8. [12 marks] The height of the water, h(x, t), in the wall of an earth dam satisfies the equations; ah ah ah with h(0,t) = 0, (1, t) = 0, h(x,0) = r. at ax2 ac (a) Draw a sketch of the situation in the (2,t) plane, as described in lectures, indicating the equation, boundary and initial conditions. (b) Use separation of variables and Fourier series to find the full solution for h(x, t). ar sin ax + cosas - ax cos ax + sin ax Note: /ro * cos ax dx +C, 1 sin ax da +C 8. [12 marks] The height of the water, h(x, t), in the wall of an earth dam satisfies the equations; ah ah ah with h(0,t) = 0, (1, t) = 0, h(x,0) = r. at ax2 ac (a) Draw a sketch of the situation in the (2,t) plane, as described in lectures, indicating the equation, boundary and initial conditions. (b) Use separation of variables and Fourier series to find the full solution for h(x, t). ar sin ax + cosas - ax cos ax + sin ax Note: /ro * cos ax dx +C, 1 sin ax da +C