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The motion of a finite string, fixed at both ends, was
Chapter 16, Problem 16.9(choose chapter or problem)
The motion of a finite string, fixed at both ends, was determined by the wave equation (16.19) and the boundary conditions (16.20). We solved these by looking for a solution that was sinusoidal in time. A different, and rather more general, approach to problems of this kind is called separation of variables. In this approach, we seek solutions of (16.19) with the separated form u(x , t) = X (x)T (t), that is, solutions that are a simple product of one function of x and a second of t. [As usual, there's nothing to stop us trying to find a solution of this form. In fact, there is a large class of problems (including this one) where this approach is known to produce solutions, and enough solutions to allow expansion of any solution.] (a) Substitute this form into (16.19) and show that you can rewrite the equation in the form T"(t)I T (t) = c2X"(x)/ X (x). (b) Argue that this last equation requires that both sides of this equation are separately equal to the same constant (call it K). It can be shown that K has to be negative.26 Use this to show that the function T (t) has to be sinusoidal which establishes (16.21) and we're back to the solution of Section 16.3. The method of separation of variables plays an important role in several areas, notably quantum mechanics and electromagnetism.
Questions & Answers
QUESTION:
The motion of a finite string, fixed at both ends, was determined by the wave equation (16.19) and the boundary conditions (16.20). We solved these by looking for a solution that was sinusoidal in time. A different, and rather more general, approach to problems of this kind is called separation of variables. In this approach, we seek solutions of (16.19) with the separated form u(x , t) = X (x)T (t), that is, solutions that are a simple product of one function of x and a second of t. [As usual, there's nothing to stop us trying to find a solution of this form. In fact, there is a large class of problems (including this one) where this approach is known to produce solutions, and enough solutions to allow expansion of any solution.] (a) Substitute this form into (16.19) and show that you can rewrite the equation in the form T"(t)I T (t) = c2X"(x)/ X (x). (b) Argue that this last equation requires that both sides of this equation are separately equal to the same constant (call it K). It can be shown that K has to be negative.26 Use this to show that the function T (t) has to be sinusoidal which establishes (16.21) and we're back to the solution of Section 16.3. The method of separation of variables plays an important role in several areas, notably quantum mechanics and electromagnetism.
ANSWER:Step 1 of 3
We are certainly free to try for a solution of the form \(u(x,t) = X(x)T(t)\). If we substitute this
into Eq. (16.19), \(\partial ^2u/\partial t^2=c^2\partial ^2u/\partial x^2\), we find \(X(x)T’’(t) = c^2T(t)X’’x)\), from which it follows that