(4-10)  ①  Start with our first order wave equation, for a wave moving to the right:

Make the same substitution you did before,  in Question 4-7: (q = x – vt). Differentiate again to find expressions for the partial second derivatives, ∂2y/∂x2 and ∂2y/∂t2.

Now use the first order equation again to rewrite your equation for ∂2y/∂t2 in terms of ∂y/∂x.  This will allow you to establish the relationship between ∂2y/∂x2 and ∂2y/∂t2.

②  This time start with the equation you obtained for a left-travelling wave, in Question 4-8. Follow the same steps, remembering that this time the wave will have an equation of the form y = f(x + vt), so the appropriate substitution is q = x + vt.

Compare the second order wave equation with the one you obtained for the right-travelling wave, in ①.