Two-Dimensional Waves. A stretched string lies along the x-axis. The string is displaced along both the y- and z-directions, so that the transverse displacement of the string is given by
Y(x, t) = Acos (kx - wt) z(x, t) = Asin (kx - wt)
(a) Draw a graph of z versus y for a particle on the string at x = 0. This shows the trajectory of the particle as seen by an observer on the +x-axis looking back toward x = o. Indicate the position of the particle at t = 0, t = 'IT/2w, t = 'IT/w, and t = 3'IT/2w.
(b) Find the velocity vector of a particle at an arbitrary position x on the string. Show that this represents the tangential velocity of a particle moving in a circle of radius A with angular velocity w, and show that the speed of the particle is constant (i.e., the particle is in uniform circular motion). (See Problem 3.75.)
(c) Find the acce1era1ion vector of the particle in part (b). Show that the acceleration is always directed toward the center of the circle and that its magnitude is a = w2A. Explain these results in terms of uniform circular motion. Suppose that the displacement of the string was instead given by
Y(x, t) = Acos (kx - wt) z(x, t) = -Asin (kx - wt)
Describe how the motion of a particle at x would be different from the motion described in part (a).