be solved by expanding and in the eigenfunctions of the Laplacian in , which satisfy the boundary conditions. Thus the eigenfunction satisfies in , and on . In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary . If is a circle, then