Once again we nd the solution to our problem in two

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: th a large or even infinite number of unknowns can give an effective model for the same physical phenomenon. This partially explains, for example, why quantum mechanics possesses two superficially different formulations, via Schr¨dinger’s partial difo ferential equation or via “infinite matrices” in Heisenberg’s “matrix mechanics.” 4.4 The vibrating string Our next goal is to derive the equation which governs the motion of a vibrating string. We consider a string of length L stretched out along the x-axis, one end of the string being at x = 0 and the other being at x = L. We assume that the string is free to move only in the vertical direction. Let u(x, t) = vertical displacement of the string at the point x at time t. We will derive a partial differential equation for u(x, t). Note that since the ends of the string are fixed, we must have u(0, t) = 0 = u(L, t) for all t. It will be convenient to use the “configuration space” V0 described in Section 3.3. An element u(x) ∈ V0 represents a configuration of the string at some 3 For further discussion of this method one can refer to numerical analysis books, such as Burden and Faires, Numerical analysis , Seventh edition, Brooks...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online