A wave propagates on a string in the positive x-direction at a velocity v. The shape of the string at t = t0 is given by g(x, t¬0) = A sin(x/a). Write the wave equation for a general time t.

Sol. At t = t0, g(x, t0) = A sin (x/a) …(1) For a wave traveling in the positive x-direction, the general equation is given by y = f(x/a – t/T) Putting t = –t0 and comparing with equation (1), we get ⇒ g(x, 0) = A sin {(x/a) + (t0/T)} ⇒ g(x, t) = A sin {(x/a) + (t0/T) – (t/T)} As T = a/v (a = wave length, v = speed of the wave) ⇒ y = A sin (x/a + t0/(a/v) – t/(a/v)) = A sin (x + v(t0 - t)/a) ⇒ Y = A sin [x – v(t – t0/a)]