Answer to Question #152911 in Physics for Nabel Shah

Question

Show that the functions f(vt-x) represen a plane progressivee wave propagating in the positive X direction

Answer 1

Consider a disturbance y which propagates along the x-axis with velocity v. The disturbance y may refer to the elevation of water wave or the magnitude of the y-displacement of a string. Since the disturbance is moving y will depend on x and t. When t = 0 y will be some function of x which we may call f(x). We assume that the wave propagates without change of shape. At a later time t the wave profile will be identical with that at t = 0 except that the wave profile has moved a distance vt in the positive x-direction. If we take a new origin O′ at the point x = vt and denote distances measured from O′ by X then x = X + vt and the equation of the wave profile referred to this new origin is

$$f(vt-x)$$

This equation is the most general expression of a wave moving with constant velocity v without change of shape along the positive direction of x.