could you please answer the question with explanation

4. This questions is about the wave equation in one dimension Ox2 2 at2 (1) Last week you showed that o(x, t) = f(x - ct) + g(x + ct) where f and g are arbitrary functions. (a) Using o as the ordinate and x as the abscissa, draw a picture of the p at t = 0 and at t = 2 for initial conditions $(2, 0) = exp(-202) (2) at (2, 0) = -2cx exp(-202) (3) The characteristics of a partial differential equation are defined as lines along which infor- mation is transported. For a differential equation of the form AS 2x2 + 2B axdy + . (x, y, ax' ax ( 4) the characteristics are given by the solutions of the equation A dy dy dx 2B dx + D =0 (5) (c) Find the characteristics of the wave equation and draw a picture of these on the xt- plane. Hint: Think of Eq. 5 as a quadratic; solve it and integrate the result. (d) Explain the statement "characteristics of a partial differential equation are defined as lines along which information is transported" by considering the relationship between the coordinates (n, $) and the characteristics