true for any component, regardless of its angular frequency of . The limiting case of corresponds to a "fixed end" that doesn't move, whereas the limiting case of corresponds to a "free end". The Sturm–Liouville formulation A flexible string that is stretched between two points and satisfies the wave equation for and . On the boundary points, may satisfy a variety of boundary conditions. A general form that is appropriate for applications is where and are non-negative. The case where is required to vanish at an endpoint (i.e. "fixed end") is the limit of this condition when the respective or approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form A consequence is that The eigenvalue must be determined so that there is a non-trivial solution of the boundary-value problem This is a special case of the general problem of Sturm–Liouville theory. If and are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for and can be obtained from expansion of these functions in the appropriate trigonometric series. Investigation by numerical methods Approximating the continuous string with a finite number of equidistant mass points, one gets the following physical model: If each mass point has the mass , the tension of the string is , the separation between the mass points is , and are the offset of these points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string), the vertical component of the force towards point is and the vertical component of the force towards point is Taking the sum of these two forces and dividing with the mass , one gets for the vertical motion: As the mass density is this can be written The wave equation is obtained by letting , in which case takes the form , where is continuous function of two variables, takes the form , and But the discrete formulation () of the equation of state with a finite number of mass point is just the suitable one for a numerical propagation of the string motion. The boundary condition where is the length of the string, takes in the discrete formulation the form that for the outermost points and the equations of motion are and while for where . If the string