A wave travels on a string in the positive x -direction at a velocity v . The shape of the string at t = t₀ is described by g(x, t₀) = A (x/a) . Determine the wave equation for a general time t .

Step-by-step solution

The general equation of a wave traveling in the positive x -direction with velocity v is given by f(x, t) = h(x - vt) . At t = t₀ , the wave equation is given as f(x, t₀) = h(x - vt₀) = A ( x a ) . Let z = x - vt₀ , which implies x = z + vt₀ . Substituting this into the function gives h(z) = A ( z + vt₀ a ) . For a general time t , replace z with x - vt : f(x, t) = h(x - vt) = A ( x - vt + vt₀ a ) f(x, t) = A ( x - v(t - t₀) a )