Consider a semi-infinite string, fixed at the origin x = 0

QUESTION:

Consider a semi-infinite string, fixed at the origin x = 0 and extending far out to the right. Let f () be a function that is localized around the origin, such as the function of Figure 16.4(a). (a) Describe the wave given by the function f (x ct) for a large negative time to. (b) One way to solve for the subsequent motion of this wave on the semi-infinite string is called the method of images and is as follows: Consider the function u = f (x ct) f (x ct). (The second term here is called the "image." Can you explain why?) Obviously this satisfies the wave equation for all x and t. Show that it coincides with the given wave of part (a) at the initial time to and everywhere on the semi-infinite string. Show also that it obeys the boundary condition that u = 0 at x = 0. (c) It is a fact that there is a unique wave that obeys the wave equation and any given initial and boundary conditions. Therefore the wave of part (b) is the solution for all times (on our semi-infinite string). Describe the motion on the semi-infinite string for all times.