(a) The red and blue waves in Fig. 15.20 combine so that the displacement of the string at 0 is always zero. To show this mathematically for a wave of arbitrary shape, consider a wave moving to the right along the string in fig. 15.20 (shown in blue) that, at time T, is given by y1(x, T) = .f(x), where is some function of x. (The form of .f(x) determines the shape of the wave.) IT the point 0 corresponds to x = 0, explain why, at time T, the wave moving to the left in fig. 15.20 (shown in red) is given by the function y2(x,t)= -.f( -x) . (b) Show that the total wave function y(x, T) = y, (x, T) + Y2(X, T) is zero at 0, independent of the form of the function .f(x).
(c) The red and blue waves in fig. 15.21 combine so that the slope of the string at 0 is always zero. To show this mathematically for a wave of arbitrary shape, again let the wave moving to the right in Fig. 15.21 (shown in blue) be given by y, (x, T) = .f(x) at time T. Explain why the Wave moving to the left (shown in red) at this same time This given by Y2(X, T) =.f( -x).
(d) Slow that the total wave function y(x, T) = y, (x, T) + y,(x, T) has zero slope at 0, independent of the form of the function .f(x), as long as .f(x) has a finite first derivative.